function
integration
logarithms
calculus
derivative
complex analysis
complex analysis unsolved

First, read [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?t=87025]this topic.[/url]

Claim: Suppose $G$ is a convex open region in $\mathbb{C}$ and $f$ is analytic on $G.$ Suppose that there exists $w\ne0$ such that for all $z\in G,\ |f'(z)-w|<|w|.$ Then $f$ is injective from $G$ to $\mathbb{C}.$

The proof of this claim is essentially my post in the other topic, with slight modifications. That's not my question. Here's my challenge, which is aimed at limiting the claim:

Find an example of a connected open region $G$ (not convex, of course) and an analytic function $f$ on $G$ such that $\exists\,w\ne0$ such that  $|f'(z)-w|<|w|\ \forall\,z\in G$ but $f$ is not injective.

Here is a counterexample:
\[ f(z) = \frac12 (z + 1/z).  \]
\[ f(i) = f(-i) = 0.  \]
The region is a small neighborhood of the path $i \to 2i \to 2i + 2 \to -2i + 2 \to -2i \to -i$.

BTW your condition gives the following condition on $f'$ in the limit (as $w$ tends to infinity):

(*) There exists $h$ s.t. $\mathrm{Re}\ hf'(z) > 0$ for all $z\in G$ (i.e. $f'(z)$ belongs to a halfplane).

This condition is slightly more general than yours. It is sufficient since for every $z_1 \neq z_2$ we have
\[ f(z_2) - f(z_1) = \int_{z_1}^{z_2} f'(z) dz.  \]
Therefore,
\[ \mathrm{Re} (h (f(z_2) - f(z_1))/(z_2 - z_1) ) > 0.  \]
Therefore, $f(z_2) \neq f(z_1)$.

The half-plane condition is a little weaker than the one I started with, so it's a little easier to find examples - but clearly, we are talking about essentially the same things. And there's room for more than one example.

Here's my example. Let $G$ be the open unit ball but not the nonpositive real axis. Choosing the principal branch of the logarithm on this region, define $f(z)=\frac34z^{\frac43}.$

The images of points near the negative real axis are duplicated on both sides, so the function is not injective.

The derivative is $f'(z)=z^{\frac13}.$ The image of this is a $120^{\circ}$ wedge of the unit circle. We can see that $|f'(z)-1|<1$ on $G.$

[quote="Kent Merryfield"]The half-plane condition is a little weaker than the one I started with, so it's a little easier to find examples - but clearly, we are talking about essentially the same things. And there's room for more than one example.
[/quote]

My example satisfies your condition. :) In fact, any example that satisfies the half-plane condition satisfies your condition, if we take a small neighborhood of a path connecting two points $z_1$ and $z_2$ (where $f(z_1) = f(z_2)$). Indeed, consider a counterexample to the half-plane condition: a function $f$ and a region $G$. Let $f(z_1) = f(z_2)$. Take a neighborhood $N$ of a path from $z_1$ to $z_$ s.t. the boundary of $N$ lies in $G$. Then the set ${f'(z) : z\in N}$ lies in a half-plane, and is bounded away from its boundary line. Therefore, some circle touching the origin covers it (${f'(z) : z\in N}$). We conclude that your condition is satisfied.


Let a,b,c be the sides of an acute angle triangle. Prove that

 \sqrt (-a^2+b^2+c^2) \sqrt (a^2-b^2+c^2)  +

  \sqrt (a^2-b^2+c^2) \sqrt (a^2+b^2-c^2) +

  \sqrt (a^2+b^2-c^2) \sqrt (-a^2+b^2+c^2)  \leq ab+bc+ca

This beautiful inequality is an exciting challenge for all mathematicians
who love inequalities. Good work!!

b 2 +c 2 -a 2 =2bc cosA then by Koshy-Bunyakovski inequality left side \leq 2(c^2ab+a^2bc+b^2ac)^{1/2} (cos A cos B+cos B cos C+cos C cos A)^{1/2}\leq 2(c^2ab+a^2bc+b^2ac)^{1/2}(3/4)^1/2=(3(c^2ab+a^2bc+b^2ac))^{1/2}\leq ((ab+bc+ca)^2)^{1/2}=ab+bc+ca.

An other solution for this big, but cude, inequlity.

Let's first prove the inequality :

(*) \sqrt ab + \sqrt cd <= \sqrt (a+c)(b+d)

(*) is equivalent to (**) \sqrt a/a+c b/b+d + \sqrt c/b+d d/a+c <= 1

Let 0 < x,y < \pi/2 such as sinx = a/a+c, siny = b/b+d

Then (**) is : sinx siny + cosx cosy = sin(x-y) <= 1 which is true.

Back to the original inequality, let S be the LHS.

Then it suffices to apply (*) to the 6 terms of 2*S to get the wanted inequality  :D

CRUX : Does it mean that it comes from the Crux magazine ?

Yes, CRUX means that it comes from the mathematical review CRUX MATHEMATICORUM.

Fedor Bakharev, I don't remember how to prove that

 cosAcosB + cosBcosC + cosCcosA  \leq 3/4

Have you a simple solution. Thank you.

Hello.
I can prove this inequality as follows.

2(cos A cos B+cos B cos C+cos C cos A)= (cos A + cos B +cos C)^2 -
(cos^2 A + cos^2 B + cos^2 C). Now the inequality follows from next two.

1) cos A + cos B + cos C = (R+r)/R\leq 3/2

2) cos^2 A + cos^2 B + cos^2 C = 1 - 2cos A cos B cos C.

cos A cos B cos C= sin (\pi-2A)/2 sin (\pi-2B)/2 sin (\pi-2C)/2

A'=(\pi-2A)/2,B'=(\pi-2B)/2,C'=(\pi-2C)/2 --- three angles of some triangle.

Then sin (A'/2) sin (B'/2) sin (C'/2)=r'/(4R')\leq 1/8.

Best regards, Fedor.

Hello,

I suggest the way to shorten the Fedor's solution:

cosAcosB + cosB.cosC + cosC.cosA  \leq (cosA+cosB+cosC) 2

Now, using the fact that  - 1 < cosA + cosB + cosC  \leq 3/2 we have disired inequality.

Namdung


counting
distinguishability
geometry
3D geometry
octahedron
rotation

Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)

[url]http://www.artofproblemsolving.com/Wiki/index.php/2000_AMC_12_Problems/Problem_25[/url]

Is this explaination plausible?
[hide]There are in total of $ 8!$ ways of placing the colors counting repetition.

Picking any corner, each figure can rotate 4 times. There are a total of 6 corners, so in total 24 repetition for each distinguishable way

$ 8!/24\equal{}1680$
[/hide]

Yeah, it works. It's a little funny how that solution isn't on the wiki...it looks like the quickest way.  :huh:


geometry
3D geometry
function
ratio
pyramid
rectangle
sphere

3 - 2 - 1 - 1 + 2 + 3 + 5 + 8

BISMILLAH

By Poga Humayun Dundiwala

Heavy heart and weight of gratitude
Counting the blessing with extravagant attitude
Number of blessing and one too many
The IMAM of want and chant and Psalm of content phony

Heavy heart and weight of gratitude
Counting the cost under apparent magnitude
Number of blessing and needful suffice
Weight of beholden and audit office

Poga Say's in a fooligton maze

Weight of a being and measure of existence
A Eternal machine refueling with instance
A overflowing fullness that remain empty
Occasional outrage and permanent serenity

Heavy heart and weight of gratitude
Counting the beads with blessed blind attitude
Number of chant and want of man
Lightness of mark and weight of the pen

Mr Barzakh Fitrath Ullah: The dark square niche with glowing cube crystal
The dark square footstool with the glowing circle
The four legged dark KURSI and the eight headed glowing ARSH
The first dimensional lines of the audible pen and the second dimensional planes of the visual brush
These are the silent blessing of watery niche with the sounding clay crystal
Square up of all SAWAB with rolling down KHOAB circle
These are payment of silent blessing with the grateful AWAZ
These are payment to silent water from vibrating QUARTZ

Mrs Be Aql Khan Usta : This life is not some unreal Khawab
But this life is the place to gain the real Sawab
This life is where
I will my awareness to submit in sincere
To will of GOD
This life is where i sing the DURUD OF MUHAMMAD
Sallel La Hu Alahi Wa Sallaim
This life is blessing this life where we drink the living breathing Tasnim

Mr Barzakh Fitrath Ullah : ALLAH is manifest ALLAH is hidden and he have no partner
Therefore this life is not real nor unreal but reflectional SANSAR
Yet this life not the blind mutation of two
But the clear water with the NURANI view
This life is like this living breathing vibration
The WORD of GOD that occupies the whole without GOD being present in part portion or section

