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Solafidefarms and I are having a friendly argument, and he invited others to join in. To see the backstory please visit his blog.

I wrote:
[quote]
Sorry for not taking up your challenge, my ISP has been acting funny the last few days, and learning Perl is too much fun.

I have found one good article on biblical contridictions, it sort of glorifies atheism, sorry for being offended a bit, I was too.

Contradictions
The Bible is riddled with repetitions and contradictions, things that the Bible bangers would be quick to point out in anything that they want to criticize. For instance, Genesis 1 and 2 disagree about the order in which things are created, and how satisfied God is about the results of his labors. The flood story is really two interwoven stories that contradict each other on how many of each kind of animal are to be brought into the Ark--is it one pair each or seven pairs each of the "clean" ones? The Gospel of John disagrees with the other three Gospels on the activities of Jesus Christ (how long had he stayed in Jerusalem--a couple of days or a whole year?) and all four Gospels contradict each other on the details of Jesus Christ's last moments and resurrection. The Gospels of Matthew and Luke contradict each other on the genealogy of Jesus Christ' father; though both agree that Joseph was not his real father. Repetitions and contradictions are understandable for a hodgepodge collection of documents, but not for some carefully constructed treatise, reflecting a well-thought-out plan.

Of the various methods I've seen to "explain" these:
1. "That is to be taken metaphorically" In other words, what is written is not what is meant. I find this entertaining, especially for those who decide what ISN'T to be taken as other than the absolute WORD OF GOD--which just happens to agree with the particular thing they happen to want...

2. "There was more there than...." This is used when one verse says "there was a" and another says "there was b", so they decide there was "a" AND "b"--which is said nowhere. This makes them happy, since it doesn't say there WASN'T "a+b". But it doesn't say there was "a+b+litle green martians". This is often the same crowd that insists theirs is the ONLY possible interpretation (i.e. only "a") and the only way. I find it entertaining they they don't mind adding to verses.

3. "It has to be understood in context" I find this amusing because it comes from the same crowd that likes to push likewise extracted verses that support their particular view. Often it is just one of the verses in the contradictory set is suppose to be taken as THE TRUTH when if you add more to it it suddenly becomes "out of context". How many of you have goten JUST John 3:16 (taken out of all context) thrown up at you?

4. "there was just a copying/writing error" This is sometimes called a "transcription error", as in where one number was meant and an incorrect one was copied down. Or that what was "quoted" wasn't really what was said, but just what the author thought was said when he thought it was said. And that's right--I'm not disagreeing with events, I'm disagreeing with what is WRITTEN. Which is apparently agreed that it is incorrect. This is an amusing misdirection to the problem that the bible itself is wrong.

5. "That is a miracle". Naturally. That is why it is stated as fact.

6. "God works in mysterious ways" A useful dodge when the speaker doesn't understand the conflict between what the bible SAYS and what they WISH it said.

The following is from the KJ version. If you have something different, let me know.
Quote

God good to all, or just a few?
PSA 145:9 The LORD is good to all: and his tender mercies are over all his works.

JER 13:14 And I will dash them one against another, even the fathers and the sons together, saith the LORD: I will not pity, nor spare, nor have mercy, but destroy them.
War or Peace?
EXO 15:3 The LORD is a man of war: the LORD is his name.

ROM 15:33 Now the God of peace be with you all. Amen.
Who is the father of Joseph?
MAT 1:16 And Jacob begat Joseph the husband of Mary, of whom was born Jesus, who is called Christ.

LUK 3:23 And Jesus himself began to be about thirty years of age, being (as was supposed) the son of Joseph, which was the son of Heli.
Who was at the Empty Tomb? Is it:
MAT 28:1 In the end of the sabbath, as it began to dawn toward the first day of the week, came Mary Magdalene and the other Mary to see the sepulchre.

MAR 16:1 And when the sabbath was past, Mary Magdalene, and Mary the mother of James, and Salome, had bought sweet spices, that they might come and anoint him.

JOH 20:1 The first day of the week cometh Mary Magdalene early, when it was yet dark, unto the sepulchre, and seeth the stone taken away from the sepulchre.
Is Jesus equal to or lesser than?
JOH 10:30 I and my Father are one.

JOH 14:28 Ye have heard how I said unto you, I go away, and come again unto you. If ye loved me, ye would rejoice, because I said, I go unto the Father: for my Father is greater than I.
Which first--beasts or man?
GEN 1:25 And God made the beast of the earth after his kind, and cattle after their kind, and every thing that creepeth upon the earth after his kind: and God saw that it was good.
GEN 1:26 And God said, Let us make man in our image, after our likeness: and let them have dominion over the fish of the sea, and over the fowl of the air, and over the cattle, and over all the earth, and over every creeping thing that creepeth upon the earth.

GEN 2:18 And the LORD God said, It is not good that the man should be alone; I will make him an help meet for him.
GEN 2:19 And out of the ground the LORD God formed every beast of the field, and every fowl of the air; and brought them unto Adam to see what he would call them: and whatsoever Adam called every living creature, that was the name thereof.
The number of beasts in the ark
GEN 7:2 Of every clean beast thou shalt take to thee by sevens, the male and his female: and of beasts that are not clean by two, the male and his female.

GEN 7:8 Of clean beasts, and of beasts that are not clean, and of fowls, and of every thing that creepeth upon the earth, GEN 7:9 There went in two and two unto Noah into the ark, the male and the female, as God had commanded Noah.
How many stalls and horsemen?
KI1 4:26 And Solomon had forty thousand stalls of horses for his chariots, and twelve thousand horsemen.

CH2 9:25 And Solomon had four thousand stalls for horses and chariots, and twelve thousand horsemen; whom he bestowed in the chariot cities, and with the king at Jerusalem.
Is it folly to be wise or not?
PRO 4:7 Wisdom is the principal thing; therefore get wisdom: and with all thy getting get understanding.

ECC 1:18 For in much wisdom is much grief: and he that increaseth knowledge increaseth sorrow.

1 Cor.1:19: "For it is written, I will destroy the wisdom of the wise, and wil bring to nothing the understanding of the prudent."
Human vs. ghostly impregnation
ACT 2:30 Therefore being a prophet, and knowing that God had sworn with an oath to him, that of the fruit of his loins, according to the flesh, he would raise up Christ to sit on his throne;

MAT 1:18 Now the birth of Jesus Christ was on this wise: When as his mother Mary was espoused to Joseph, before they came together, she was found with child of the Holy Ghost.
The sins of the father
ISA 14:21 Prepare slaughter for his children for the iniquity of their fathers; that they do not rise, nor possess the land, nor fill the face of the world with cities.

DEU 24:16 The fathers shall not be put to death for the children, neither shall the children be put to death for the fathers: every man shall be put to death for his own sin.
The bat is not a bird
LEV 11:13 And these are they which ye shall have in abomination among the fowls; they shall not be eaten, they are an abomination: the eagle, and the ossifrage, and the ospray,
LEV 11:14 And the vulture, and the kite after his kind;
LEV 11:15 Every raven after his kind;
LEV 11:16 And the owl, and the night hawk, and the cuckow, and the hawk after his kind,
LEV 11:17 And the little owl, and the cormorant, and the great owl,
LEV 11:18 And the swan, and the pelican, and the gier eagle,
LEV 11:19 And the stork, the heron after her kind, and the lapwing, and the bat.

DEU 14:11 Of all clean birds ye shall eat.
DEU 14:12 But these are they of which ye shall not eat: the eagle, and the ossifrage, and the ospray,
DEU 14:13 And the glede, and the kite, and the vulture after his kind,
DEU 14:14 And every raven after his kind,
DEU 14:15 And the owl, and the night hawk, and the cuckow, and the hawk after his kind,
DEU 14:16 The little owl, and the great owl, and the swan,
DEU 14:17 And the pelican, and the gier eagle, and the cormorant,
DEU 14:18 And the stork, and the heron after her kind, and the lapwing, and the bat.
Rabbits do not chew their cud
LEV 11:6 And the hare, because he cheweth the cud, but divideth not the hoof; he is unclean unto you.

'Gerah', the term which appears in the MT means (chewed) cud, and also perhaps grain, or berry (also a 20th of a sheckel, but I think that we can agree that that is irrelevant here). It does *not* mean dung, and there is a perfectly adequate Hebrew word for that, which could have been used. Furthermore, the phrase translated 'chew the cud' in the KJV is more exactly 'bring up the cud'. Rabbits do not bring up anything; they let it go all the way through, then eat it again. The description given in Leviticus is inaccurate, and that's that. Rabbits do eat their own dung; they do not bring anything up and chew on it.
Insects do NOT have four feet
LEV 11:21 Yet these may ye eat of every flying creeping thing that goeth upon all four, which have legs above their feet, to leap withal upon the earth;
LEV 11:22 Even these of them ye may eat; the locust after his kind, and the bald locust after his kind, and the beetle after his kind, and the grasshopper after his kind.
LEV 11:23 But all other flying creeping things, which have four feet, shall be an abomination unto you.
Snails do not melt
PSA 58:8 As a snail which melteth, let every one of them pass away: like the untimely birth of a woman, that they may not see the sun.
Fowl from waters or ground?
GEN 1:20 And God said, Let the waters bring forth abundantly the moving creature that hath life, and fowl that may fly above the earth in the open firmament of heaven.
GEN 1:21 And God created great whales, and every living creature that moveth, which the waters brought forth abundantly, after their kind, and every winged fowl after his kind: and God saw that it was good.

GEN 2:19 And out of the ground the LORD God formed every beast of the field, and every fowl of the air; and brought them unto Adam to see what he would call them: and whatsoever Adam called every living creature, that was the name thereof.
Odd genetic engineering
GEN 30:39 And the flocks conceived before the rods, and brought forth cattle ringstraked, speckled, and spotted.
The shape of the earth
ISA 40:22 It is he that sitteth upon the circle of the earth, and the inhabitants thereof are as grasshoppers; that stretcheth out the heavens as a curtain, and spreadeth them out as a tent to dwell in:

MAT 4:8 Again, the devil taketh him up into an exceeding high mountain, and sheweth him all the kingdoms of the world, and the glory of them;

Astromical bodies are spherical, and you cannot see the entire exterior surface from anyplace. The kingdoms of Egypt, China, Greece, Crete, sections of Asia Minor, India, Maya (in Mexico), Carthage (North Africa), Rome (Italy), Korea, and other settlements from these kingdoms of the world were widely distributed.
Snakes, while built low, do not eat dirt
GEN 3:14 And the LORD God said unto the serpent, Because thou hast done this, thou art cursed above all cattle, and above every beast of the field; upon thy belly shalt thou go, and dust shalt thou eat all the days of thy life:
Earth supported?
JOB 26:7 He stretcheth out the north over the empty place, and hangeth the earth upon nothing.

JOB 38:4 Where wast thou when I laid the foundations of the earth? declare, if thou hast understanding.

Heaven supported too
JOB 26:11 The pillars of heaven tremble and are astonished at his reproof.
The hydrological cycle
ECC 1:7 All the rivers run into the sea; yet the sea is not full; unto the place from whence the rivers come, thither they return again.

JOB 38:22 Hast thou entered into the treasures of the snow? or hast thou seen the treasures of the hail,

Storehouses are not part of the cycle

Order of creation
Here is the order in the first (Genesis 1), the Priestly tradition:

Day 1: Sky, Earth, light
Day 2: Water, both in ocean basins and above the sky(!)
Day 3: Plants
Day 4: Sun, Moon, stars (as calendrical and navigational aids)
Day 5: Sea monsters (whales), fish, birds, land animals, creepy-crawlies (reptiles, insects, etc.)
Day 6: Humans (apparently both sexes at the same time)
Day 7: Nothing (the Gods took the first day off anyone ever did)

Note that there are "days", "evenings", and "mornings" before the Sun was created. Here, the Deity is referred to as "Elohim", which is a plural, thus the literal translation, "the Gods". In this tale, the Gods seem satisfied with what they have done, saying after each step that "it was good".

The second one (Genesis 2), the Yahwist tradition, goes:

Earth and heavens (misty)
Adam, the first man (on a desolate Earth)
Plants
Animals
Eve, the first woman (from Adam's rib)
How orderly were things created?
#1: Step-by-step. The only discrepancy is that there is no Sun or Moon or stars on the first three "days".
#2: God fixes things up as he goes. The first man is lonely, and is not satisfied with animals. God finally creates a woman for him. (funny thing that an omniscient god would forget things)
How satisfied with creation was he?
#1: God says "it was good" after each of his labors, and rests on the seventh day, evidently very satisfied.
#2: God has to fix up his creation as he goes, and he would certainly not be very satisfied with the disobedience of that primordial couple. (funny thing that an omniscient god would forget things)
Moses' personality
Num.12:3: "Now the man Moses was very meek, above all the men which were upon the fact of the earth."