Mrs Be Aql Khan Usta : The best business is religious SAWDAH
The best of innovation is religious BIDAH
It is the business where profits multiplies by SUNNAH sanction
It is the TIJARAH where SAWAB multiplies by Fibonacci function
Therefore count this blessing where one is many
It is one present JULM where punishment multiplies without losing its previous tyranny

Barzakh : He gave Fibbonacci formula in the HADITH of BIDAH E HASANAH and the sequence in the DEEN of ISLAM
In the First KALAMA of ALLAH chose from his infinite names and infinite prophets he chose ALLAH 1 and MUHAMMAD 1 SALLEL LA HU ALAHI WA SALLAM
Now he who chooses the first sequence 1 + 1 may continue toward the two
With the QURAN and SUNNAH by his body and RUH 1 + 1 = 2
Then toward the third run
With IMAN AMAL and EHSAAN 1 + 1 + 1 = 3
He continues to ascend toward the infinite area
He continues to grow By five KALAMA E SHAHADA and eight AHKAM E SHARRIAH 5 + 8 = 1 + 3 + 4 + 7 + 1 + 1 + 2 + 3 + 5
This is the ASL asset of Islamic Bidathi innovation
And the inheritance SAWAB of its NAQL repetition
This is the real ISLAMIC enigmatic BIDAH and the ASL concept of ISLAMIC AWL
The RUHANI solution to all mental BE AQL

Be Aql : Prophet MUHAMMAD SALLEL LA HU ALAHI WA SALLAM said i warn you of the newly invented matter
And every newly invented matter is an innovation of its creator
And every innovation is misguidance
And every misguidance is in the hell fire residence
According to HADITH An Nasaee
All Ulema Say's this HADITH is Kullu shaee
Now doe's all newly invented matters leads to hell
Ancient water and modern well
I don't know that but i know this
When you neglect 13 Arkan and Ahkam and 21 SALAH SUNNAH your AHZAB increases while your SAWAB decrease 13 + 21

Barzakh : Now who else but you to tell me about 13 SALATH E MUSTAHAB and 21 SALATH SUNNAH SAWAB 13 + 21 = 34
Who else but you to tell me about 13 SALATH E MUSTAHAB and 21 SALATH E SUNNAH what can nullify SALATH by 21 acts MOQRUH AHZAB 13 + 21 + 21 = 55
Who else but you to tell me about 40 SUNNATH E MUHAQADA and 44 SUNNAH E GAIR E MUHAQADA in our 5 daily SALAH 40 + 44 + 5 = 89
Who else but you to tell me about SUNNAH OF MUHAMMAD UR RASUL ALLAH
Yes i agree i know very little about AL QURAN and AL HADIS
But i know lot about every other which what that and this
I know he is under ALLAH but above AL QURAN
I know he is the spiral stair case through what AL QURAN descended to the creation by Fibonacci function
I know he is above AL QURAN but under SUNNAH
I know he is the highest Father of every lowly Munnah
I know he is under ALLAH but above AL QURAN
I know he is above AL FURQAN but under its FORMAN 8 + 13 + 21 + 34 + 55 + 89

General | Scientific Miracles in the Quran

--------------------------------------------------------------------------------

Fibonacci Numbers: A Measure of Beauty

By Harun Yahya
 March 02, 2005

The Golden Number is not mathematical imagination, but a natural principle related to the laws of equilibrium
[Allah has appointed a measure for all things.] (Qur’an, 65: 3)

The Wikipedia encyclopaedia describes beauty as ‘the phenomenon of the experience of pleasure, through the perception of balance’. Everyone admires beauty in nature and the unique balance found in it. Some say that this balance and perception of beauty is due to the Golden number or the ratio that gives certain things their exquisitness.

If a pleasing or exceedingly balanced form is achieved in terms of elements of application or function, it is there that we may look for a function of the Golden Number. The Golden Number is a product not of mathematical imagination, but of a natural principle related to the laws of equilibrium.[1]

What do the pyramids in Egypt, Leonardo da Vinci’s portrait of the Mona Lisa, sunflowers, the snail, the pine cone and your fingers all have in common?

The answer to this question lies hidden in a sequence of numbers discovered by the Italian mathematician Fibonacci. The characteristic of these numbers, known as the Fibonacci numbers, is that each one consists of the sum of the two numbers before it.[2]

Fibonacci numbers

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, …

Fibonacci numbers have an interesting property. When you divide one number in the sequence by the number before it, you obtain numbers very close to one another. In fact, this number is fixed after the 13th number in the series. This number is known as the "golden ratio."

GOLDEN RATIO = 1.618

233 / 144 = 1.618

377 / 233 = 1.618

610 / 377 = 1.618

987 / 610 = 1.618

1597 / 987 = 1.618

2584 / 1597 = 1.618

The Golden Ratio

When conducting their researches or setting out their products, artists, scientists and designers take the human body, the proportions of which are set out according to the golden ratio, as their measure. Leonardo da Vinci and Le Corbusier took the human body, proportioned according to the golden ratio, as their measure when producing their designs. The human body, proportioned according to the golden ratio, is taken as the basis also in the Neufert, one of the most important reference books of modern-day architects.

The Golden Ratio in the Human Body

Click to Enlarge

The "ideal" proportional relations that are suggested as existing among various parts of the average human body and that approximately meet the golden ratio values can be set out in a general plan as follows:[3]

The M/m level (shown in the table to the right) is always equivalent to the golden ratio. M/m = 1.618

The first example of the golden ratio in the average human body is that when the distance between the navel and the foot is taken as 1 unit, the height of a human being is equivalent to 1.618. Some other golden proportions in the average human body are:

The distance between the finger tip and the elbow / distance between the wrist and the elbow

The distance between the shoulder line and the top of the head / head length

The distance between the navel and the top of the head / the distance between the shoulder line and the top of the head

The distance between the navel and knee / distance between the knee and the end of the foot

The Human Hand

Lift your hand from the computer mouse and look at the shape of your index finger. You will in all likelihood witness a golden proportion there.

Our fingers have three sections. The proportion of the first two to the full length of the finger gives the golden ratio (with the exception of the thumbs). You can also see that the proportion of the middle finger to the little finger is also a golden ratio.[4]

You have two hands, and the fingers on them consist of three sections. There are five fingers on each hand, and only eight of these are articulated according to the golden number: 2, 3, 5, and 8 fit the Fibonacci numbers.

The Golden Ratio in the Human Face

There are several golden ratios in the human face. Do not pick up a ruler and try to measure people’s faces, however, because this refers to the "ideal human face" determined by scientists and artists.

For example, the total width of the two front teeth in the upper jaw over their height gives a golden ratio. The width of the first tooth from the centre to the second tooth also yields a golden ratio. These are the ideal proportions that a dentist may consider. Some other golden ratios in the human face are:

Length of face / width of face

Distance between the lips and where the eyebrows meet / length of nose

Length of face / distance between tip of jaw and where the eyebrows meet

Length of mouth / width of nose

Width of nose / distance between nostrils

Distance between pupils / distance between eyebrows

Golden Proportion in the Lungs

In a study carried out between 1985 and 1987,[5] the American physicist B. J. West and Dr. A. L. Goldberger revealed the existence of the golden ratio in the structure of the lung. One feature of the network of the bronchi that constitutes the lung is that it is asymmetric. For example, the windpipe divides into two main bronchi, one long (the left) and the other short (the right). This asymmetrical division continues into the subsequent subdivisions of the bronchi.[6] It was determined that in all these divisions the proportion of the short bronchus to the long was always 1/1.618.

The Golden Rectangle and the Design in the Spiral

A rectangle, the proportion of whose sides is equal to the golden ratio is known as a "golden rectangle"
A rectangle, the proportion of whose sides is equal to the golden ratio is known as a "golden rectangle." A rectangle whose sides are 1.618 and 1 unit long is a golden rectangle. Let us assume a square drawn along the length of the short side of this rectangle and draw a quarter circle between two corners of the square. Then, let us draw a square and a quarter circle on the remaining side and do this for all the remaining rectangles in the main rectangle. When you do this you will end up with a spiral.

The British aesthetician William Charlton explains the way that people find the spiral pleasing and have been using it for thousands of years stating that we find spirals pleasing because we are easily able to visually follow them.[7]

The spirals based on the golden ratio contain the most incomparable designs you can find in nature. Examples we can give of this are the spiral sequences on the sunflower and the pine cone.

The Design in Sea Shells

When investigating the shells of mollusks, which live at the bottom of the sea, the form and the structure of the internal and external surfaces of the shells attracted scientists’ attention.