Num.31:14, 17, 18: "And Moses was wroth...And Moses said unto them, "Have ye saved all the women alive? ... Now therefore kill every male among the little ones, and kill every woman, ... But all the women children ... keep alive for yourselves."
Righteous live?
Ps.92:12: "The righteous shall flourish like the palm tree."

Isa.57:1: "The righteous perisheth, and no man layeth it to heart."
Acts 1:18: "Now this man (Judas) purchased a field with the reward of iniquity; and falling headlong, he burst asunder in the midst, and all his bowels gushed out."

Matt. 27:5-7: "And he (Judas) cast down the pieces of silver in the temple, and departed, and went and hanged himself. And the chief priests...bought with them the potter's field."
Jesus' first sermon plain or mount?
Matt.5:1,2: "And seeing the multitudes, he went up into a mountain: and when he was set, his disciples came unto him: And he opened his mouth, and taught them, saying...."
Luke6:17,20: "And he came down with them, and stood in the plain, and the company of his disciples, and a great multitude of people...came to hear him.. And he lifted up his eyes on his disciples and said..."
Jesus' last words
Matt.27:46,50: "And about the ninth hour Jesus cried with a loud voice, saying, "Eli, eli, lama sabachthani?" that is to say, "My God, my God, why hast thou forsaken me?" ...Jesus, when he cried again with a loud voice, yielded up the ghost."

Luke23:46: "And when Jesus had cried with a loud voice, he said, "Father, unto thy hands I commend my spirit:" and having said thus, he gave up the ghost."

John19:30: "When Jesus therefore had received the vinegar, he said, "It is finished:" and he bowed his head, and gave up the ghost."
Years of famine
II SAMUEL 24:13: So God came to David, and told him, and said unto him, shall SEVEN YEARS OF FAMINE come unto thee in thy land? or will thou flee three months before thine enemies, while they pursue. thee?

I CHRONICLES 21:11: SO God came to David, and said unto him, Thus saith the LORD, Choose thee. Either THREE YEARS OF FAMINE or three months to be destryed before thy foes, while that the sword of thine enemies overtaketh thee;
Moved David to anger?
II SAMUEL 24: And again the anger of the LORD was kindled against Israel, and he moved David against them to say, Go, number Isreal and Judah.

I CHRONICLES 21: And SATAN stood up against Isreal, and provoked David to number Israel.
The GENEALOGY OF JESUS?
In two places in the New Testament the genealogy of Jesus son of Mary (PBUH) is mentioned. Matthew 1:6-16 and Luke 3:23-31. Each gives the ancestors of Joseph the CLAIMED husband of Mary and Step father of Jesus(PBUH). The first one starts from Abraham(verse 2) all the way down to Jesus. The second one from Jesus all the way back to Adam. The only common name to these two lists between David and Jesus is JOSEPH, How can this be true? and also How can Jesus have a genealogy when all Muslims and most Christians believe that Jesus had/has no father.
God be seen?
Exod. 24:9,10; Amos 9:1; Gen. 26:2; and John 14:9
God CAN be seen:
"And I will take away my hand, and thou shalt see my backparts." (Ex. 33:23)
"And the Lord spake to Moses face to face, as a man speaketh to his friend." (Ex. 33:11)
"For I have seen God face to face, and my life is preserved." (Gen. 32:30)

God CANNOT be seen:
"No man hath seen God at any time." (John 1:18)
"And he said, Thou canst not see my face; for there shall no man see me and live." (Ex. 33:20)
"Whom no man hath seen nor can see." (1 Tim. 6:16)
CRUEL, UNMERCIFUL, DESTRUCTIVE, and FEROCIOUS or KIND, MERCIFUL, and GOOD:
"I will not pity, nor spare, nor have mercy, but destroy." (Jer. 13:14) "Now go and smite Amalek, and utterly destroy all that they have, and spare them not, but slay both man and woman, infant and suckling."

"The Lord is very pitiful and of tender mercy." (James 5:11)
"For his mercy endureth forever." (1 Chron. 16:34)
"The Lord is good to all, and his tender mercies are over all his works." (Ps. 145:9)
"God is love." (1 John 4:16)
Tempts?
"And it came to pass after these things, that God did tempt Abraham." (Gen 22:1)

"Let no man say when he is tempted, I am tempted of God; for God cannot be tempted with evil, neither tempteth he any man." (James 1:13)
Judas died how?
"And he cast down the pieces of silver into the temple and departed, and went out and hanged himself." (Matt. 27:5)

"And falling headlong, he burst asunder in the midst, and all of his bowels gushed out." (Acts 1:18)
Ascend to heaven
"And Elijah went up by a whirlwind into heaven." (2 Kings 2:11)

"No man hath ascended up to heaven but he that came down from heaven, ... the Son of Man." (John 3:13)
What was Jesus' prediction regarding Peter's denial?
Before the - crow - Matthew 26:34

Before the - crow twice - Mark 14:30
How many times did the - crow?
MAR 14:72 And the second time the - crew. And Peter called to mind the word that Jesus said unto him, Before the - crow twice, thou shalt deny me thrice. And when he thought thereon, he wept.

MAT 26:74 Then began he to curse and to swear, saying, I know not the man. And immediately the - crew.
MAT 26:75 And Peter remembered the word of Jesus, which said unto him, Before the - crow, thou shalt deny me thrice. And he went out, and wept bitterly.

LUK 22:60 And Peter said, Man, I know not what thou sayest. And immediately, while he yet spake, the - crew.
LUK 22:61 And the Lord turned, and looked upon Peter. And Peter remembered the word of the Lord, how he had said unto him, Before the - crow, thou shalt deny me thrice.

JOH 13:38 Jesus answered him, Wilt thou lay down thy life for my sake? Verily, verily, I say unto thee, The - shall not crow, still thou hast denied me thrice.

JOH 18:27 Peter then denied again: and immediately the - crew.
Who killed Saul
SA1 31:4 Then said Saul unto his armourbearer, Draw thy sword, and thrust me through therewith; lest these uncircumcised come and thrust me through, and abuse me. But his armourbearer would not; for he was sore afraid. Therefore Saul took a sword, and fell upon it.
SA1 31:5 And when his armourbearer saw that Saul was dead, he fell likewise upon his sword, and died with him.
SA1 31:6 So Saul died, and his three sons, and his armourbearer, and all his men, that same day together.
SA2 1:15 And David called one of the young men, and said, Go near, and fall upon him. And he smote him that he died.
How many beatitudes in the Sermon on the Mount
MAT 5:3 Blessed are the poor in spirit: for theirs is the kingdom of heaven.
MAT 5:4 Blessed are they that mourn: for they shall be comforted.
MAT 5:5 Blessed are the meek: for they shall inherit the earth.
MAT 5:6 Blessed are they which do hunger and thirst after righteousness: for they shall be filled.
MAT 5:7 Blessed are the merciful: for they shall obtain mercy.
MAT 5:8 Blessed are the pure in heart: for they shall see God.
MAT 5:9 Blessed are the peacemakers: for they shall be called the children of God.
MAT 5:10 Blessed are they which are persecuted for righteousness' sake: for theirs is the kingdom of heaven.
MAT 5:11 Blessed are ye, when men shall revile you, and persecute you, and shall say all manner of evil against you falsely, for my sake.

LUK 6:20 And he lifted up his eyes on his disciples, and said, Blessed be ye poor: for yours is the kingdom of God.
LUK 6:21 Blessed are ye that hunger now: for ye shall be filled. Blessed are ye that weep now: for ye shall laugh.
LUK 6:22 Blessed are ye, when men shall hate you, and when they shall separate you from their company, and shall reproach you, and cast out your name as evil, for the Son of man's sake.
LUK 6:23 Rejoice ye in that day, and leap for joy: for, behold, your reward is great in heaven: for in the like manner did their fathers unto the prophets.
Does every man sin?
KI1 8:46 If they sin against thee, (for there is no man that sinneth not,) and thou be angry with them, and deliver them to the enemy, so that they carry them away captives unto the land of the enemy, far or near;

CH2 6:36 If they sin against thee, (for there is no man which sinneth not,) and thou be angry with them, and deliver them over before their enemies, and they carry them away captives unto a land far off or near;

PRO 20:9 Who can say, I have made my heart clean, I am pure from my sin?

ECC 7:20 For there is not a just man upon earth, that doeth good, and sinneth not.

JO1 1:8 If we say that we have no sin, we deceive ourselves, and the truth is not in us.
JO1 1:9 If we confess our sins, he is faithful and just to forgive us our sins, and to cleanse us from all unrighteousness.
JO1 1:10 If we say that we have not sinned, we make him a liar, and his word is not in us.

JO1 3:9 Whosoever is born of God doth not commit sin; for his seed remaineth in him: and he cannot sin, because he is born of God.
Who bought potter's field
ACT 1:18 Now this man purchased a field with the reward of iniquity; and falling headlong, he burst asunder in the midst, and all his bowels gushed out.
ACT 1:19 And it was known unto all the dwellers at Jerusalem; insomuch as that field is called in their proper tongue, Aceldama, that is to say, The field of blood.

MAT 27:6 And the chief priests took the silver pieces, and said, It is not lawful for to put them into the treasury, because it is the price of blood.
MAT 27:7 And they took counsel, and bought with them the potter's field, to bury strangers in.
MAT 27:8 Wherefore that field was called, The field of blood, unto this day.
Who prophesied the potter's field?
Matthew 27:9-10 (mentions Jeremy but no such verse in Jeremiah) is in Zechariah 11:12-13
Who bears guilt?
GAL 6:2 Bear ye one another's burdens, and so fulfil the law of Christ.

GAL 6:5 For every man shall bear his own burden.
Do you answer a fool?
PRO 26:4 Answer not a fool according to his folly, lest thou also be like unto him.

PRO 26:5 Answer a fool according to his folly, lest he be wise in his own conceit.
How many children did Michal, the daughter of Saul, have?
SA2 6:23 Therefore Michal the daughter of Saul had no child unto the day of her death.

SA2 21:8 But the king took the two sons of Rizpah the daughter of Aiah, whom she bare unto Saul, Armoni and Mephibosheth; and the five sons of Michal the daughter of Saul, whom she brought up for Adriel the son of Barzillai the Meholathite:
How old was Jehoiachin when he began to reign?
KI2 24:8 Jehoiachin was eighteen years old when he began to reign, and he reigned in Jerusalem three months. And his mother's name was Nehushta, the daughter of Elnathan of Jerusalem.

CH2 36:9 Jehoiachin was eight years old when he began to reign, and he reigned three months and ten days in Jerusalem: and he did that which was evil in the sight of the LORD.
Marriage?
Proverbs 18:22
1 Corinthians 7 (whole book. See 1,2,27,39,40)
Did those with Saul/Paul at his conversion hear a voice?
ACT 9:7 And the men which journeyed with him stood speechless, hearing a voice, but seeing no man.

ACT 22:9 And they that were with me saw indeed the light, and were afraid; but they heard not the voice of him that spake to me.
Where was Jesus three days after his baptism?
MAR 1:12 And immediately the spirit driveth him into the wilderness.

JOH 1:35 Again the next day after John stood, and two of his disciples;

(various trapsing)
How many apostles were in office between the resurection and ascention?
1 Corinthians 15:5 (12)
Matthew 27:3-5 (minus one from 12)
Acts 1:9-26 (Mathias not elected until after resurrection)

MAT 28:16 Then the eleven disciples went away into Galilee, into a mountain where Jesus had appointed them.
Judging
1 Cor 3:15 " The spiritual man makes judgments about all things, but he himself is not subject to any man's judgment:" (NIV)

1 Cor 4:5 " Therefore judge nothing before the appointed time; wait till the Lord comes. He will bring to light what is hidden in darkness and will expose the motives of men's hearts. At that time each will receive his praise from God."
Good deeds
Matt 5:16 "In the same way, let your light shine before men, that they may see your good deeds and praise your Father in heaven." (NIV)

Matt 6:3-4 "But when you give to the needy, do not let your left hand know what your right hand is doing, so that your giving may be in secert. Then your Father, who sees what is done in secret, will reward you." (NIV)
For or against?
MAT 12:30 He that is not with me is against me; and he that gathereth not with me scattereth abroad.
(default is against)

MAR 9:40 For he that is not against us is on our part.
(default is for)

LUK 9:50 And Jesus said unto him, Forbid him not: for he that is not against us is for us.
(default is for)
Whom did they see at the tomb?
MAT 28:2 And, behold, there was a great earthquake: for the angel of the Lord descended from heaven, and came and rolled back the stone from the door, and sat upon it.
MAT 28:3 His countenance was like lightning, and his raiment white as snow:
MAT 28:4 And for fear of him the keepers did shake, and became as dead men.
MAT 28:5 And the angel answered and said unto the women, Fear not ye: for I know that ye seek Jesus, which was crucified.

MAR 16:5 And entering into the sepulchre, they saw a young man sitting on the right side, clothed in a long white garment; and they were affrighted.

LUK 24:4 And it came to pass, as they were much perplexed thereabout, behold, two men stood by them in shining garments:

JOH 20:12 And seeth two angels in white sitting, the one at the head, and the other at the feet, where the body of Jesus had lain.
God change?
malachi 3:6
james 1:17
1 samuel 15:29
jonah 3:10
genesis 6:6
Destruction of cities (what said was jeremiah was zechariah)
MAT 27:9 Then was fulfilled that which was spoken by Jeremy the prophet, saying, And they took the thirty pieces of silver, the price of him that was valued, whom they of the children of Israel did value;

zechariah 11:11-13
(nothing in Jeremiah remotely like)
Who's sepulchers
acts 7:16
genesis 23:17,18
Strong drink?
proverbs 31:6,7
john 2:11-11
When second coming?
MAT 24:34 Verily I say unto you, This generation shall not pass, till all these things be fulfilled.

MAR 13:30 Verily I say unto you, that this generation shall not pass, till all these things be done.

LUK 21:32 Verily I say unto you, This generation shall not pass away, till all be fulfilled.

1 thessalonians 4:15-18
Solomon's overseers
550 in I Kings 9:23
250 in II Chron 8:10
The mother of Abijah:
Maachah the daughter of Absalom 2 Chron 9:20

Michaiah the daughter of Uriel 2 Chron 13:2
When did Baasha die?
26th year of the reign of Asa I Kings 16:6-8

36th year of the reign of Asa I 2 Chron 16:1
How old was Ahaziah when he began to reign?
22 in 2 Kings 8:26

42 in 2 Chron 22:2
Who was Josiah's successor?
Jehoahaz - 2 Chron 36:1

Shallum - Jeremiah 22:11
The differences in the census figures of Ezra and Nehemiah.
What was the color of the robe placed on Jesus during his trial?
scarlet - Matthew 27:28

purple John 19:2
What did they give him to drink?
vinegar - Matthew 27:34

wine with myrrh - Mark 15:23
How long was Jesus in the tomb?
Depends where you look; Matthew 12:40 gives Jesus prophesying that he will spend "three days and three nights in the heart of the earth", and Mark 10:34 has "after three days (meta treis emeras) he will rise again". As far as I can see from a quick look, the prophecies have "after three days", but the post-Resurrection narratives have "on the third day".

Please give counter arguments that we can discuss. I am in no way attempting to make a cheap shot or smunge christianity. I am just trying to examine the faith under the vision of logic and other peoples views.
[/quote]

He replied
[quote]
Could we please make this a topic in G&FF? I'm completely willing to post the topic, and this to be a reply.

You have made a lot of accusations. I feel that I cannot answer all of them. If I cannot post a public topic, may I enlist the help of some fellow believers on this site?(MathFiend, JesusFreak, darktreb, etc.)

Billy
[/quote]

I have done what Billy has requested. Let the thought provoking discourse begin.

I have moved this thread to Round Table as I felt the topic was a little heavy for G&FF.

this is very interesting. please post more information. :)

bleumoose, are you talking to me, or penguinintegral? :?

Thank you for showing your interest, hl78reg. I will post more later. :)

[quote="bleumoose"]I tend to go with the transcription error folks, as it is pretty well known that the Bible has many versions, all of them say different things, and none of them are in the original language. Your quotes on a "wrathful" verses a "merciful" God are, if you look carefully, from the Old (wrathful) and the New (merciful) Testaments. It is widely believed in the Christian world that the coming of Jesus heralded a new, more forgiving version of the Jewish (etc) faith, as can be seen in these comparisons. Outside of that, I really don't have too much to say, other than my beliefs are that the Bible isn't meant to be taken literally in most senses. Who cares in what order God created the universe, or how long it took? The important message is that God created the universe. Who cares how long Jesus stayed in one place or another? The important messages are the ones he imparted on how to live your life: "Love your neighbor" and "Love the Lord thy God with all thy heart, with all thy soul, and with all thy mind" and so on. THAT'S what's important; not the small details of His time here on earth.[/quote]

So, in otherwords, you now wish me to prove things using ideas now?

Also, jesus wasn't the only prophet/messiah trying to make a name for themselves. There were as many messiahs back then as there are crazy people today claiming to prove fermat's last theorm with pre-calculus methods.(note the parallel drawn)

Please, do not resort to name calling. Every serious discussion on this topic I have seen has quickly disinigrated into a flame war.

[quote="chess64"]I would like to make a very bold statement:

[u][b]The second coming of Jesus has already taken place![/b][/u]

[hide]
First we must understand the prophecy about his first coming.

[b]ELIJAH AND HIS SECOND COMING[/b]

2,000 years ago, all Isrealites were waiting for the return of the prophet Elijah, to be followed by the Messiah. (Matt. 11:14; 17:10; John 1:21) It was commonly believed, however, that Elijah would descend bodily from heaven, just as he was lifted up.

When Jesus claimed to be the Messiah, the question arose: Where was Elijah?

"And Jesus answered and said unto them, Elias truly shall first come, and restore all things. But I say unto you, that Elias is come already... Then the disciples understood that he spake unto them of John the Baptist." (Matt. 17:10-13)

Notice that John the Baptist was not the same person as the earlier prophet Elijah, and he did not descend from the sky on a chariot of fire, as was commonly expected (II Kings 2:9-11) However, John the Baptist was identified as a person who had come [b]in the power and spirit[/b] of Elijah (Luke 1:13-17)

If the original Prophet, Elijah, had actually bodily gone up, then we should have expected the self-same person to also descend from heaven to fulfill the prophecy of his second advent. The fact that someone else was appointed to represent Elijah in power and spirit proves that the ascent of Elijah to heaven should be taken in a spiritual sense, rather than a physical sense.

It is not ironic that history has repeated itself and yet a majority of the Christians are not willing to learn from history!

[b]THE RETURN OF JESUS CHRIST[/b]

There are many parallels in the stories of the return of Elijah and the return of the Messiah. It is commonly believed that Jesus Christ was bodily lifted up to heaven (just like Elijah was) and he will return from heaven in great glory (like Elijah was supposed to come in a chariot of fire).

[b]What if the return of the Messiah is also very different from expectations?[/b] Will people reject him also like John the Baptist was rejected by most of his people?

Suppose the return of the Messiah is not a coming in a cloud with power and glory. (In a cloud he will be too far for us to be able to see him; it obviously has a different interpretation.) Suppose he comes suddenly and very quietly, unobserved by most people, as the Bible says:

"Watch therefore: for you know not what hour your Lord doth come... therefore be ye also ready: for in such an hour as ye think not, the son of man cometh." (Matt. 24:42-44)

"But the day of the Lord will come as a thief in the night." (2 Peter 3:10)

"...know perfectly that the Day of the Lord so cometh as a thief in the night." (1 Thess. 5:2)

The Bible says that when the Messiah comes, he will be rejected by the world:

"But first must he suffer many things, and be rejected this generation. And as it was in the days No-e, so shall it be also in the days of the Son of man." (Luke, 17:25-26)

[b]PUNISHMENT FOR HIS REJECTION[/b]

Rejection of Noah by his people resulted in a punishment from God in the form of the flood. Rejection of the Messiah in our age has also to result in punishment from God. The punishment was predicted by Jesus himself. This was to be wars, floods, earthquakes and pestilence.

The 20th century was full of wars, earthquakes, floods and pestilience like plague and AIDS. There was a terrible earthquake in San Fransisco in 1906. There was a similar devastating earthquake in India around the same time. Then came World War I in 1914, followed by many more wars.

God is just and merciful! Why would God inflict all these punishments for no reason at all. [b]The Messiah had to come first and be rejected.[/b] Then and only then would the world be justifiably punished. [b]We must look for the Messiah before the beginning of the twentieth century.[/b]

[b]HE HAS ALREADY COME[/b]

Thus it came to pass. The Messiah came quietly near the end of the nineteenth century; unobserved by the world. He did not make a grand entrance with power and great glory. He came very quietly.

The original Elijah did not return. John the Baptist appeared in the power and spirit of Elijah. In exactly the same manner, the original Jesus of Nazareth did not return. Mirza Ghulam Ahmad of Qadian, India, came [b]in the power and spirit of Jesus[/b]. But we did not learn from history. The Jews had rejected John the Baptist because he was not the same original Elijah; and so the world rejected the new Messiah because he was not the same Jesus. Jesus himself said:

"Ye shall not see me henceforth, till ye shall say, [b]blessed is he that cometh in the name of the Lord.[/b]" (Matt. 23:39)

The Messiah of our age proclaimed that he had been sent in the name of Jesus, but he was rejected by the majority just as predicted. (Luke 17:25)
[/hide]

If anyone is interested, I can post some more information.[/quote]

I can make the same argument about the 21th century. 9/11, london bombings, madrid
bombings, plane crashes, infomercials, etc.

Bad things do not equal messiah.

[quote="chess64"][quote="PenguinIntegral"]I can make the same argument about the 21th century. 9/11, london bombings, madrid bombings, plane crashes, infomercials, etc.[/quote]

Very true! But think about it. Does God stop punishing the world after one century? Has the whole world accepted him yet? No! You are actually just supporting my claim. ;)[/quote]

Um, give me a point at which the comming happened, and I will give you some really bad stuff that happened before then.

[quote="bleumoose"]I tend to go with the transcription error folks, as it is pretty well known that the Bible has many versions, all of them say different things, and none of them are in the original language. Your quotes on a "wrathful" verses a "merciful" God are, if you look carefully, from the Old (wrathful) and the New (merciful) Testaments. It is widely believed in the Christian world that the coming of Jesus heralded a new, more forgiving version of the Jewish (etc) faith, as can be seen in these comparisons. Outside of that, I really don't have too much to say, other than my beliefs are that the Bible isn't meant to be taken literally in most senses. Who cares in what order God created the universe, or how long it took? The important message is that God created the universe. Who cares how long Jesus stayed in one place or another? The important messages are the ones he imparted on how to live your life: "Love your neighbor" and "Love the Lord thy God with all thy heart, with all thy soul, and with all thy mind" and so on. THAT'S what's important; not the small details of His time here on earth.[/quote]

So the bible is not infallible?

This is not meant to be an argument against Christianity, but at what point do we stop blaming God or giving God credit for natural disasters, all of which have modern, verifiable, scientific explanations?  I understand that it's one thing to give credit to the creator for [i]everything[/i], but why do we intertwine our actions with the existence of natural phenonema?  Can we not view the creator with a modern, scientific, skeptical eye?

If religion does not jive with what we actually observe, do we try harder to observe, or do we alter the tenets of our religion?

actually, thousands of years ago, i think it was okay in some polytheistic religions to be angered by a god and just pray to another. can anyone confirm that?

[quote="chess64"][quote="PenguinIntegral"]Um, give me a point at which the comming happened...[/quote]

http://www.alislam.org/library/messiah.html[/quote]

Well, for a start, natural disasters, wars, and sickness?

It's not a subjectivist like me who is interested in truth.  I'm not.  I'm interested in what affects me and everything I subjectively value.  It is religious people who seem interested in truth.  That is why I ask the questoins I ask, to explore both their psychological need for truth and also question their assumptions.

If suppose that god will sent some people to the hell fire is the "TRUTH", doesn't you value your own safety???

So in my opinion without knowing the "TRUTH" then we can not know if our action indeed will benefit or against our interest.

I think we should all be good and live life like we're supposed to  :)

[quote="MCrawford"][quote="oldman"]
So in my opinion without knowing the "TRUTH" then we can not know if our action indeed will benefit or against our interest.[/quote]

Exactly.  That's why I don't worry about religious axioms.  How could I ever establish their truth?

If anyone had ever been smart enough to establish a verifiable ontological argument (something that could be checked by computer were it valid), it would be quite famous.  Then I would start paying attention.