The internal surface is smooth, while the outside surface is fluted. The mollusk’s body is inside the shell. The outside edges of the shell augment its rigidity and, thus, increase its strength. The shell’s form is astonishing in its perfection highlighting the beauty of its creation. The spiral idea in shells is expressed in the perfect geometrical form, in a surprisingly beautiful, "sharpened" design.[8]

The shells of most mollusks grow in a logarithmic spiral manner. There can be no doubt, of course, that these animals are unaware of even the simplest mathematical calculation, let alone logarithmic spirals. So how is it that the creatures in question can know that this is the best way for them to grow? How do these animals, that some scientists describe as "primitive," know that this is the ideal form for them? It is impossible for growth of this kind to take place in the absence of a consciousness or intellect. That consciousness exists neither in mollusks nor, despite what some scientists would claim, in nature itself. It is totally irrational to seek to account for such a thing in terms of chance. This design can only be the product of the Almighty Allah.

An example of perfect geometry
Growth of this kind was described as "gnomic growth" by the biologist Sir D’Arcy Thompson, an expert on the subject, who stated that it was impossible to imagine a simpler system, during the growth of a seashell, than which was based on widening and extension in line with identical and unchanging proportions. As he pointed out, the shell constantly grows, but its shape remains the same.[9]

One can see one of the best examples of this type of growth in a nautilus, just a few centimetres in diameter. C. Morrison describes this growth process, which is exceptionally difficult to plan even with human intelligence, stating that along the nautilus shell, an internal spiral extends consisting of a number of chambers with mother-of-pearl (calcium carbonate and conchiolin secreted by the mantle of mollusks) lined walls. As the animal grows, it builds another chamber at the mouth of the spiral shell  larger than the one before it, and moves forward into this larger area by closing the door behind it with a layer of mother-of-pearl.[10]

Growth in a spiral form in the animal world is not restricted to the shells of mollusks. Animals such as antelopes, goats and rams complete their horn development in spiral forms based on the golden ratio.[11]

The Golden Ratio in the Hearing and Balance Organ

The cochlea in the human inner ear serves to transmit sound vibrations. This bony structure, filled with fluid, has a logarithmic spiral shape with a fixed angle of α=73°43´ containing the golden ratio.

The Golden Ratio in DNA

The molecule in which all the physical features of living things are stored, too, has been created in a form based on the golden ratio. The DNA molecule, the very program of life, is based on the golden ratio. DNA consists of two intertwined perpendicular helixes. The length of the curve in each of these helixes is 34 angstroms and the width 21 angstroms. (1 angstrom is one hundred millionth of a centimetre.) 21 and 34 are two consecutive Fibonacci numbers.

The Golden Ratio in Snow Crystals

The golden ratio also manifests itself in crystal structures. Most of these are in structures too minute to be seen with the naked eye. Yet you can see the golden ratio in snow flakes. The various long and short variations and protrusions that comprise the snow flake all yield the golden ratio.[12]

The Golden Ratio in Space

In the universe there are many spiral galaxies containing the golden ratio in their structures.

The Golden Ratio in Physics

You encounter Fibonacci series and the golden ratio in fields that fall under the sphere of physics. When a light is held over two contiguous layers of glass, one part of that light passes through, one part is absorbed, and the rest is reflected. What happens is a "multiple reflection." The number of paths taken by the ray inside the glass before it emerges again depends on the number of reflections it is subjected to. In conclusion, when we determine the number of rays that re-emerge, we find that they are compatible with the Fibonacci numbers.[13]

The fact that a great many unconnected animate or inanimate structures in nature are shaped according to a specific mathematical formula is one of the clearest proofs that these have been specially designed. The golden ratio is an aesthetic rule well known and applied by artists. Works of art based on that ratio represent aesthetic perfection. Plants, galaxies, micro-organisms, crystals and living things designed according to this rule imitated by artists are all examples of Allah’s superior artistry. Allah reveals in the Qur’an that He has created all things with a measure. Some of these verses read:

[Allah has appointed a measure for all things.] (Qur’an, 65: 3)

[Everything has its measure with Him.] (Qur’an, 13: 8)

--------------------------------------------------------------------------------

**The author, who writes under the pen-name Harun Yahya, has published many books on political, faith-related and scientific issues. Some of the books of the author have been translated into English, German, French, Spanish, Italian, Portuguese, Albanian, Arabic, Polish, Russian, Bosnian, Indonesian, Turkish, Tatar, Urdu and Malay and published in the countries concerned. Visit his website at http://www.harunyahya.com  or contact him

[1] Mehmet Suat Bergil, Doğada/Bilimde/Sanatta, Altın Oran (The Golden Ratio in Nature/Science/Art), Arkeoloji ve Sanat Yayinlari, 2nd Edition, 1993, p. 155.

[2] Guy Murchie, The Seven Mysteries of Life, First Mariner Boks, New York, pp. 58-59.

[3] J. Cumming, Nucleus: Architecture and Building Construction, Longman, 1985.

[4] Mehmet Suat Bergil, Doğada/Bilimde/Sanatta, Altın Oran (The Golden Ratio in Nature/Science/Art), Arkeoloji ve Sanat Yayinlari, 2nd Edition, 1993, p. 87.

[5] A. L. Goldberger, et al., "Bronchial Asymmetry and Fibonacci Scaling." Experientia, 41 : 1537, 1985.

[6] E. R. Weibel, Morphometry of the Human Lung, Academic Press, 1963.

[7] William Charlton, Aesthetics: An Introduction, Hutchinson University Library, London, 1970.

[8] "The ‘Golden’ spirals and ‘pentagonal’ symmetry in the alive Nature," online at:

[9] D’Arcy Wentworth Thompson, On Growth and Form, C.U.P., Cambridge, 1961.

[10] C. Morrison, Along The Track, Withcombe and Tombs, Melbourne.

[11] "The ‘Golden’ spirals and ‘pentagonal’ symmetry in the alive Nature," online at

[12] Emre Becer, "Biçimsel Uyumun Matematiksel Kuralı Olarak, Altın Oran" (The Golden Ratio as a Mathematical Rule of Formal Harmony), Bilim ve Teknik Dergisi (Journal of Science and Technology), January 1991, p.16.

[13] V.E. Hoggatt, Jr. and Bicknell-Johnson, Fibonacci Quartley, 17:118, 1979.

The works posted on this page reflect solely the opinions of the authors.

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                             Prayer (Salat),
                According to Five Islamic Schools of Law

                                (Part I)

                 By: 'Allamah Muhammad Jawad Maghniyyah
              Translated from the Arabic by Mujahid Husayn

Salat (prayer) is either obligatory (wajib) or supererogatory (mandub).
The most important of prayers are the obligatory prayers performed daily
five times, and there is consensus among Muslims that a person who denies
or doubts their wujub is not a Muslim, even if he recites the shahadah,
for these prayers are among the 'pillars' (arkan) of Islam. They are the
established necessity of the faith (al-Din) that doesn't need any
ijtihad or study, taqlid.

The Daily Supererogatory Prayers (Rawatib):

    Supererogatory prayers are of various kinds, and among them are those
which are performed along with the obligatory daily prayers (fara'id). The
schools differ regarding the number of their rak'ahs. The Shafi'is consider
them to be eleven rak'ahs: two before the morning (subh) prayer, two before
the noon (zuhr) prayer and two after it, two after the sunset (maghrib)
prayer,
two after the night ('isha') prayer and a single rak'ah called 'al-watirah'

     The Hanbalis consider them to be ten rak'ahs; two rak'ahs before and
after the noon prayer, two after the sunset and The night prayer, and two
rak'ahs before The morning prayer.

     According to the Malikis there is no fixed number for the
supererogatory (nawafil) prayers performed with the obligatory salat,
Though it is best to offer four rak'ahs before the zuhr and six after the
maghrib prayer.

     The Hanafis classify the nawafil performed along with the fara'id
into 'masnunah' and 'mandubah'.(1)   The 'masnunah' are five: two rak'ahs
before the subh; four before the zuhr, and two after it, except on Friday;
two after the maghrib and two after the 'isha' prayer.
The 'mandubah' are four: four -or two- rak'ahs before the 'asr, six
after the maghrib, and four before and after the 'isha' prayer.

     The Imamis observe:  The rawatib are 34 rak'ahs: eight before the
zuhr, eight before the 'asr, four after the maghrib, two after the 'isha'
(recited while sitting and counted as a single rak'ah; it is called 'al
watirah'), eight rak'ahs of the midnight prayer (salat al-layl), two
rak'ahs of al-shaf', a single of al-watr,(2)  and two rak'ahs before the
morning prayer, called 'salat al-fajr'.

The time of Zuhr and 'Asr Prayers:

    The fuqaha' begin with salat al-zuhr, because it was the first salat
to be declared obligatory, followed by the 'asr, the maghrib, the 'isha'
and the subh prayer, in that order. All the five prayers were made
obligatory on the night of Prophet's cosmic journey (al 'Isra'),
nine years after the beginning of his mission (bi'thah). Those who hold
this opinion cite as proof verse 78 of the Surat al-'Isra' which
stipulates all the five prayers:

   " Perform salat from the declining of the sun to the darkening of the
night and the recital of the dawn; surely the recital of the dawn is
witnessed."  (17:78)

    The schools concur that salat is not valid if performed before its
appointed time and that the time of the zuhr prayer sets in when the sun
passes the meridian. They differ concerning its duration.

     The Imamis say: The specific period of the zuhr prayer extends from
the moment the sun crosses the meridian up to a period required to perform
it, and the specific period of the 'asr prayer is the duration required to
perform it just before sunset. The time between these two specific periods
is the common period for the two salats.  This is the reason they consider
it valid to perform both the prayers successively during their common
period.(3) But if the time remaining for the end of the day is sufficient
only for performing the zuhr prayer, the 'asr prayer will be offered first
with the niyyah of ada' and later the zuhr prayer will be performed as
qada'.

     The four Sunni schools observe: The time of the zuhr prayer begins
when the sun crosses the meridian and continues till the shadow of an
object becomes as long as its height; and when the length of the shadow
exceeds the height of the object, the time for the zuhr prayer comes to an
end. Here the Shafi'is and the Malikis add: These limits are for an
unconstrained person (mukhtar), and for one who is constrained (mudarr),
the time for zuhr prayer extends even after an object's shadow equals its
height. The Imamis consider the time when an object's shadow equals its
height as the end of the time of fadilah (honor) for the zuhr, and when it
equals twice the height of the object as the time of fadilah for the 'asr
prayer.

     The Hanafis and the Shafi'is state: The time of 'asr prayer begins
when the length of an object's shadow exceeds its height and continues up
to sunset.

     The Malikis say: For the 'asr prayer there are two times, the first
for ordinary circumstances and the second for exigencies. The former
begins with an object's shadow exceeding its height and lasts until the
sun turns pale. The latter begins from when the sun turns pale and
continues until sunset.

     The Hanbalis observe: One who delays offering the 'asr prayer till
after an object's shadow exceeds twice its height, his salat will be
considered ada' if performed before sunset, though he will have sinned
because it is haram to delay it until this time. They are alone in all the
schools in holding this opinion.

The Time of Maghrib and 'Isha' Prayers:

     The Shafi'i and the Hanbali schools (in accordance with the view of
their respective Imams) state: The time for the maghrib prayer begins when
the sun sets and ends when there radish after glow on the western horizon
vanishes.

     The Malikis say: The duration for the maghrib prayer is narrow
and confined to the time required after sunset to perform the maghrib
prayer along with its preliminaries of taharah and adhan, and it is not
permissible to delay it voluntarily. But in an emergency, the time for the
maghrib prayer extends until dawn. The Malikis are alone in considering it
impermissible to delay the maghrib prayer beyond its initial time.

    The Imamis observe: The period specific to the maghrib prayer extends
from sunset(4) for a duration required to perform it, and the specific
period of the 'isha' prayer is the duration required to finish it before
midnight. The time between these two specific periods is the common time
for both maghrib and 'isha' prayers. Hence they allow the joint
performance of these two salats during this common time.

    That was with respect to someone who is in a position to act out of
free choice (mukhtar), but as to a person constrained by sleep or
forgetfulness, the time for these two salats extend until dawn, with the
period specific for the 'isha' prayer becoming the time required to
perform it just before dawn and the specific period for the maghrib prayer
becoming the time required to perform it just after midnight.

The Time of subh Prayer:

     There is consensus among The schools, with the exception of the
Maliki, that the time for the morning prayer begins at day-break (al fajr
al-sadiq) and lasts until sunrise. The Malikis say: The subh prayer has
two times: for one in a position to act out of free choice it begins with
daybreak and lasts until there is enough twilight for faces to be
recognized; for one in constrained circumstances it begins from the time
when faces are recognizable and continues up to sunrise.
____________
NOTE: THE NUMBERS FOLLOWING SOME TERMS ARE MARKS FOR FOOTNOTES THAT WILL
BE APPENDED IN SHAA ALLAH IN THE NEXT PART.
**************************************************************************
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               |  o_,_7 _||   . _o_7 _||  4_|_||   o_w_,  |
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                             Prayer (Salat),
                According to Five Islamic Schools of Law

                               (Part II)

                 By: 'Allamah Muhammad Jawad Maghniyyah

Ignorance of the Qiblah (direction):

  It is wajib for a person ignorant of the qiblah to inquire and
strive to determine its exact or approximate direction, and in case
neither of the two is possible, the four Sunni schools and a group from
among the Imamis say: He may perform salat in any direction; his salat
will be valid and it will not be wajib for him to repeat it except in the
opinion of the Shafi'is. Most Imamis observe: He will perform Salat in
four directions to comply with the command for salat and to ascertain its
proper performance.  But if there isn't sufficient time for performing
salat four times or if one is incapable of performing it in four
directions, he may perform, salat in the directions that he can.

A Subsidiary Issue:

  If a person prays not facing the qiblah and comes to know about
his mistake, the Imamis state: If the error is known during the salat and
the correct qiblah lies between his two hands, the part of the salat
already performed will be valid and he will have to correct his direction
for the remaining part of the salat. But if it is known that he has been
praying facing the right or the left (90 degree off the direction) or his
back towards the qiblah (180 degree off the direction), the salat will be
invalid and he will perform it anew.   If the error is known after
performing the salat, it should be performed again if its time
is still there, not otherwise. Some Imamis say: The salat will not be
repeated if there is only a little deviation from the qiblah, irrespective
of whether its time is still there or not. But if it has been performed
facing the right or the left (90 degree off), it should be repeated if its
time is there, not otherwise. If the salat is performed with one's back to
the qiblah (180 degree off), it should be repeated regardless of whether
its time is still there or has passed

  The Hanafis and the Hanbalis observe: If after inquiring and
striving to find the qiblah one is unable to ascertain its approximate
direction and performs salat in a direction which turns out to be wrong,
he must change his direction accordingly if the mistake is known during
the salat, and if it is known afterwards his salat is valid and he has no
further obligation.

  The Shafi'is say: If it becomes certain that there has been a
mistake in determining the qiblah, it is wajib to repeat the salat, but if
there is only a likelihood of mistake, the salat is valid Irrespective of
whether the probability arises during the salat or after it.

  As to one who neither makes an inquiry nor an effort to determine
the qiblah, but by chance performs the salat in the right direction, the
Malikis and Hanbalis consider his salat to be invalid. The opinion of the
Imamis and the Hanafis is that his salat is valid provided he has no
doubts while praying and was sure about the direction of the qiblah at the
time of starting the salat, because, as pointed out by the Imamis, in such
a situation it is correct for him to make the niyyah of acquiring nearness
(qurbah) to God.

________________
The Numbers that follow some of the words
refer to notes that will appended in the coming issues in shaa Allah.
                __________________________________________
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               |         ||           ||    | ||          |
               |  o_,_7 _||   . _o_7 _||  4_|_||   o_w_,  |
               | ( :   /     (_)    /             (   .   |
               |__________________________________________|

                             Prayer (Salat),
                According to Five Islamic Schools of Law

                               (Part III)

                 By: 'Allamah Muhammad Jawad Maghniyyah

Wajib Covering During Salat:

     The schools concur that it is wajib (necessary) upon both men and
women
to cover those parts of their bodies during salat which should ordinarily
be
kept covered before 'strangers'. Beyond that their positions differ. Is it
wajib for a woman to cover, fully or partly, her face and hands during
salat, although she is not required to do so outside salat? Is it wajib for
a man to cover other parts of his body during salat apart from the area
between the navel and the knees, though it is not wajib to do so outside
salat?

     The Hanafis observe: It is wajib upon a woman to cover the back
of her hands and the soles of her feet as well, and upon a man to cover
his knees in addition to the area between the navel and the knees.
The Shafi'is and Malikis say: It is permissible for a woman to keep
her face and both the palms and the back of her hands uncovered
during salat.

    The Hanbalis state: It is not permissible for her to expose any
part except the face.

    The Imamis observe: It is wajib for both men and women to cover only
those parts of their body during salat which they are supposed to cover
ordinarily in the presence of a 'stranger'. Hence it is permissible for a
woman to expose during salat that part of her face which is washed
during wudu'; her hands up to the wrists, and her feet up to the ankles
both the back as well as the palms of hands and the soles of feet. For
a man, it is wajib to cover the rear and the private parts, though better
to cover the entire area between the navel and the knees.