Until then, I feel quite comfortable setting my own standards of action and I am comfortable that they are good standards -- to my own subjective set of values.[/quote]

For my self ontological argument will always have a problem. First, we can not make a well definittion for the word "god", because people who believe in him only know Him by his attributes (all knowing, omnipresent, etc). Once we make a god definition then we bound god subject to our definition, which is not good if indeed that god is boundless.

(here is an attempt on impossibility of succesfull ontological argument, http://cogprints.org/2492/01/ontarg-en.pdf)

The second, any ontological argument will always based on certain axiom that might not all people agree that they are really an axiom (btw have you see Godel ontological argument, some logician says that if the axiom is really accepted than the conclusion follows).

Maybe knowing of god can only obtained through gnosis. The necessay condition for this if you really really seek the god, and the sufficient condition is if god willing to revealed himself.
And since only a few who really seek him very very hard, then there isn't a surprise that only a few who believe in him.

Just for thought exercise: suppose a three dimensional man is consider a higher being by a one dimensional bacteria. How do you make the bacteria aware of the three dimensional receptivity???

[quote="oldman"]
And since only a few who really seek him very very hard, then there isn't a surprise that only a few who believe in him.
[/quote]

I disagree that few seek God very very hard.  In fact, I think almost everyone does at some point who has any inkling of belief.

Has anyone heard of the oscillating universe theory?

[quote="hwenterprise"]no.

not to spam or anything. :)

i think this religion topic is getting out of control. :D[/quote]

Well, take a look at oscillating universes, and it can show just how  this universe is not  the only or the first of the force's creation  :)  Or just read my blog...

[quote="jhollenbeck"]I believe in God and Jesus as our savior(christian)  This thread is getting out of hand, don't you think :D[/quote]

There was a little spamming, which is rude, but I think the serious participants kept the discussion at a respectful level.  Was there any post in particular you want me to take a look at as a moderator?

I did not have time to read through the entire discussion (just first 4 and a half pages) but it seems many of the arguments in this thread have gone off topic to the existence of a creator, which we already have a thread for, and that is not what we are discussing here. For a discussion on whether there is a creator or not go to that thread. Religion is separate from their being a creator, because religion implies that the already existing creator wants worshippers. The thread started out as a debate over contradictions in holy books and I believe it should stay that way.

Also, none of the contradictions that I read in the Old Testament (or for me, the Torah) work, and if they would be brought up instead of in a list, in an organized fashion..

Also, by Abraham and Isaac, couldn't it also be that God is setting an example that if he tells you to do something you do it? It is still Abraham's will that he did it, God didnt force him, God only knew that he would choose to do it, so it is still in the merit of Abraham.

By Jephtah I believe there are also sources which say he exiled her for 3 months, which is biblically considered like killing them. The "never marry" and "die a virgin" emphasis is also important. In other reports, I dont believe it even says that he sacrificed her..

Also, being Jewish, I believe in the Torah (Old Testament for Christians) which encompasses the 5 books and many others, about 40, of course, none related to Jesus. Also, I believe in the original Hebrew text (which is original because they found scripts thousands of years old that are the exact same. The english translation of the Old Testament is I believe not 100% accurate, any of them, since they are (or atleast some of them I think, I think all of them) based on the Saptaguent, the Greek translation of the Torah which was translated by 70 rabbis who translated parts differently so that the Greeks wouldnt take those parts as offensive against them (the greeks) and attack them (the rabbis). I'm not 100% sure on that though, that it is based on the Saptaguent. However, I am sure that the english translation isnt 100% because Hebrew cant fully be translated into english.. there are problems w/ it.

(P.S. I love the Round Table arguments.. 'dunno why' (heh) but it seems that its so much better to argue w/ people who actually know what they're talking about than w/ "lots of other people" who dont)

I think the discussion moved to tangents because those tangents are relevant to the discussion.

In fact, I think the discussion of holy books was tangential to the discussion of religion in the first place, though I'll let the original poster correct me if I am mistaken.

Well, not really, because the first post is about contradictions in holy books, so discussing holy books is somewhat tangential, but discussing contradictions in holy books is on the topic.l

So, I was just wondering. After reading a bunch of tangents I have one question I must ask because I wish to participate in this discussion.

Are we talking about generally, the validity/infallibility of the Holy Bible? Or just of the concept of religion?


1.) Who teaches the online classes?

2.) I heard the classes are in "online classrooms/chatrooms". If you have a slow internet connection, would it take a long time to "load" the classroom/chatroom.

3.) If the chatroom/classroom suddenly gets spammed by some person (like they keep on posting the same thing over and over again, so it gets really annoying), would the person get kicked out? (I find this really annoying in AIM chatrooms)

Thanks!  :)

1) Depends on what class it is. See the Enroll page, I think it says who's teaching each class.

2) It shouldn't take [i]that[/i] long...

3) The classrooms are moderated. You should go to MathJams to see how it is. Whatever you type goes to the teacher, who then makes it available to the rest of the class if he wishes.

Hope that helped. :)

[quote="Drunken_Math"]1.) Who teaches the online classes?
[/quote]

The instructors are listed on the classes list.

[quote="Drunken_Math"]1.) Who teaches the online classes?[/quote]The classes are all taught by either Richard Rusczyk, Mathew Crawford, Naoki Sato, or myself.  [url=http://www.artofproblemsolving.com/AboutUs/AoPS_A_Company.php]Read our bios.[/url]

[quote="Drunken_Math"]2.) I heard the classes are in "online classrooms/chatrooms". If you have a slow internet connection, would it take a long time to "load" the classroom/chatroom.[/quote]The classroom itself is a Java applet -- it may be initially slow to load at the start of class, but once the class starts it shouldn't matter whether you're on dialup or broadband.  You should test it out by attending one of our free [url=http://www.artofproblemsolving.com/Community/AoPS_Y_Math_Jams.php]Math Jams[/url].

[quote="Drunken_Math"]3.) If the chatroom/classroom suddenly gets spammed by some person (like they keep on posting the same thing over and over again, so it gets really annoying), would the person get kicked out? (I find this really annoying in AIM chatrooms)[/quote]Our classroom is moderated, meaning that you only see what the instructor chooses to let you see.  So no "spamming" of the classroom is possible.

[quote="Drunken_Math"]Thanks!  :)[/quote]You're welcome.  Thanks for being a part of AoPS!

Yes, when I was on dialup, the chatroom would take several minutes to load.

I highly recommend MCrawford and RRusczyk because I've taken classes under them. I can't say for the others. I hope to take a class under Naoki later this year

No spam allowed.

[quote="mel"]I must echo solafidefarms's sentiments. Mr. Crawford, Mr. Rusczyk, and Mr. Patrick are all fantastic instructors. I've had the pleasure of working with all of them during my IS, and each session is always engaging, lots of fun, and even uplifting sometimes.

Unless you are extremely fortunate, I can't imagine anything short of a math summer program parallelling the opportunities provided here.[/quote]

Oh, for this summer Intro B IS... I got to work with wdragyn and joejia!!! Like mel said "each session is always engaging, lots of fun, and even uplifting sometimes"!!!!

I got a question though... ok... if I would like to send them thank you cards   :D  ... where should I sent it to? I thought that the instructors are from different places and I hope that my thank-yous are received. I think I should sent it to the AoPS central place but I'm not sure.

If you send it to AoPS central, we'll make sure it finds the right home.

AoPS Incorporated
PO Box 2185
Alpine, CA 91903-2185


Suppose we have the letters M, I and U we can use
to form "words". The challenge is, based on the
"word" MI, transforming it in word MU applying in each
step, one of the following rules:

1) If the word ends I add a U (eg MI -> MIU)
2) Duplicate all the text after an M (eg MIU -> MIUIU)
3) Replace any sequence III by U (eg MUIIIU -> MUUU)
4) Remove any UU (eg MUUU -> MU)

 :)

Is it possible?
Suppose I=1, U=3, M=0
01>03
1: Add 3 (no change mod 3)
2: Duplicate (if original string is not 0 mod 3, next will neither)
3: Replace 111 by 3 (no change mod 3)
4: remove 33 (no change mod 3)

So, with an original string "not 0 mod 3" (like 01), you can't arrive to a string "0 mod 3" (like 03)

Did I commit a mistake? :(

This is a puzzle from Godel, Escher, Bach, which happens to be a good book.
See: http://en.wikipedia.org/wiki/MU_puzzle
Pacman's solution is pretty much the same thing as the wikipedia solution.


[quote="GameBot"]The mean of five numbers is 5. If one of the numbers is removed from the group, the mean of the remaining four numbers is 4.25. What number was removed from the group?[/quote]

Since originally there were 5 numbers, and the mean was 5, that means the sum was $ 5 \times 5 \equal{} 25$. When one of the numbers was removed, there were four numbers, with a mean of 4.25, so a sum of $ 4 \times 4.25 \equal{} 17$. The difference between the sums is 8, so that's the number that was removed.


search
ceiling function
logarithms
induction
algorithm
combinatorics proposed
combinatorics

We have two players, a question asker and a question answerer.  The question answerer has selected a subset of size $ k$ of an $ n$-set, and the question asker's questions take the form of arbitrary subset $ S$ of the $ n$-set.  In all cases, we want to know the minimum number of questions for the question-asker to be sure of learning the selected $ k$-set.

1) $ k \equal{} 2$; answerer answers "yes" if the selected pair of elements are both contained in $ S$

2) $ k \equal{} 2$; answerer answers "yes" if at least one of the selected elements is contained in $ S$

3) $ k \equal{} 2$; answerer answers "yes" if exactly one of the selected elements is contained in $ S$

4) $ k \equal{} 3$; answerer answers "yes" if at least two of the selected triple of elements is contained in $ S$

5) arbitrary (fixed) $ k$; answerer answers "yes" if the selected subset is contained in $ S$

6) arbitrary (fixed) $ k$; answerer answers "yes" if the selected subset has nonempty intersection with $ S$

7) arbitrary (fixed) $ k$; answerer answers with the number of selected elements contained in $ S$

Approximate solution to question 1:
First, we have a search space of size $ \binom{n}{2}$ and we're asking yes-no questions, so we certainly can't do better than $ \left\lceil \log_2 \binom{n}{2} \right\rceil$ questions.  We can achieve $ 2 \left\lceil \log_2 n\right\rceil$, as follows: divide the $ n$-set as $ A \cup B$ with $ |A|$ and $ |B|$ both equal to $ \frac{n}{2}$.
First ask, "Is the pair in $ A$?".
If the answer is "yes," by induction you need only $ 1 \plus{} 2 \left\lceil \log_2 \frac{n}{2}\right\rceil \leq 2 \left\lceil \log_2 n\right\rceil$ questions total.
If the answer is "no," then ask "Is the pair in $ B$?"
If the answer is "yes," by induction you need only $ 1 \plus{} 2 \left\lceil \log_2 \frac{n}{2}\right\rceil \leq 2 \left\lceil \log_2 n\right\rceil$ questions total.
If the answer is "no," then we have one number in each pair.  Now do binary search in each half (for example, asking "is the pair in this half of $ A$ or in $ B$?" as the first question) to get $ 2 \plus{} 2 \left\lceil \log_2 \frac{n}{2}\right\rceil \leq 2 \left\lceil \log_2 n\right\rceil$ questions total.

The difference $ \left\lceil \log_2 \binom{n}{2} \right\rceil \minus{} 2 \left\lceil \log_2 n\right\rceil$ is never more than 2 (and in fact this algorithm gives exactly 1 more than our theoretical bound more than half the time), and it seems likely that this is the true value (though maybe the distance from the theoretical bound can be reduced to 1 in all cases?).

The answer to (2) is identical to the answer from (1).

(3) is cute, and an answer within a constant of the theoretical bound is still possible (was discussing this with D. Speyer and P. McNamara, and I don't think we worked out the precise details).

Even for $ k \equal{} 2$, the answer to question 7 seems challenging.

One further variant:

8) $ k \equal{} 3$; answerer answers "yes" if exactly two of the selected triple of elements is contained in $ S$.


number theory
relatively prime
real analysis
real analysis theorems

Does the set $ \{a\sqrt {p} \plus{} b\sqrt {q}|a,b\in \mathbb{Q}\}$ dense in $ \mathbb{R}$ for relatively prime integers $ p$ and $ q$?

thx.

I assume the question is meant to say $ a,b\in\mathbb{Z}.$ Allowing either $ a$ or $ b$ to take on all rational values doesn't make for an interesting question.