The Material Used for Covering During Salat:

    The covering should meet the following requirements where the
ability and freedom to meet them exist:

1. Taharah: The purity of The covering and the body are necessary for
the validity of salat in the opinion of all the schools, although each of
them concedes certain exceptions in accordance with the following
details:

    The Imamis state: Blood from wounds and sores, irrespective of
its quantity, is considered excusable on The dress as well as the body if
its removal entails difficulty and harm (mashaqqah and haraj). A blood spot
smaller than the size of a dirham coin, regardless of its being due to
one's blood or that of someone else, is also excusable provided that: it is
in a single place and not in different places; it is not the blood of hayd,
nifas and istihadah; it is not the blood of anything intrinsically najis,
such as dog and pig, or the blood of a dead body. Also excusable is the
impurity of anything that does not constitute part of essential dress
during salat, e.g. a sash, cap, socks, shoes, ring, anklet and that which
one carries with oneself, e.g. knife or currency. The impurity of the dress
of a woman rearing a child, irrespective of whether she is the mother or
someone else, is exempted on condition that it be difficult for her to
change it and that she washes it once every day. In other words, in their
opinion every najasah on dress or body is exempted in conditions of
emergency (idtirar).

    The Malikis observe: Cases of uncontrolled discharge of urine or
excrement, as well as piles, are excusable; so is any impurity on the
body or clothes of a woman suckling an infant that may be soiled by
the infant's urine or feces. So also are exempted the body and clothes
of a butcher, surgeon and scavenger. Also exempted is: blood -even
that of a pig- if it is less than the size of a dirham coin: the discharge
from boils, the excrement of fleas, and other things which need not be
mentioned because they occur rarely.

    The Hanafis say: Najasah, blood or anything else, if less than the
size of a dirham coin is exempted. Also exempted in emergencies is the
urine and excrement of a cat and mouse. Tiny splashes- as small as the
point of a needle- of urine, the blood that unavoidably stains a butcher,
and the mud on roads- even if it is usually mixed with najasah and
provided the najasah itself is not visible- are exempted. Consequently,
they consider najasah in a small quantity as exempted, such as the urine
of an animal eating which is halal, if it covers a fourth of the clothes
and less than one-fourth of the body.

    According to the Shafi'is, every najasah which is in such a small
quantity that the eye cannot see it is exempted. So is the mud on roads
which is mixed with a small quantity of najasah, worms present in
fruits and cheese, najis liquids added in medicines and perfumes,
excrements of birds, najis hair in small quantity if they do not belong
to a dog or a pig, and other things as well which are mentioned in
detailed works.

        The Hanbalis say: Minute quantities of blood and pus are ex-
empted, and so is the mud on roads whose najasah is certain, as well as
the najasah that enters the eyes and washing which is harmful.

2. Wearing Silk: There is consensus among the schools that wearing silk
and gold is haram for men both during and outside salat, while it is
permissible for women. This is in accordance with this statement of the
Prophet (S):

Wearing silk and gold is unlawful for the men of my ummah, while it is
lawful for its women.

   Accordingly, the Imamis observe: A man's salat is not valid if he wears
pure silk and any clothing embroidered with gold during it, regardless
of whether it is a waistband, cap, socks, or even a gold ring. They allow
wearing silk during salat in times of illness and during war.

The Shafi'is state: If a man performs salat while wearing silk or over
something made of it, it will be considered a haram act, though his salat
will be valid (al-Nawawi, Sharh al-Muhadhdhab, iii, 179). I have not found
an express statement in the books of the remain- ing schools concerning the
validity or invalidity of salat performed in silk, though the Hanafis as
well as the Hanbalis (in accordance with one of two narrations) concur with
the Shafi'is regarding the general rule that if there is any command
prohibiting something which is not directly connected with salat such as
the command prohibiting usurpa- tion- the salat will be valid if it is not
observed and The person will be considered as having performed a wajib and
a haram act together. Accordingly the salat performed in a dress of silk is
valid. The author of al-fiqh 'ala al-madhahib al-'arba'ah reports a
consensus for the Sunni schools that that it is valid for a man constrained
to perform salat while wearing silk, and it is not wajib for him to repeat
it.

3. Lawfulness of the Clothing: The Imamis consider it necessary that
the clothing worn to be lawfully owned. Hence if a person performs salat
in usurped clothes with the knowledge of their being so, his salat is
invalid. This is also the opinion of Ibn Hanbal in one of the two state-
ments narrated from him.

     The other schools regard salat in usurped clothes as valid on the
grounds that the prohibition does not directly relate to salat so as to
invalidate it.

     The Imamiyyah are very strict concerning usurpation, and some
of them even observe: If a person performs salat in clothes in
which a single thread is usurped, or carries with him an usurped knife,
dirham, or any other thing, his salat will not be valid. But they also
say: If one performs salat in usurped clothes out of ignorance or
forgetfulness, his salat is valid.

4. The Skin of Uneatable Animals: The Imamis are alone in holding that
it is invalid to perform salat while wearing the skin (even if tanned)
of an animal whose flesh is not allowed to be eaten, as well as anything
consisting its hair, wool, fur or feathers. The same is true of clothes
bearing any secretion from its body- eg. sweat and saliva- as long as
it is wet. Hence, even if a single hair of a cat or any such animal happens
to be present on the dress of a person performing salat and if he performs
it with the knowledge of its presence, his salat is invalid.
They exclude wax, honey, the blood of bugs; lice, fleas and other
insects which have no flesh, as well as the hair, sweat and saliva of
human beings.

      They also consider salat invalid if any part of a dead animal
(maytah) happens to be on the clothes irrespective of whether the
animal is one used for food or not, whether its blood flows when cut
or not, and its skin is tanned or not.

  A Subsidiary Issue: If there is only a single clothing to cover the body
and that too is najis to an extent that is not excusable, what should
one do if he has no alternative other than either performing salat in the
najis clothing or in the state of nature?

    The Hanbalis say: He should perform salat in the najis clothing,
but it is wajib upon him to repeat it later.

    The Malikis and a large number of Imamis observe: He should
perform salat in the najis clothing and its repetition is not wajib upon
him.

    The Hanafis and the Shafi'is state: He should perform salat naked
and it is not valid for him to cover himself with the najis clothing.

The Place of Salat:

   An Usurped Place: The Imamis consider salat performed in an usurped
place and usurped clothing as invalid provided it is done voluntarily and
with the knowledge of the usurpation. The other schools observe: The
salat performed in an usurped place is valid, though the person per-
forming it will have sinned, since the prohibition does not relate directly
to salat; rather, it relates to dispensations (of property). Their position
in this regard is the same as in the case of usurped clothing. Furthermore
in the opinion of the four schools the Salat of  usurper himself is valid
in usurped property.

      The Imamis also consider as valid the salat of the true owner and
anyone whom he permits, and regard as invalid the salat of the usurper
and anyone whom the owner has not granted permission. The Imamis however
permit salat in vast stretches of (owned) land which are either impossible
or difficult for people to avoid, even if the permission of the owner has
not been acquired.

Taharah (purity) of the Place:  The four Sunni schools observe: The place
should be free from both wet and dry najasah (impurity). The Shafi'is
overdo by saing: The taharah of all that which touches and comes into
contact with the body or clothes of the performer is wajib. Therefore, if
he rubs himself against a najis wall or cloth or holds a najis object or a
rope laying over najasah, his salat will be invalid. The Hanafis require
only the location of the feet and the forehead to be tahir.   The Imamis
restrict it to the loca- tion of the forehead, i.e. the place of sajdah. As
to the najasah of other locations, the salat will not be invalid unless the
najasah is transmitted to the body or clothing of the performer (the person
performing salat).

Salat Performed on a Mount: The Hanafis and the Imamis require the
place to be stationary; hence it is not valid in their opinion to perform
salat while riding an animal or something that swings back and forth,
except out of necessity, because one who has no choice will perform
salat in accordance with his capacity.

     The Shafi'is, Malikis and Hanbalis observe: Salat performed on a
mount is valid even during times of peace and despite the ability to
perform it on the ground, provided it is performed completely and
meets all the requirements.

Salat Inside the Ka'bah: The Imamis, Shafi'is and Hanafis state: It is
valid to perform salat, faridah or nafilah, inside the Ka'bah.
The Malikis and the Hanbalis say: Only nafilah, not faridah, is
valid therein.