But to answer that question: a necessary and sufficient condition for $ \{au\plus{}bv\mid a,b\in\mathbb{Z}\}$ to be dense in $ \mathbb{R}$ is for $ \frac{u}{v}$ to be irrational.

upps  :blush: , it is related to the combinatorial problems below:

If $ x$ is irrational and $ n \in \mathbb{N}$, from the sequences $ x,2x,\cdots,nx$, two of them  with fractional part less than $ \frac{1}{n}$.

thx for your reply.


I'm not sure what the book is talking about two holidays.

The first holiday is mentioned as:

[quote="My book"]
The fifteenth birthday for many Hispanic girls is a coming of age celebration with a party at home. This party is called [b]una fiesta de quinceanera[/b](n in cean has ~ symbol above it) and can range from a small, informal gathering to a celebration resembling a wedding.  In most cases a local [b]conjunto[/b](group of musicians) plays. It is customary for the [b]padrino[/b] or [b]madrina[/b] to present the honoree with a special gift.[/quote]

The second holiday is mentioned as:

[quote="My book"]
In small towns and cities throughout the Spnaish speaking world, there are special celebrations for all kinds of occasions.  You've already learned that in Spanish-speaking countries many people celebrate not only their birthday, but also their [b]Dia de santo[/b](I didn't learn this.. What is this??). Many cities and countries have saint's days which they often celebrate as holidays.  Spain's patron saint, for example, is Santiago. July 25 is the Feast of Santiago and is an important holiday in Spain.  Juan Carlos, the king of Spain, has his saint's day on June 24, [b]el Dia de San Juan[/b], making it a national holiday[/quote]

Maybe some parts are descriptions of holidays but I still don't get it. :?

Oh and the bold parts are just bolded because that's how it's shown on the book.

Muchas gracias! :lol:  :lol:  :lol:  :lol:

[quote="Silverfalcon"]I'm not sure what the book is talking about two holidays.

The first holiday is mentioned as:

[quote="My book"]
The fifteenth birthday for many Hispanic girls is a coming of age celebration with a party at home. This party is called [b]una fiesta de quinceanera[/b](n in cean has ~ symbol above it) and can range from a small, informal gathering to a celebration resembling a wedding.  In most cases a local [b]conjunto[/b](group of musicians) plays. It is customary for the [b]padrino[/b] or [b]madrina[/b] to present the honoree with a special gift.[/quote]

Maybe some parts are descriptions of holidays but I still don't get it. :?

Oh and the bold parts are just bolded because that's how it's shown on the book.

Muchas gracias! :lol:  :lol:  :lol:  :lol:[/quote]

i don't kno anything about the second one, but we discussed the first one (quinceanera) in my spanish class. you know how for girls in America, their 16th birthday (or "sweet 16") is a big deal? well, for Mexicans, the 15th birthday is their big deal. however, it is even more so than for Americans.An all out party is thrown. The girl usually wears a pink dress, normally pale pink. LOTS of people are invited, one girl at my school had over 300 people attend her quinceanera. there is an organised dance that the family of the girl practices before hand, and then performs. basically, the quinceanera is a big deal.

this is what my spanish teacher told us in class, and what i have heard from the girl at my school who had her quinceneara a couple years ago. if i got anything wrong, anyone who knows about these things please correct me!

Why in this world when someone says something about spanish speaking people, some american comes with the comparation with Mexico. Mexican are just a small part of the spanish speaking population in the world.

The [b]qunceaños[/b] how it is usually called, instead of [b]fiesta de quinceañera[/b] is basically one of the biggest parties that a woman would ever have on her honor in her life, and yes, it is quite similar to a wedding party, except for the small fact that there is no groom (the wedding party, any way, is not about the groom!). Usually is a BIG BIG party. But lately in some countries there has been a change on the uses. At least in Costa Rica, some families decide to spend the money of the party paying a travel for the girl alone, or may be the girl and some adult going with her, to some place in the world. I know people who have gone to specific cities and spend a few weeks there, or having a tour, say, for all western Europe.

With respect to the [b]Día de tu Santo[/b], each day on the Catholic calendar is dedicated to some saints (today, April 15th, is Saint Anastasia and Saint Damian, and some more, you can check some [url=http://www.terra.es/personal/angerod/calendar2.htm]here[/url]), then the chances that your name has some saint in the calendar is pretty high, at least if your name is a common name, as in my case, David, is in six different days, because there have been six different saints with my name and each of them has its own day (of course, it is not the only saint on each of those days).

By the way, non of those is what I would call A HOLIDAY, it is not a day you don't work or something like that. Both are somehow similar to any birthday, just that a little different  :D

I think that what Silverfalcon migth be talking about the Santo Patrono´s day of a city or a Country. My city, for example, has it Santo Patrono´s Days on January 20th and a movile date on May, and no one works on that days. México´s Santa Patrona is the Virgen de Guadalupe (December 12) and almost no one works on that day. (by the way it is my birthday and I have never gone to my school on my birthday)

[quote="Silverfalcon"]I get the first one now. Thanks.

For the second one, I still don't quite get what those "saint days" do. Are they just no working days? I mean all of them? Or just some of them?

Is it like Martin Luther King Day(well, there are days with president but this is one I can think of with person's name)?

Or is it just like a day with someone's name with nothing really special?

I'm bit confused. :blush:[/quote]

OK, I thought that you were talking about the [b]Dia de tu santo[/b] instead of [b]Dia del Patron(a)[/b] of the city, but it has sence (hehe, thanks Dave).

It is not quite like Martin Luther King Day. This day is in honor to an specific person and it is celebrated at the same time in all the US (Even when here in Georgia is a lot bigger the celebration than basically everywhere else). El [b]dia del patron[/b] is quite different. Remember: LatinAmerica was conquered by Spain, that was mainly catholic in that time. Therefore, many many places in Latin America has name of catholic saints (I am from Costa Rica, and the capital of the country is in the city called San Jose, that means, Saint Joseph). Hence, in the Saint Joseph day, it was a holiday, just in that city (it was, but no longer is, at least not for the goverment employees, the goverment no longer respect the eclesiastic holidays, except for, may be, Hollyweek). For around ten years I lived in a city called San Juan, then, in Saint John's day, I had hollyday, and also, in Saint Peter Day (because I use to travel a few miles and study in San Pedro!).

Does it make more sence? if not, I can try to reexplain.

[quote="MathFiend"][quote="djimenez"]Why in this world when someone says something about spanish speaking people, some american comes with the comparation with Mexico. Mexican are just a small part of the spanish speaking population in the world. [/quote]

my bad...I guess I keep forgetting that there is more than just Mexico out there. My spanish teacher married a Mexican, so everything she tells us is in the context of Mexico, rather than Spain etc. (and yes I kno there's more, I just can't think atm). So, I guess that whenever someone mentions Spanish, I think of Senora Castillo, and in return think of Mexico. my bad.  :blush:[/quote]

No problem about that, somehow it is not exactly your fault. There are 23 countries where Spanish spoken for at least half a million people, and the native spanish speaking population, depending on the source, varies from 350 to 500 millions. The most populated spanish countries is Mexico, no doubht about that, the population there is around 75 millions,  but anyway, that is around 20% of us, who are hispanic.

Nevertheless, the hispanic culture is very wide, nevertheless, most of the hispanic here in the United states feel that we are all thought to be Mexican. Do not misunderstand me, I don't have anything against mexicans, but I am not mexican, nevertheless, the hispanic television on the US is mainly mexican, the "hispanic food places" usually sell mexican style food, the "hispanic products" that you find in most of the supermarkets (at least Kroger and Publix) use to be from the mexican culture, and many of us don't use it on our home countries.

I remember the son of a friend that was complaining because the first time he went to the school (fourth grade) when they arrived to the US (his father was a visitor here) the teacher noted his hispanic accent and (T: teacher, K: kid)
[list]T: Then, where in Mexico are you from?
K: No, I am not from Mexico, I am from Spain!
T: Yeah, yeah! it is the same thing![/list]
The kid standed up and went to the World Map in one side of the classroom and said
[list]K: No... you see, this one is Mexico, this one is Spain... THERE IS AN OCEAN IN THE MIDDLE! (My adendum: AND VERY WIDE BY THE WAY!!!).
T: Yes, but you are hispanic, you are all the same.... [/list]
That kind of mentality is the one that sometimes offend to several hispanic people (and I include myself!).

Hi I lived in Spain all my childhood so I have a pretty good sense of what [b]el dia del santo[/b] is like.

When it was my saints' day I usually got a phonecall from everybody that I knew wishing me a "happy saints day" . It's also a good way to catch up on friendships if you haven't spoken with someone for a while.

Within families people usally get together and have a big family lunch or dinner, sometimes on the weekend (Sunday) because everybody is quite busy.

Little kids sometimes get gifts from family members but friends do not bring gifts and a party with freinds is not customary either.

Yes there are some important saints days like St. Joseph, St George orf St. Jhon. The  big festival and celebrations attached to those days are due to pagan reasons often not related to catholicity at all. Alot of this festivals were present before and someone (probably one of the hundreds of popes in the middle ages) named that particular festival after a saint.

Hope this helps you understand "el dia del santo" in Spain. [/b]


let $u_{n}$ be sequence determinate by :
    $u_{1}=1 and u_{n+1}= \sqrt{u_{n}^{2}+\frac{1}{2^{n}}}$forall$n \geq 1$
set a limit $lim_{n \to \infty}u_{n}$

We have $u_{n+1}^{2}-u_{n}^{2}=\frac{1}{2^{n}}$, now use telescopic sum.


[quote="Mathcounts State Sprint 2005"]What is the number of ways to completely cover a 2x5 rectangle with 2x1 rectangles?[/quote]

Could someone please explain how to solve this kind of problem and be sure to count every case? I missed a point because of this problem.

I hate those too. You have to visiualize it, but I sometimes get it wrong too.  :(


algebra
polynomial
Ring Theory
induction
superior algebra
superior algebra unsolved

show that if $ f\equal{}a_{0}\plus{}a_{1}x\plus{}...\plus{}a_{n}x^{n}$is nipotent then $ a_{0},a_{1},...,a_{n}$ are nilpotents. Ring R is commutative with 1

$ R[x]$ is a subring of $ R[[X]]$ hence the claim follows from the statement of your last thread: http://www.mathlinks.ro/Forum/viewtopic.php?t=305331

I could but why? Everything else would only complicate matters here and I doubt that it would give any significant insight.

[quote="wiosna"]could you give me any other way to prove it[/quote]Induction on $ n$ - but for the induction step you will certainly bump into some kind of identity given in -oo-'s proof in one of your other postings.


In triangle $ABC$, angle $B$ is a right angle.  Points $P$ and $Q$ are on side $BC$, with $P$ between $B$ and $Q$, so that segments $AP$ and $AQ$ trisect angle $BAC$.  The measure of angle $APC$ is $114^\circ$.  What is the number of degrees in the measure of angle $ACB$?

[hide="Solution"]The number of degrees in angle APB is 180-114=66. The number of degrees in angle BAP is 90-66=24. The number of degrees in angle BAC is 3*24=72. The number of degrees in angle ACB is 90-72=18.[/hide]

[hide="My Answer."]We are given $\angle BAP=\angle PAQ=\angle QAC$.  If we find one of these angles, we can find $\angle BAC.$  We know that $\angle APC=114$, so $\angle APB=66$, because of a straight angle.  Now, we know $\angle BAP=24$, because the angles of a triangle add to 180.  Now we know $\angle BAQ=3\cdot 24=72$.  Therefore, $\angle ACB=180-(72+90)=180-162=18$.[/hide]


Let ABC be a triangle with hypotenuse 4.  On each side of triangle ABC an equilateral triangle is constructed outward ("as shown in the figure" but not really).  Find the sum of the the areas of the equilateral triangles.

my algebraically devoid answer:
[hide]  I just thought of the pythagorean theorem and extended it.  The two triangles on the unknown sides should equal the area of the triangle on the known side.  So, I got $(1/2*4*4*2) = 16$.  I'm not sure if that is correct though... oops I meant *2...[/hide]

[hide]
If we denote sides as $a,b,c=4$ then areas of constructed triangles are:
$a^2{\sqrt 3\over 4},b^2{\sqrt 3\over 4},c^2{\sqrt 3\over 4}$

So we have sum equal to ${\sqrt 3\over 4}(a^2+b^2+c^2)={\sqrt 3\over 4}(c^2+c^2)=8\sqrt 3$
[/hide]

How did you get the $\sqrt(3)$ ?  I would love to know what I did wrong.  Please explain  :)

If we have equilateral triangle $\Delta ABC$ with side length $x$, then area of the triangle is $\frac{x^2\sqrt 3}{4}$.