A Woman's Prayer Beside a Man: A group of Imami legists observe: If
a man and a woman perform salat in a single place so that she is either
in front of him or beside him, and there is neither any screen between
them nor does the distance between the two exceed 10 cubits, the
salat of the one who starts earlier will not be invalid, and if both star
simultaneously, the salat of both will be invalid.

The Hanafis say: If the woman is in front or beside a man, the
salat will be invalid if performed in a single place with no screen at
least
a cubit high between them, the woman has sex appeal, her shanks and
ankles are adjacent to his, the salat is not a funeral prayer, and the
salat is being jointly performed, i.e. either she is following him or both
are following a single imam.

The Shafi'is, the Hanbalis and most Imamis are of the view that
the salat is valid, though the manner of performance is makruh.

The Locale of Sajdah: The schools concur that the place where the
forehead is placed during prostration should be stationary and should
not be inordinately higher than the location of the knees (during
sajdah). They differ regarding that on which sajdah is valid.

The Imamis state: It is valid to perform sajdah only on earth and those
things which grow on it which are not used for food or clothing.
Therefore, a person cannot perform sajdah on wool, cotton, minerals
and that which grows on the surface of water, for water is not earth.
They permit sajdah on paper because it is made of a material
which grows on earth. They argue their position by pointing out that
sajdah is an 'ibadah (obedience) prescribed by the Shari'ah that depends
for
its particulars on textual evidence (nass). The legists of all the schools
concur regarding the validity of sajdah on earth and that which grows
on it, thus Imamis restrict it to that because there is certainty. They
offer as further evidence these traditions of the Prophet (S):

   The salat of any of you will not be valid unless he performs wudu' as in-
   structed by God and then performs sajdah by placing his forehead on the
   earth.

   The earth has been created a masjid (a place for performing sajdah) and
   a purifier .

   Khabbab says: "We complained to The Prophet (S) regarding the
   excessive heat of sun-baked ground on our foreheads, but he did not
   accept our complaint."

Had it been valid to perform sajdah on carpets, why would they have
complained?!   However Imamis permit sajdah on cotton and linen in the case
of emergency.

The four schools observe: It is valid to perform sajdah on anything,
including even a part of one's turban, provided it is tahir. Rather,
the Hanafis permit sajdah on one's palm even without an emergency,
though it is considered as makruh.

To be continued-insha'Allah

_________
FOOTNOTES:

  1. The Hanafis use two terms ('fard' and 'wajib') for something whose
performance is obligatory and whose omission is impermissible. Hence
they divide obligation into two kinds: fard and wajib. 'Fard 'is a duty
for which there is definite proof, such as Qur'anic text, mutawatir
sunnah, and ijma' (consensus).  'Wajib' is a duty for which there is a
Dhanni (non-definite) proof, such as qiyas (analogy) and khabar
al-wahid (isolated tradition). That whose performance is preferable to its
omission is also of two kinds: 'masnun' and 'mandub'. 'Masnun' is an act
which the Prophet (S) and the 'Rashidun' caliphs performed regularly, and
'mandub' is an act ordered by the Prophet (S) though not performed
regularly by him (S). That which it is wajib to avoid and whose
performance is not permissible is 'muharram' if it is established by a
definite proof. If based on a Dhanni proof, it is 'makruh', whose
performance is forbidden.

  2. According to the Hanafis, the salat al-watr consists of three rak'ahs
with a single salam. Its time extends from the disappearance of twilight
after sunset to dawn. The Hanbalis and Shafi'is say: At minimum it is one
rak'ah and at maximum eleven rak'ahs, and its time is after the 'isha'
prayer. The Malikis observe: It has only one rak'ah.

   3. There are among 'ulama' of the Sunni schools those who agree with
the Imamis on performing the two salats together even when one is not
travelling. al-Shaykh Ahmad al-Siddiq al-Ghumari has written a book on
this topic, Izalat al khatar 'amman jama'a bayn al-salatayn fi al-hadar.

  4. There is no difference regarding the definition of sunset between the
Imamis and the other four schools. But the Imamis say that the setting of
the sun is not ascertained simply by the vanishing of the sun from sight,
but on the vanishing of the reddish afterglow from the eastern horizon,
for the east overlooks the west and the eastern afterglow, which is a
reflection of sun's light,

@ poga

This is the Proposed & Own Problems subsection of the Number Theory section of the Olympiad forum.
I see you are new to AoPS, but please try to post in the appropriate places. Your post has very little if at all anything to do with a proposed olympiad-level number theory problem.

Could a mod please move this?


algebra
polynomial
vector
linear algebra
matrix

1) The set of all polynomials of degree bigger than or equal n is a vector space.

2.) Let M denote the set of m x n matrices whose entries are rational numbers. M is a vector space over the real numbers( ie. when the set of scalars is the real numbers).

Thanks

1)  Go through the vector space axioms.  Do they all hold?  (This problem really should specify "real polynomials" and "real vector space.")

2)  Go through the vector space axioms.  Do they all hold?  (You want to look at scalar multiplication in particular.)

1) i think this question is  "false", but i am not sure. just want to double check my answer.
2) i think the answer is "true". Am i right about it?

thanks

1) Does $ (x^n\plus{}1)\minus{}x^n$ belong to your set?

2) Take any nonzero matrix $ A$ that belongs to $ M.$ Take a real number that you know is irrational - let's say $ \sqrt{2}.$ is $ (\sqrt{2})A\in M?$

In other words, they're both false.

The example in 1) is not closed under either addition or scalar multiplication (try multiplication by $ 0.$) But the example in 2) is interesting because it is closed under addition - it's only scalar multiplication that trips it up. $ M$ would be a vector space if you were willing to let the field of scalars be $ \mathbb{Q}.$


geometry
rhombus
incenter
parallelogram
perpendicular bisector
geometry proposed

A quadrilateral has an inscribed circle (I). For each vertex, an incircle is drawn for the triangle formed by the vertex and the two points at which the circle (I) touches the adjacent sides. Each pair of adjacent incircles has two common external tangents, one a side of the quadrilateral. Show that the other four tangents form a rhombus.

let $I$ be incenter of $\triangle PQS$ cause $O_1O_2$ is perpendicular bisector of $AI$  ,hence $I$ lies on common tangent of these two circle.
$\angle XIP=\angle API=\angle PQS+\frac{\angle P}{2},\angle SPI=\frac{\angle P}{2} \Longrightarrow \angle(\ell_1,PS)=\angle PSQ\longrightarrow \ell_1 \parallel QS$

similarity $\ell_1 \parallel \ell_2 \parallel QS,\ell_3\parallel \ell_4 \parallel PR$ therefore $EFGH$ is a parallelogram.

To Yetti :thanks for this nice problem .sorry,I have no time to complete the solution,now.If I find time I'll post rest of it,Mayebe on April 21.

My solution was different. A simple angle chase reveals that the other four common external tangents of the 4 circle pairs form a parallelogram, just the way amir2 did it. Then I used the following general theorem to show that this parallelogram is also tangential, i.e., a rhombus:

[b]Theorem:[/b] Let (K), (L), (M), (N) be four arbitrary circles centered inside a tangential quadrilateral ABCD with an incircle (I), such that the circle (K) touches the sides DA, AB, the circle (L) touches the sides AB, BC, the circle (M) touches the sides BC, CD, and the circle (N) touches the sides CD, DA. Then the common external tangents a, b, c, d to the circle pairs (K), (L); (L), (M); (M), (N); and (N), (K) different from AB, BC, CD, DA also form a tangential quadrilateral.

Let the other tangent pairs intersect at points $P \equiv d \cap a,$ $Q \equiv a \cap b,$ $R \equiv b \cap c,$ and $S \equiv c \cap d,$ so that $a \equiv PQ,$ $b \equiv QR,$ $c \equiv RS,$ $d \equiv SP.$ Denote the tangency points of the circle (K) with the lines DA, AB, SP, PQ as $A_1, A_2, P_1, P_2$ and similarly, the tangency points of the circle (L) with the lines AB, BC, PQ, QR as $B_1, B_2, Q_1, Q_2,$ the tangency points of the circle (M) with the lines BC, CD, QR, RS as $C_1, C_2, R_1, R_2,$ and the tangency points of the circle (N) with the lines CD, DA, RS, SP as $A_1, A_2, P_1, P_2.$ Since the quadrilateral ABCD is tangential,

AB + CD = BC + DA

Subtracting the segments $AA_2, BB_1, CC_2, DD_1$ from the left side and equal segments $AA_1, BB_2, CC_1, DD_2$ from the right side of this equation,

$A_2B_1 + C_2D_1 = B_2C_1 + D_2A_1$

The segment lengths of the common external tangents $A_2B_1 = P_2Q_1,$ $B_2C_1 = Q_2R_1,$ $C_2D_1 = R_2S_1,$ $D_2A_1 = S_2P_1$ of the circle pairs (K), (L); (L), (M); (M), (N); (N), (K); respectively, are equal, hence,

$P_2Q_1 + R_2S_1 = Q_2R_1 + S_2P_1$

Subtracting the segments $PP_2, QQ_1, RR_2, SS_1$ from the left side and equal segments $PP_1, QQ_2, RR_1, SS_2$ from the right side of this equation,

$PQ + RS = QR + SP$

As a result, the quadrilateral PQRS is also tangential.