Proof:
Denote with $D$ intersection of $AB$ and perpendicular from $C$.
We know that area is equal to $\frac{AB\cdot CD}{2}$ and since we have $x=AB$ it suffices to show that $CD={x\sqrt 3\over 2}$.

Using pythagorean theorem we see that $CD^2=x^2-(\frac{x}{2})^2=x^2-\frac{x^2}{4}=\frac{3x^2}{4}$.
Thus, $CD=\frac{x\sqrt 3}{2}$


The surface area of a particular cube is 600 square inches. When the edges of the cube are doubled in length, what is the volume of the new cube, in cubic inches?

[hide] We have $ 6e^2\equal{}600 \implies e\equal{}10$. Twice this is $ 20$, so $ 20^3\equal{}\boxed{8000}$.[/hide]


Most people think that [tex]0^2 = 0 \times 0 = 0.[/tex]  Zero squared is also
[tex]\displaystyle{0^{5-3} = \frac{0^5}{0^3}}[/tex] and if zero to a power is zero, then the expression equals 0/0, and division by zero is undefined!  So what does 0 squared equal?

You can't use laws of exponents if the base is 0.

To elaborate a little -- while it is true that 02 = 05 - 3, it is [i]not[/i] true that this also equals 05/03.  Any time you arrive by arithmetic procedure at an expression with a denominator of 0, it means you've done something wrong.


geometry
combinatorics proposed
combinatorics

On a plane, call the band with width $d$ be the set of all points which are away the distance of less than or equal to $\frac{d}{2}$ from a line.Given four points $A,\ B,\ C,\ D$ on a plane.If you take three points among them, there exists the band with width $1$ containing them. Prove that there exist the band with width $\sqrt{2}$ containing all four points .

Ok,I'll try to give it a shot.
It's easy to see that if a triangle with three perpendicular with length >1,then there's no band with width 1 containing them.
We suppose that triangle ABC has the max area among the four triangles.
We consider that Hb=1 ( Why? the reason is obvious ... )
E is belong to the line AC and BE is perpendicular to AC
if AE<=1 or CE<=1 then AB<=$\sqrt{2}$ or CB<=$\sqrt{2}$
it's easy to see that there exist a band with width 1 containg them
so we consider that AE>1 or CE>1 now we just consider BD>$\sqrt{2}$ ( Why? Quite simply ... ) then either triangle BAD or  triangle BCD has three perpendicular with length >1(Here is a little bit complicated),which means that one of them cannot be covered by a band with width 1. so we done.

A solution is given on page 266 of the March 2008 issue of the American Mathematical Monthly.


linear algebra
matrix
vector
quadratics
algebra
linear algebra unsolved

Can someone help me prove this:

Given a complex mxm matrix A and complex m-vector b, prove that the complex n-vector x is the solution of the least squares problem if and only if A*(Ax-b)=0.

Thanks in advance.

e.g., write $ f(\bar{x})\equal{}\parallel{}A\bar{x}\minus{}\bar{b}\parallel{}^2\equal{}\parallel{}b\parallel{}^2\plus{}\bar{x}^TA^TA\bar{x}\minus{}2\bar{x}^TA^T\bar{b}$ as a quadratic equation, then determine the gradient as $ \nabla f(\bar{x})\equal{}2[A^TA\bar{x}\minus{}A^T\bar{b}]$ and the Hesse matrix as $ H_f(\bar{x})\equal{}2A^TA$. Since the Hesse matrix is positive semi-definite, we have a minimum at $ \nabla f(\bar{x})\equal{}\bar{0}\Longleftrightarrow A^T(A\bar{x}\minus{}\bar{b})\equal{}\bar{0}$

Note: JimmyT did say his matrices were complex valued. And that asterisk in what he wrote as "A*(Ax-b)" isn't a multiplication symbol, it's the Hermitian conjugate.

It looks like hjbrasch answered the question is a real context. Some adjustments in language are needed.


Let $ABCDE$ be a convex pentagon, such that $AB=BC$, $CD=DE$, $\angle B+\angle D=180^{\circ}$, and it's area is $\sqrt2$.

a) If $\angle B=135^{\circ}$, find the length of $[BD]$.

b) Find the minimum of the length of $[BD]$.

Put $ \angle D \equal{}\alpha \Rightarrow \angle B\equal{} \pi \minus{} \alpha$, $ AB\equal{}BC\equal{}x$,$ CD\equal{}DE\equal{}y$ and $ \angle ACE \equal{}\theta$.

We have: $ [ABCDE ]\equal{} [ABC]\plus{}[ACE]\plus{}[CDE]$.

$ [ABC]\equal{}\frac{1}{2}\cdot x^2\cdot\sin(\pi \minus{}\alpha)$ and $ [CDE]\equal{}\frac{1}{2}\cdot y^2 \cdot \sin\alpha$.
$ AC\equal{} 2x \cos\frac{\alpha}{2}$ and $ CE\equal{}2y\sin\frac{\alpha}{2}$.
Therefore $ [ACE]\equal{}\frac{1}{2}\sin\theta\cdot(2x\cos\frac{\alpha}{2}) \cdot (2y\sin\frac{\alpha}{2}) \equal{} xy\sin \theta \cdot\sin\alpha$.
Hence $ [ABCDE]\equal{}\frac{\sin\alpha}{2}\cdot(x^2\plus{}y^2\plus{}2xy\sin\theta)$.
But $ \angle BCD \equal{} \frac{\pi}{2}\plus{}\theta$, so : $ [ABCDE]\equal{} \frac{\sin\alpha}{2}\cdot(x^2\plus{}y^2\minus{}2xy\cos\angle BCD)\equal{}\frac{BD^2}{2}\cdot \sin \alpha$.

So we have the equation: $ BD^2\equal{}\frac{2\sqrt{2}}{\sin \alpha}$

a) if $ \angle B \equal{} \frac{3\pi}{4} \Rightarrow \alpha\equal{}\frac{\pi}{4}\Rightarrow BD\equal{}2$.
b) $ BD$ is minimal if $ \frac{2\sqrt{2}}{\sin \alpha}$ is minimal, i.e. if $ \alpha\equal{}\frac{\pi}{2}$. In this case $ BD\equal{}\sqrt{2\sqrt{2}}$.


[quote="iamagenius"]a train traveling from aytown to beetown meets with an accident after 1 hour. the train is stopped for 30 minutes, after which it proceeds at four-fifths of its usual rate, arriving at beetown 2 hours late. if the train had covered 80 miles more before the accident, it would have been just one hour late. what is the usual rate of the train?[/quote]

There are two ways, I think

[hide="slower way"]Let's say it traveled total $d$ miles and that its normal speed is $a$ mph.

We can make the time equations:

$\frac{d}{a}+2=1+0.5+\frac{d-a}{\frac{4}{5}a}$

$\frac{d}{a}+1=1+\frac{80}{a}+0.5+\frac{d-80-a}{\frac{4}{5}a}$

Solve and get $a=20$ and $d=140$. So the usual rate is $\boxed{20}$ mph.[/hide]

[hide="faster way"]Let's say the usual speed is $a$ mph. If it travels at four-fifths speed, it takes five-fourth of usual time, which is one-fourth more than usual.

So travelling $80$ miles in 1/4 more time than what it used to made a difference of $1$ hour, which is equivalent to travelling $a$ miles. So

$\frac{80}{4}=a$

And $a=\boxed{20}$(mph).[/hide]

Let the normal speed be x miles/hr
             normal time duration be y hrs
total distance= xy miles
in the unfortunate day everything changed but the total distance remained constant.
In the 1st 1hr it travelled x miles
In the next 30 mins it travelled 0 miles
in the 1st 1.5 hrs it travelled x miles
total time taken that day=y+2
after 1st 1.5 hrs time left=(y+2)-1.5=y+0.5, which it travelled with speed 4/5x miles/hr
distance covered=4/5x(y+0.5)
total distance= x+4/5x(y+0.5)
x+4/5x(y+0.5)=xy-------------------(1)
similarly we get 2nd equation
(x+80)+4/5x(y-0.5)=xy ---------(2)
solve to get the answer.


Three circles of radius r are inscribed in a circle
of radius 2 + (radical 3). Find r.  (excuse my lack of LaTeX)

What approaches would you recommend?

If those three circles are mutually externally tangent, then all you have to do is connect their centers to form an equilateral triangle whose centroid happens to be the center of the larger circle.

It doesn't say they're mutually tangent so....

1) Let's assume that they're tangent how do we solve?  I've got a diagram drawn but I don't know what to do next.

2) Assume that they're not tangent then what?

2) It's impossible to solve without more information

Since it is inscribed, it's mutually tangent.

Try connecting the centers of the circles and drawing the radii of the bigger circle.

Well, I've tried messing around with different diagrams, etc but I'm not getting anywhere.  So feel free to post a solution  :P

Did you know that $ 2/3$ times the altitude of the equilateral triangle sunehra told you to draw is equal to the big circle's radius?  :wink:

No I didn't know that :)  How can we prove that?

Draw all three medians and use $ 30\minus{}60\minus{}90$ triangles. The rest will follow through.  :)

What answer did you get?

I got r = $ 2\sqrt{3} \plus{} 3$ $ /4$

But the answer is r = $ /sqrt{3}$  (why didn't that come out right?)

So how did you solve it?  I would post my solution, but I suck at LeTax.
Basically, I split up the equilateral triangle into 5 congruent 30-60-90 triangles by drawing the medians and dropping down a perpendicular line.

My above hint was incorrect (I was thinking of something else), but by using $ 30\minus{}60\minus{}90$ angles, the answer is extremely easy. I got $ \sqrt{3}$ also. I'll post a solution a few minutes later.

[quote="Evolved"]But the answer is r = $ / sqrt3$ (why didn't that come out right?)[/quote]

the code is [code]$\sqrt{3}$[/code]

[geogebra]59302c7085fd4e0be10fc8543343ac2439b17acb[/geogebra] Now, we have the equation $ \frac {2r\sqrt {3} \plus{} 3r}{3} \equal{} 2 \plus{} \sqrt {3}\Rightarrow{r(2\sqrt {3} \plus{} 3)} \equal{} 6 \plus{} 3\sqrt {3}$. We get $ r \equal{} \frac {6 \plus{} 3\sqrt {3}}{2\sqrt {3} \plus{} 3}$. This ultimately simplifies to $ \sqrt {3}$.

Nice diagram :)  My diagram was pretty much the same expect I mistakenly labeled r as r$ \sqrt{3}$

[quote="Evolved"]What answer did you get?

I got r = $ 2\sqrt {3} \plus{} 3$ $ /4$

But the answer is r = $ /sqrt{3}$  (why didn't that come out right?)

So how did you solve it?  I would post my solution, but I suck at LeTax.
Basically, I split up the equilateral triangle into 5 congruent 30-60-90 triangles by drawing the medians and dropping down a perpendicular line.[/quote]

It's supposed to be : $ \sqrt{3}$.
Not /, it's  \.


this is kinda random, but i was wondering... how do you prove Archimedes' Lemmas? i've tried, but they seem to be nigh impossible!  :huh:
you can find them with diagrams here:

http://www.gogeometry.com/ArchBooLem00.htm

how do you prove this stuff? man, i feel like an idiot,
cuz so old guy who lived more than 2,000 years ago is beating the crap outta my brain... T_T

Think of vectors and similar triangles(I don't have enough time to prove this right now).


geometry
circumcircle
trapezoid
trigonometry
trig identities
Law of Sines
geometry solved

My English is not well.So I translated the problem on net.Maybe there are some mistakes in it.

There's a point O in equilateral triangle ABC.Angle AOB =110 degrees. Angle BOC =135 degrees.
Questions:
(1)Can it form a triangle to regard OA , OB , OC as? If can,ask not appearing this triangle reading of every mao; If cannot, please give reasons.
(2)If horn size of AOB keep intact, act as horn when how many degrees BOC mean , is the triangle taking OA , OB , OC as side a right triangle?

Just rewording the post:

$O$ is a point in an equilateral triangle $ABC$ such that $\angle AOB = 110^\circ$, $\angle BOC = 135^\circ$.

1) Can $OA$, $OB$, $OC$ be the sides of a triangle?  If so, what are the angles of the triangle formed?  If not, explain why.

2) If we keep the size of $\angle AOB$ as $110^\circ$ and allow the size of $\angle BOC$ to vary, what must the size of $\angle BOC$ be when the triangle with side lengths $OA$, $OB$, $OC$ is a right-angled triangle?