Using directed lines and directed circles, the above lemma can be stated without any reference to the common tangents being external or internal or to a figure:

[b]Theorem:[/b] Let (K), (L), (M), (N) be four arbitrary directed circles. If four directed common tangents of the directed circle pairs (K), (L); (L), (M); (M), (N); and (N), (K) touch a single directed circle (I), then the other four directed common tangents of these directed circle pairs touch a single directed circle (I’).


The operators $ ``\plus{}$,'' $ ``\minus{}$,'' and $ ``\times$,'' are each used once to fill in the blanks in the expression $ 5\_\_\_\_2\_\_\_\_7\_\_\_\_3.$ Find the minimum value of the expression.

The minimum is $ 5 \minus{} 2\times7\times3 \equal{} \fbox{ \minus{} 37}$. Our strategy was to subtract from only the first number the largest possible answer formed from the last three numbers, and since the three are all greater than $ 1$, we multiply for largest results.

EDIT: Oh wow! I phail again. Ok, $ 5 \minus{} 2\times7 \plus{} 3 \equal{} \fbox{ \minus{} 6}$ is the answer, [b]not[/b] the $ \minus{} 37$ answer above.

EDIT2: Oh whoops. Bad day, must be.

[quote="GameBot"]The operators $ `` \plus{}$,'' $ `` \minus{}$,'' and $ ``\times$,'' are [b]each used once[/b] to fill in the blanks in the expression $ 5\_\_\_\_2\_\_\_\_7\_\_\_\_3.$ Find the minimum value of the expression.[/quote]

:wink:

It said the answer was -14.... Is it an error?

I AGREE WITH ALL OF YOU

 :gathering:  :gathering:  :gathering:  :gathering: AN AoPS SOCIAL GATHERING !!!


real analysis
function
integration
real analysis unsolved

Prove that in any real measurable set with positive measure we can find arbitrarily long arithmetic progressions.

It is enough to prove for bounded set $ E $ ( because for some $
n \in \mathbb{Z} $ we will have $ m( E \cap [n,n+1] ) > 0 $. So
suppose that $ E $ is bounded. Fix some $ n \geqslant 1 $.
Consider $ E_{h} = E \cup E+h \cup \ldots \cup E+nh $, for every $
h > 0 $. Let us prove that $ m(E_{h}) \rightarrow m(E) $, when $ h
\rightarrow 0 $. Indeed, it is true when $ E $ is a finite union
of open intervals. For the general case, for every $ \varepsilon >
0 $ there exists set $ U $, which is a finite union of open
intervals, such that $ m( E \vartriangle U ) < \varepsilon $. Then
$ m( E_{h} \vartriangle U_{h} ) < (n+1) \varepsilon $, for every $
h > 0 $. We have that $ m(U_{h}) \rightarrow m(U) $, when $ h
\rightarrow 0 $, therefore $ limsup_{ h \rightarrow 0 } m(E_{h})
\leqslant m(U) + (n+1) \varepsilon \leqslant m(E) + (n+2)
\varepsilon $, for every $ \varepsilon > 0 $, so $ m(E_{h})
\rightarrow m(E) $, when $ h \rightarrow 0 $. Now take $ f(x):=
\chi_{E}(x) $ - the characteristic function of $ E $, and define $
f_{h}(x):= f(x) + f(x-h) + f(x-2h) + \ldots + f(x-nh) $. We know
that $ f_{h}(x) = 0 $ if $ x \notin E_{h} $, and $ \int f_{h}(x)
dx = (n+1)m(E) $. Therefore $ (n+1)m(E) = \int f_{h}(x) dx
\leqslant ( max f_{h}(x) ) m(E_{h}) $, so $ max f_{h}(x) \geqslant
(n+1) \frac{m(E)}{m(E_{h})} $. We know that $ f_{h}(x) \leqslant
n+1 $, and $ f_{h}(x) $ is always an integer, and also that $
m(E_{h}) \rightarrow m(E) $, when $ h \rightarrow 0 $. So for $ h
$ small enough we will have that $ max f_{h}(x) = n+1 $, so there
exists $ x \in \mathbb{R} $, such that $ f_{h}(x) = n+1 $, which
means that $ x,x+h,x+2h, \ldots , x+nh \in E $.

I hope it is correct.


geometry
search
algebra unsolved
algebra

A=(1+x).(1+x^2).(1+x^3).........(1+x^63)

When all terms are multiplied, what is coefficient of x^2008

Hope I'm not wrong, I think that the coefficient of $ x^{2008}$ should be $ 6$.  :)

edit: I just saw that this is in geometry section...maybe some moderator should move it...

[quote="telemat"]Can you explain clearly[/quote]
Sure I can:
The maximum power of $ x$ that we get after multiplying out is $ x^{\sum_{i\equal{}1}^{63}i}\equal{}x^\frac{63\cdot 64}{2}\equal{}x^{2016}$. Thus, we have to search the number of ways we can "leave out" powers of $ x$ that multiply up to $ x^8$, i.e. we need the number of ways $ 8$ can be expressed as a sum of (different) positive integers. This is easy to find ($ 8\equal{}1\plus{}7\equal{}1\plus{}2\plus{}5\equal{}1\plus{}3\plus{}4\equal{}2\plus{}6\equal{}3\plus{}5$). Therefore, the coefficient is $ 6$.

It's not a very good explanation, sorry  :blush: , but I hope you understand what I mean  :P .


let $a,b,c,d$be positive numbers prove it:
$\frac{a}{b+2c+3d}+\frac{b}{c+2d+3a}+\frac{c}{d+2a+3b}+\frac{d}{a+2b+3c}\ge\frac{2}{3}$

Use Cauchy in Engel form to get \[\sum_{\textrm{cyc}}\frac{a}{b+2c+3d}\geq \frac{(a+b+c+d)^{2}}{4(ab+ac+ad+bc+bd+cd)}.\] Let $t=ab+ac+ad+bc+bd+cd$ to make this readable  :lol: So we would like to prove that $3(a+b+c+d)^{2}\geq 8t$, i.e. $3(a^{2}+b^{2}+c^{2}+d^{2}+2t) \geq 8t$, i.e. $3(a^{2}+b^{2}+c^{2}+d^{2}) \geq 2t$, i.e. \[(a-b)^{2}+(a-c)^{2}+(a-d)^{2}+(b-c)^{2}+(b-d)^{2}+(c-d)^{2}\geq 0.\] (Or just AM-GM it of course  :D)

This one comes straight out of the IMO Shortlist 1993
I remember because coincidentally, I did this one yesterday (in exactly the same way Christophe.  :lol: )


Billy Bob's job pays a lot. The first day of each month, he gets 100 dollars. The second day, he gets 200 dollars. The third day, he gets 400 dollars. The fourth day, he gets 800 dollars, and so on. He doesn't have to work on Saturday or Sunday. How much does Billy Bob earn in June (assuming that Billy Bob does not have to pay income tax)?

I'm sorry I didn't specify. I meant this June.

[hide]There are 4 Sundays and Saturdays, so there's 30-8=22 days.

There answer is therefore $(100(2^{22}))-100=\$419430300$ [/hide]

[quote="SplashD"][hide]There are 4 Sundays and Saturdays, so there's 30-8=22 days.

There answer is therefore $(100(2^22))-100=\$419430300$ [/hide][/quote]
[hide] This comes from the fact that $1+2+4+8+16+...+2^n=2^{n+1}-1$. [/hide]

[quote="Arvind_sn"]Do you know if there is a rigorous proof for that? Is it in one of the AoPS volumes?[/quote]

I don't know if you would call this rigorous...
Using the formula for sum of a finite geometric sequence: $\frac{t_1(1-r^n)}{1-r}$, where $t_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. $\frac{1(1-2^{n+1})}{1-2}=\frac{1-2^{n+1}}{-1}=2^{n+1}-1$.


This is a homework question, so please don't do it for me. All I want is for someone to explain the general way to approach this problem - ei, how to find the intersection of curves:

Show that the curve of intersection of $ x^2 \plus{} 2y^2 \minus{} z^2 \plus{} 3x \equal{} 1$ and $ 2x^2 \plus{} 4y^2 \minus{} 2z^2 \minus{} 5y \equal{} 0$ lies on a plane.