Pompeiu's Theorem states that when $ABC$ is an equilateral triangle and $O$ is a point in the plane, $OA,OB,OC$ are the sides of a triangle, which is degenerate only when $O$ lies on the circumcircle of $ABC$, so the answer is yes.

If $O$ lies inside the triangle $ABC$, then the angles of the triangle formed by $OA,OB,OC$ are $\angle BOC-60^{\circ},\angle COA-60^{\circ},\angle AOB-60^{\circ}$. This is found by considering points $P,S\in BC,\ N,R\in CA,\ Q.M\in AB$ s.t. $O\in MN,PQ,RS$ and $MN\|BC,PQ\|CA,RS\|AB$. We now use the isosceles trapezoids $OMBP,OPCR,ORAM$ to construct our triangle: $OA=RM,OB=MP,OC=PR$, so the triangle we want is $RMP$. It's easy to compute its angles in terms of $\angle BOC,\angle COA,\angle AOB$ now.

Try to show that the locus of $O$ s.t. $OA^2=OB^2+OC^2$ is an arc subtending an angle of $150^{\circ}$ and passing through $B,C$.

Just for the sake of completeness (it's clear that similar problems will regularly occur on ML) let me state a more general fact:

[color=blue][b]Theorem 1.[/b] Let O be a point inside an equilateral triangle ABC. Then, there exists a triangle XYZ with the sidelengths YZ = OA, ZX = OB and XY = OC; moreover, the angles of such a triangle XYZ are

< ZXY = < BOC - 60°;
< XYZ = < COA - 60°;
< YZX = < AOB - 60°.

Particularly, this triangle XYZ is right-angled if and only if one of the angles < BOC, < COA and < AOB equals 150°.[/color]

[i]Proof.[/i] Let D, E, F be the orthogonal projections of the point O on the sides BC, CA, AB of triangle ABC. Then, since < OEA = 90° and < OFA = 90°, the points E and F lie on the circle with diameter OA. In other words, the circumcircle of triangle AEF has diameter OA. But by the extended law of sines, a side of a triangle equals the diameter of the circumcircle, multiplied with the sine of the opposite angle; thus, in triangle AEF, we have $EF=OA\cdot\sin\measuredangle EAF$. Since < EAF = < CAB, and < CAB = 60° (since triangle ABC is equilateral), this becomes $EF=OA\cdot\sin 60^{\circ}$. Similarly, $FD=OB\cdot\sin 60^{\circ}$ and $DE=OC\cdot\sin 60^{\circ}$. Hence, EF : FD : DE = OA : OB : OC. Thus, there exists a triangle XYZ with the sidelengths YZ = OA, ZX = OB and XY = OC, and any such triangle XYZ must be similar to the triangle DEF. Hence, < XYZ = < DEF.

Now, since the points E and F lie on the circle with diameter OA, the quadrilateral AEOF is cyclic, so that < FEO = < FAO. In other words, < FEO = < BAO. Similarly, < DEO = < BCO. Thus,

  < DEF = < FEO + < DEO = < BAO + < BCO
              = (180° - < ABO - < AOB) + (180° - < CBO - < BOC)

by the sum of angles in triangles AOB and BOC. After a rearrangement of terms, this becomes

  < DEF = (360° - < AOB - < BOC) - (< ABO + < CBO) = < COA - < ABC = < COA - 60°

(since < ABC = 60°, as triangle ABC is equilateral). Hence, < XYZ = < DEF = < COA - 60°. Similarly, < YZX = < AOB - 60° and < ZXY = < BOC - 60°.

Finally, the triangle XYZ is right-angled if and only if one of its angles < ZXY = < BOC - 60°, < XYZ = < COA - 60°, < YZX = < AOB - 60° equals 90°. This holds if and only if one of the angles < BOC, < COA, < AOB equals 60° + 90° = 150°.

Thus, Theorem 1 is completely proven.

Of course, Theorem 1 easily solves the problem.

  Darij


Let $f(x)$ be a continuous and decrease function on interval [0, b] and a$\in$[0, b]. Prove that:
$b.\int\limits_{0}^{a}f(x)dx\geq a.\int\limits_{0}^{b}f(x)dx$

We have that $ \int_{0}^{a}f(x)dx=\frac{a}{b}\int_{0}^{b}f\left(\frac{a}{b}x\right)dx$ so it remains to prove that $a\int_{0}^{b}f\left(\frac{a}{b}x\right)dx \geq a\int_{0}^{b}f(x)dx$ which is true because $f(\frac{a}{b}x)\geq f(x)$.


calculus
integration
algebra
partial fractions
calculus computations

Now this is tricky to write...bear with me.

Integrate (2x^2 - 9x - 6)/(x(x^2 - x - 6)) with respect to x between the lower limit x = 4 and upper limit x = 6.  Given that this definite integral = Ln(m/n) determine the values for the inegers m and n.

[quote="sludgethrower"]
\[\int_{4}^{6}\frac{2x^{2}-9x-6}{x(x^{2}-x-6)}\ dx=\ln \frac{m}{n}\]
Determine the values for the integers $m$ and $n$.
[/quote]
[hide="hint"]
$x^{2}-x-6=(x-3)(x+2)$ so partial fractions can easily be used
[/hide]

I know the answer...so i don't need hints...

If you don't need hints why have you posted that problem?


inequalities
limit
real analysis
real analysis unsolved

Let we are given the sequence of nonnegative real numbers $ a_n$ and assume the following inequality holds for all $ n\in N$:
$ a_n\geq\frac{n^2a_{n\plus{}1}\minus{}1}{n^2\minus{}n}$.
Show that $ a_n$ is converges.

[quote="Ibrogimov"]Let we are given the sequence of nonnegative real numbers $ a_n$ and assume the following inequality holds for all $ n\in N$:
$ a_n\geq\frac {n^2a_{n \plus{} 1} \minus{} 1}{n^2 \minus{} n}$.
Show that $ a_n$ is converges.[/quote]
$ (n \minus{} 1)a_n\geq na_{n \plus{} 1} \minus{} \frac {1}{n}$
Let $ x_n \equal{} (n \minus{} 1)a_n$
so
$ x_n\geq x_{n \plus{} 1} \minus{} \frac {1}{n}$
$ x_{n \plus{} 1} \minus{} x_n\leq \frac {1}{n}$
So
$ x_{n \plus{} 1} \minus{} x_1\leq \sum_{i \equal{} 1}^n\frac {1}{i}$
$ \Rightarrow x_{n \plus{} 1}\leq x_1 \plus{} \sum_{i \equal{} 1}^n\frac {1}{i}$
But $ \lim_{n\to\infty}\frac {1}{n}.({\sum_{i \equal{} 1}^n\frac {1}{i}} \plus{} x_1) \equal{} 0$
But $ x_n\geq 0$
So $ \lim_{n\to\infty}x_n \equal{} 0$


linear algebra
matrix
trigonometry
induction
algebra
polynomial

Q1 show that DETERMINANT

tan (A+P)    ,   tan (B+P)   ,    tan( C+P )
tan  (A+Q)  ,    tan ( B+Q)  ,   tan( C+Q  )   =O
tan ( A+R )    , tan ( B+R)  ,   tan( C+R )

where A+B+C+P+Q+R=0

Pretty version:
[quote="ayush_2008"]Choose $ A,B,C,P,Q,R$ such that $ A\plus{}B\plus{}C\plus{}P\plus{}Q\plus{}R\equal{}0$. Show that
$ \det\begin{bmatrix}\tan(A\plus{}P)&\tan(B\plus{}P)&\tan(C\plus{}P)\\ \tan(A\plus{}Q)&\tan(B\plus{}Q)&\tan(C\plus{}Q)\\ \tan(A\plus{}R)&\tan(B\plus{}R)&\tan(C\plus{}R)\end{bmatrix}\equal{}0$[/quote]

I would also add that none of the pairwise sums are $ \frac{\pi}{2}$ mod $ \pi$. You get an $ \infty\minus{}\infty$ indeterminate form in that case, and you don't get any linear dependence relation unless you allow infinitesimal coefficients.

[hide="Hint"]
Hint: If three angles $ x$, $ y$, $ z$ are such that $ x \plus{} y \plus{} z \equal{} 0$, then $ \tan x \plus{} \tan y \plus{} \tan z \equal{} \tan x\tan y\tan z$.
And
$ \det\left(\begin{array}{ccc}a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3\end{array}\right) \equal{} a_1b_2c_3 \plus{} a_2b_3c_1 \plus{} a_3b_1c_2 \minus{} a_1b_3c_2 \minus{} a_2b_1c_3 \minus{} a_3b_2c_1$.[/hide]

What is way more interesting is to find a simpler proof for the canonical generalization:

[color=blue][b]Problem.[/b] Let $ n\in\mathbb{N}$ be odd, and let $ A_1$, $ A_2$, ..., $ A_n$, $ X_1$, $ X_2$, ..., $ X_n$ be $ 2n$ angles such that $ \sum_{k \equal{} 1}^nA_k \plus{} \sum_{k \equal{} 1}^nX_k \equal{} 0$. Define a matrix $ M\in\mathbb{R}^{n\times n}$ by

$ M \equal{} \left(\tan\left(A_i \plus{} X_j\right)\right)_{1\leq i\leq n,\ 1\leq j\leq n}$.

Show that $ \det M \equal{} 0$.[/color]

  darij

Better: drop the odd $ n$ condition, and make it $ \sum A_k\plus{}\sum X_k\equiv (n\minus{}1)\frac{\pi}{2}\mod \pi.$

you try with $ n \equal{} 5$
 If $ x_1 \plus{} x_2 \plus{} .. \plus{} x_5 \equal{} 0$ then $ tan((x_1 \plus{} x_2) \plus{} (x_3 \plus{} x_4) \plus{} x_5) \equal{} \frac {tan(x_1 \plus{} x_2) \plus{} tan(x_3 \plus{} x_4) \plus{} tan(x_5) \minus{} tan(x_1 \plus{} x_2)tan(x_3 \plus{} x_4)tan(x_5)}{P(x_1,..,x_5)}$
then  $ 0 \equal{} tan(x_1 \plus{} x_2) \plus{} tan(x_3 \plus{} x_4) \plus{} tan(x_5) \minus{} tan(x_1 \plus{} x_2)tan(x_3 \plus{} x_4)tan(x_5)$ ....
 $ \sum_{k \equal{} 1}^{5}{tan(x_k)} \plus{} \prod_{k \equal{} 1}^{5}tan(x_k) \minus{} \sum{tan(x_i)tan(x_j)tan(x_k)} \equal{} 0$
 That $ \sum_{k \equal{} 1}^{5}{tan(x_k)}\neq \prod_{k \equal{} 1}^{5}tan(x_k)$ it isn't easy to answer that you are correct . ..  I hope that you can give more information with $ n \equal{} 5$ and example about it . Thanks !

That's why I was asking for a simpler proof.

Mine goes, roughly, as follows (sorry for using other notations than you):

For any angles $ \phi_1$, $ \phi_2$, ..., $ \phi_n$ with tangents $ x_1$, $ x_2$, ..., $ x_n$, a straightforward induction shows that

$ \tan\left(\phi_1 + \phi_2 + ... + \phi_n\right) = \frac {\sum_{i = 0}^{\infty} \left( - 1\right)^i e_{2i + 1}\left(x_1,x_2,...,x_n\right) } {\sum_{i = 0}^{\infty} \left( - 1\right)^i e_{2i}\left(x_1,x_2,...,x_n\right) }$,

where $ e_i$ denotes the $ i$-th elementary symmetric polynomial (note that the sums are not really infinite, because $ e_m\left(x_1,x_2,...,x_n\right) = 0$ for all $ m > n$).

Thus, if $ \phi_1 + \phi_2 + ... + \phi_n\equiv \left(n - 1\right)\frac {\pi}{2}\mod\pi$, then

$ \sum_{i = 0}^{\infty} \left( - 1\right)^i e_{2i + 1}\left(x_1,x_2,...,x_n\right) = 0$ if $ n$ is odd;
$ \sum_{i = 0}^{\infty} \left( - 1\right)^i e_{2i}\left(x_1,x_2,...,x_n\right) = 0$ if $ n$ is even.

These two results can be unified: If $ \phi_1 + \phi_2 + ... + \phi_n \equiv \left(n - 1\right)\frac {\pi}{2}\mod\pi$, then

$ \sum_{j = 0}^{\infty} \left( - 1\right)^j e_{n - 2j}\left(x_1,x_2,...,x_n\right) = 0$,

where $ e_k$ is defined to be the $ 0$ polynomial for all $ k < 0$.

Now, this yields

$ e_n\left(x_1,x_2,...,x_n\right) = - \sum_{j = 1}^{\infty} \left( - 1\right)^j e_{n - 2j}\left(x_1,x_2,...,x_n\right)$

whenever $ \phi_1 + \phi_2 + ... + \phi_n\equiv \left(n - 1\right)\frac {\pi}{2}\mod\pi$.

As a consequence, if $ \sum_{k = 1}^nA_k + \sum_{k = 1}^nX_k \equiv \left(n - 1\right)\frac {\pi}{2}\mod\pi$ (I am solving Jmerry's generalization here), then

$ \det M = \sum_{\sigma\in S_n}\left( - 1\right)^{\sigma}\tan\left(A_1 + X_{\sigma\left(1\right)}\right)\tan\left(A_2 + X_{\sigma\left(2\right)}\right)...\tan\left(A_n + X_{\sigma\left(n\right)}\right)$
$ = \sum_{\sigma\in S_n}\left( - 1\right)^{\sigma}e_n\left(\tan\left(A_1 + X_{\sigma\left(1\right)}\right),\tan\left(A_2 + X_{\sigma\left(2\right)}\right),...,\tan\left(A_n + X_{\sigma\left(n\right)}\right)\right)$
$ = \sum_{\sigma\in S_n}\left( - 1\right)^{\sigma}\left( - \sum_{j = 1}^{\infty} \left( - 1\right)^j e_{n - 2j}\left(\tan\left(A_1 + X_{\sigma\left(1\right)}\right),\tan\left(A_2 + X_{\sigma\left(2\right)}\right),...,\tan\left(A_n + X_{\sigma\left(n\right)}\right)\right)\right)$
(because $ \left(A_1 + X_{\sigma\left(1\right)}\right) + \left(A_2 + X_{\sigma\left(2\right)}\right) + ... + \left(A_n + X_{\sigma\left(n\right)}\right)\equiv \left(n - 1\right)\frac {\pi}{2}\mod\pi$)
$ = - \sum_{j = 1}^{\infty}\left( - 1\right)^j \sum_{\sigma\in S_n}\left( - 1\right)^{\sigma}e_{n - 2j}\left(\tan\left(A_1 + X_{\sigma\left(1\right)}\right),\tan\left(A_2 + X_{\sigma\left(2\right)}\right),...,\tan\left(A_n + X_{\sigma\left(n\right)}\right)\right)$
$ = 0$,

because for any $ j > 1$, the term $ \sum_{\sigma\in S_n}\left( - 1\right)^{\sigma}e_{n - 2j}\left(\tan\left(A_1 + X_{\sigma\left(1\right)}\right),\tan\left(A_2 + X_{\sigma\left(2\right)}\right),...,\tan\left(A_n + X_{\sigma\left(n\right)}\right)\right)$ equals $ 0$, which in turn follows from the more general (and easy) fact that whenever $ U = \left(u_{i,j}\right)_{1\leq i\leq n,\ 1\leq j\leq n}$ is an $ n\times n$ matrix, and $ P$ is a polynomial of $ n$ variables with total degree $ < n - 1$, then

$ \sum_{\sigma\in S_n}\left( - 1\right)^{\sigma}P\left(u_{1,\sigma\left(1\right)},u_{2,\sigma\left(2\right)},...,u_{n,\sigma\left(n\right)}\right) = 0$.

  Darij

A proof isn't simple . I will read many times to understand it and I hope that I can make more ... :read:
 To add interesting If you are interested a some $ \sin,\cos,ch,sh$ (a short proof) .....
[color=blue][b]Problem.[/b] Let $ n\in\mathbb{N}$ , and let $ A_1$, $ A_2$, ..., $ A_n$, $ X_1$, $ X_2$, ..., $ X_n$ be $ 2n$ complex number :
$ M\in\mathbb{C}^{n\times n}$, $ N\in\mathbb{C}^{n\times n}$, $ P\in\mathbb{C}^{n\times n}$ , $ Q\in\mathbb{C}^{n\times n}$ by
 $ M \equal{} \left(\sin^m\left(A_i \plus{} X_j\right)\right)_{1\leq i\leq n,\ 1\leq j\leq n}$,$ N \equal{} \left(\cos^m\left(A_i \plus{} X_j\right)\right)_{1\leq i\leq n,\ 1\leq j\leq n}$,$ P \equal{} \left(ch^m\left(A_i \plus{} X_j\right)\right)_{1\leq i\leq n,\ 1\leq j\leq n}$,$ Q \equal{} \left(Sh^m\left(A_i \plus{} X_j\right)\right)_{1\leq i\leq n,\ 1\leq j\leq n}$.

Show that $ \det(M) \equal{} \det(N) \equal{} \det(P) \equal{} \det(Q) \equal{} 0$ if $ n > m \plus{} 1$.[/color]
 ........  :play_ball: ......  :play_ball:


1. find $ \tan x$ if $ \sin x \plus{}\cos x\equal{}\frac{1}{5}$ and $ \frac{\pi}{2}\le x\le\pi$.
2. A parabolic dish is 10 deep and 100 in diameter. How far away from its vertex is its focus?

[hide="1?"]Wouldn't you just divide the whole equation by $ \cos{x}$?

Then, $ \tan x\equal{}\frac{1}{5}\sec x\minus{}1$

Or did you want a numerical answer?[/hide]

[hide="Number 1"]$ \sin x \plus{} \cos x \equal{} \sqrt{2} \sin\left(x \plus{} \frac{\pi}{4}\right)$, and so we have

$ x \plus{} \frac{\pi}{4} \equal{} \pi \minus{} \arcsin\left(\frac{\sqrt{2}}{10}\right)$ => $ x \equal{} \frac{3 \pi}{4} \minus{} \arcsin\left(\frac{\sqrt{2}}{10}\right)$[/hide]

[hide="#1"]$ \sin x\plus{}\cos x\equal{}{1\over 5}$

Squaring, we get

$ 1\plus{}2\sin x\cos x\equal{}{1\over 25}\implies \sin x\cos x\equal{}\minus{}{12\over 25}$

Hence $ \sin x, \cos x$ are the solutions to the equation

$ t^2\minus{}{1\over 5}t\minus{}{12\over 25}\equal{}0\iff t\in\left\{{4\over 5},\minus{}{3\over 5}\right\}$

But as $ x$ lies in the second quadrant, its sine is positive and the cosine negative, hence $ \sin x\equal{}{4\over 5}\land\cos x\equal{}\minus{}{3\over 5}\implies\tan x\equal{}\minus{}{4\over 3}$[/hide]

[hide="maybe Im wrong for number 1"]Square and simplify to get $ sin2x = - \frac {24}{25}$. So, you know it is a 7-24-25 triangle with $ 2x$ being in quadrants III or IV. Then, $ cos2x = \frac {7}{25}$.
$ \cos^2x - sin^2x = \frac {7}{25}$
$ \cos^2x + sin^2x = 1$.Add and solve for $ cosx$, and it is $ \frac {4}{5}$. It's a 3-4-5 triangle, and $ \tanh = \frac {3}{4}$.[/hide]

Edit:Didn't read the $ \frac{\pi}{2}$ part.


[quote="Nawahd Werdna"]I just wanted to know how all of you practice for CMO. For example, I do the past contests, and I try to do some of the easier USAMOs, but are there any other olympiads that have comparable problem types? Also, what topics are usually more heavily covered for CMO? What basic theorems should I know?  :maybe:  :maybe:[/quote]

I want to know too!

Many people see CMO in fact, as a "speed contest". Compared to other olympiads obviously, it has a lot less time... for a lot more questions too! So practising speed is a huge factor.

Getting used to thinking ideas up quickly is something I do for CMO.


Let $a$ be a  real number  such that $2a^6+a^2=\frac{3}{2}+2a^4$. Prove that $a^8>1$.

[color=blue][b]Let $a$ be a real number such that $2a^6 + a^2 = \frac{3}{2} + 2a^4$ Prove that $a^8 > 1$[/b][/color]

[u][b]Solution:[/b][/u]
$2a^6 + a^2 = \frac{3}{2} + 2a^4 \Rightarrow 4a^6 - 4a^4 + 2a^2 - 3 = 0 \Rightarrow$

$\Rightarrow 4a^4 \left( {a^2 - 1} \right) + 2\left( {a^2 - 1} \right) = 1 \Rightarrow \left( {a^2 - 1} \right)\left( {4a^4 + 2} \right) = 1 \Rightarrow$

$\Rightarrow a^2 - 1 > 0 \Rightarrow a^2 > 1 \Rightarrow a^8 > 1$


Let a,b,c \in [0,1] Find the minimum value of K such that the following ineq holds:
(1-a)(1-b)(1-c) \leq K(1-(a+b+c)/3)


calculus
integration
inequalities
function
limit
real analysis
real analysis unsolved

Let $ f: [0,1]\rightarrow\mathbb{R}$ is a decrasing function. For all $ x\in(0,1)$ prove that $ x\int_{0}^{1}f(t)dt\leq\int_{0}^{x}f(t)dt.$ Thank you

Hallo friend  :)

Let consider the function $ g(x)\equal{}\frac{\int_0^xf(t)dt}{x}$ on $ (0;1]$. We have $ g'(x)\leq 0$ on $ (0;1)$. So $ g$ is decreasing on $ (0;1)$. This implies for any $ x\in(0;1)$ we have
$ g(x)\geq \lim_{t\to1^{\minus{}}}g(t)\equal{}g(1)\equal{}\int_0^1f(t)dt$


geometry
geometric transformation
angle bisector

Dạ em muốn chuyển mấy cái đề này sang tiếng anh. Ai giúp em với.
1. Cho tam giác ABC không cân. Đường tròn nội tiếp I tiếp xúc với AB,AC lần lượt tại M và N. MN cắt BC tại D. Tiếp tuyến song song với BC và tiếp xúc với (I) cắt AB,AC lần lượt tại E,F. K là trung điểm EF.CMR: KD tiếp xúc với đường tròn (I) .
2. Cho tam giác ABC có phân giác AD (D thuộc BC). (O_1) và (O_2) là 2 đường tròn ngoại tiếp tam giác ABD và ACD. AD cắt 2 tiếp tuyến chung của (O_1) và (O_2) tại P,Q.CMR: PQ^2=AB.AC.
3. Cho (O). 6 đường tròn nhỏ ngoài nhau và ở miền trong của (O) lần lượt tiếp xúc với (O). Đường tròn Oi tiếp xúc với (O) tại Ai. Và đường tròn (Ok) tiếp xúc với đường tròn (O(k+1)) (qui ước(O7)=(O1)). CMR: A1A4,A2A5,A3A6 đồng qui

1. Let be given a non-isoscles triangle ABC. The incircle I touch AB, AC at M, N. MN intersect BC at D. The tangent // and touch (I) intersect AB, AC at E, F, respectively. K is the midpoint of EF. Prove that KD is a tangent of (I).
2. Given triangle ABC, AD is angle-bisector. (O_1) and (O_2) are scribed-circle ABD and ACD, respectively. AD intersect common tangent of (O_1) and (O_2) at P, Q. Prove that: PQ^2= AB.AC
3.Given (O). 6 small-circles non-intersect are touch with (O),. (Oi) touch (O) at Ai, and (Ok) touch (Ok+1). Prove that A1A4,A2A5,A3A6 intersect at common point.!@!
Do you agree??? :lol:

I haven't learn these things (both Vietnamese and English) yet so I probably don't understand what you guys are talking about but I know that your translation has some punctuation problem.

1. Let ABC is an isosceles triangle. The incircle I touches AB and AC at M, N. MN intersects BC at D. The parallel tangent and touch (I) intersect AB, AC at E, F. K is the midpoint of EF. Prove that KD is a tangent of (I).

This sentence sounds awkward and I don't know how to change it. "The tangent // and touch (I) intersect AB, AC at E, F, respectively." :noo:

2. Given triangle ABC and AD is the angle bisector of angle A. (O_1) and (O_2) are scribed-circle ABD and ACD. AD intersects common tangent of (O_1) and (O_2) at P, Q. Prove that: PQ^2= AB.AC

3.Given (O). 6 small non-intersect circles are touch with (O),. (Oi) touch (O) at Ai, and (Ok) touch (Ok+1). Prove that A1A4,A2A5,A3A6 intersect at common point

This one souds weird to me too but I don't know how to fix it. "6 small-circles non-intersect are touch with (O),. (Oi) touch (O) at Ai, and (Ok) touch (Ok+1)." :wink:

Besides, the word "touch" you are using here is misused. I think it has to be another word.