All I need is for someone to explain how to find the curve of intersection of two surfaces in general. My textbook doesn't show an example and we haven't covered it in class.

Is it as simple as multiplying the first one by 2, subtracting the second one and obtaining:

$ 6x \plus{} 5y \equal{} 2$? This says nothing aout z, though. It would appear to me that a system of two equations in three unknowns shold have infinite sollutions, but I have never tried a non-linear system before. Help?

I am a little unsure what that result means physically, if someone could enlighten me I would appreciate it.

YES, it is that simple! Write what you obtained as $ 6\cdot x \plus{} 5\cdot y \plus{} 0\cdot z \minus{} 2 \equal{}0$. What is this? Answer: the equation of a plane! So you proved the intersection of those two curves lies in this plane. You've done it.

@mavropnevma

I don't agree with You about: 6x + 5y = 2

precise this is the proection of intersection curve of two surfaces at xOy plane. And just is a case that this is plane (with z=o).
Generally speacking, eliminating z from both initial equations gives proection of intersection curve of two surfaces at xOy plane, some curve like g(x,y)=0. And Intersection curve can be than obtained like intersection of one z surface and cilyndar  with generatrise g(x,y)=0 and generating lines paralell to z-axes

The points that lie on $ F(x,y,z)\equal{}0$ and also $ G(x,y,z)\equal{}0$, clearly also lie on any linear combination of $ F$ and $ G$. If that happens to be a plane, it means they are coplanar.
It has nothing to do with projections, has it?


Hello Everyone,

Find the largest value of y which is a solution of y^3 - 12y^2 + 11y = 0.
I have NO IDEA how to do this so please explain from the basics if you may.

akhil0422

$ y^3\minus{}12y^2\plus{}11y$ can be factored into y(y^2-12y+11)=y(y-11)(y-1). So the solutions for the equation are 0, 1, and 11. The largest is 11.

This is because of the zero product identity. (correct name?)

If you have numbers multiplied that equal $ 0$ (as many as needed), then at least one of them equals $ 0$.

So, when factored, you have $ y(y\minus{}1)(y\minus{}11)\equal{}0$, you have either $ y\equal{}0$, $ y\minus{}1\equal{}0$, or $ y\minus{}11\equal{}0$.

Solving those three equations, we have $ y$ equaling $ 0$, $ 1$, or $ 11$.

Zero product property, if you want to be nit-picky.  :)


I've been trying everything I know with this one for nearly an hour but can't make any headway. Any help would be very appreciated.

Given:

x + 2y = 2

(x + y)^2 = 9

which is a possible value of y?

-8, -5, -3/2, 3/2, 5

I know that the answer is 5, but I have no idea how to determine that without plugging in each one. I've tried everything... factoring, square roots, etc. Please help!

[hide]In this problem, substitution will bring about the result to you immediately!Here is how it works:
Substituting x=2-2y into 2nd equation, we have: (2-y)^2=9, which means y=5 or y=-1. So, we have 2 solutions:(x,y)=(-8,5),(4,-1)
Similar problem, but a little harder: Find the solutions of the following system of equation:
x+2y=a
x^2+2y^2=a^2/3[/hide]


for which k
does the inequality
 1/(x^3+k)+1/(y^3+k)+1/(z^3+k)<=1/(xyz+k)
hold for all positive reals x,y,z?

For none. ;) I think you mean $\frac1{x^{3}+k}+\frac1{y^{3}+k}+\frac1{z^{3}+k}\le \frac3{xyz+k}$ instead.

Jensen should work after making the substitution $x = e^{a}$, etc.

Edit--Yeah, I just noticed it only works when x, y, and z are small enough. I don't think the inequality is true for sufficiently large x, y, and z.

Too bad $\frac{1}{1+e^{a}}$ is only concave for $a\le0$ which would only prove it for $x,y,z\le1$... if that is what you meant at all.

[quote="Vasc"]For $k \geq 2xyz$.
Or for $k=\infty$.[/quote]

I'm sorry. But how did you arrive at this magnificent conclusion?
Could anyone post a proof?
are any generalizations available?


AMC USA(J)MO autobiographical MIT college AIME

An essay I wrote about my math experiences (including experiences with USAMO)

Please feel free to comment, positively or negatively, and share your own experiences.
However, I respectfully ask that you do not comment until you have read the entire essay.
Thank you.

I respect this probably more than anything I ever have, because it is so true and heartfelt.

It was nice working with you at AMP 08, good luck for the future.

-PY

Wow. Your 'Math Experience' was quite incredible. It contained so much raw emotion. I deeply respect this.
I'm sorry to hear of your loss. However, I find it amazing that you have so much motivation and self discipline within you to do what you did, going from your first math competition to being usamo honorable mention. All the time you spent doing problem after problem. I applaud you on that. This isn't something many people can do, especially to the extent that you did it. I understand [sort of] how disappointing it must have been to not be a usamo winner despite all the work and effort you put in the last couple of years, although I would like to point out your journey has been a great one. You started much later than most people and ended up much farther ahead than most. You should be proud of that.
I , too, went to AMSP 08. I remember you from the team contest [many congrats; that was beast btw]. I don't think we ever talked, but your mathematical talent is incredible; your efforts really shined through [I watched a bit of the team finals]
There's more to life than usamo, more to life than math competitions. Loosen up a little and have some fun :]
Don't beat yourself up too bad. Continue to pursue your passion as I'm sure you will [Congrats on MIT]. Reading this made me think about my life as well.... I'm glad you've put this in the past & moved on. Thank you for sharing ;]

I really don't read many essays on math, but when I do, I don't like them very much. Your experience is absolutely amazing. If you think you lost, just because you weren't in the top 12, you are absolutely WRONG.

Qualifying for the USAMO and the AIME takes tons and tons of hard work. Lemme make this seem realistic. About 500 kids qualify for the USAMO, therefore there are over 14 millions kids (about 27,000 high schools, 500 kids in each). You are in the top 1% of math students around the country.

I am truly sorry to hear what has happened. It must have been very hard for you, but your dad would be proud. Going to MIT, making the USAMO, and most importantly having that faith in you, is truly a perfect math student.

There aren't many kids you start problem solving in late high school, and become one of the top math students in the country. It is truly phenomenal. And don't let people bring you down ever, so what if you do math on a regular basis.

I really want you to congratulate you on this accomplishment. THIS IS NOT THE END. LET ME REPEAT THIS IS NOT THE END.

THIS IS THE BEGINNING, it is start of an amazing journey for you. I wish YOU THE BEST OF luck at MIT, you deserve it greatly. Anyone who dedicates that much time into math, is truly a winner by itself.

I wish you the best of the luck in the future. Please remember this: Problem solving, Math competitions, the AMC, are the beginning steps now take that into the world and shine!

whew...that was a looooooooong essay.
2 pieces of advice:
1) turn this essay into a book and publish it. like, i'm serious. there are plenty of math nerds (sorry if this offended you, but other people would probably be mortified if they were caught reading a book like this) out there that would love to read something like this.
2)be happy that you got into MIT, because that's my goal (and i'm sure that lots of other people also have this goal.)

you made me feel a lot better about whether i will get into MIT or not too, as in "so i don't have to get into MOP 3 times to go to MIT!!!!"

well, yeah, i made MOP. hope this doesn't make you feel like a retard. but that's just because i was fortunate enough to discover competition math in 6th grade (although i had never heard of aops before 8th grade). so i feel quite fortunate.

this essay almost brought tears to my eyes. and i'm supposed to be tough, being a black belt in tae kwon do. i can half-sympathize with your loss. in 8th grade, i got 5th place at states...i was pretty p-o-ed. but losing on the usamo like that...after two years of monk-like piety...ooohh that must hurt.

some random thoughts after reading this AWESOME essay.

Your autobiography is very moving, thoughtful, and beautifully written.

I think you have learned a lot of important lessons, some lessons that some much older than you have never learned.   Please always remember never to dismiss yourself as "lesser" than anyone based on the outcome of a test.  There is so much random noise in how a given student does on a particular test on a particular day.  In particular, the difference between your getting an honorable mention on the USAMO (an extraodinary accomplishment for anyone, especially for a student who only discovered the AMC exams in tenth grade!) and someone else placing in the top 12 is not a statistically significant distinction.

What is not random noise is all you have learned from the process of preparing and from your reflections on that process.

One of your conclusions struck me as especially worth noting:

[quote]I took some time to read over the letters my dad’s students left for him. I have always known him to be very smart, but what his students remember him for most is how he goes to great lengths to help everyone understand, how his door was always open when they needed help.

Then, I resolved that when I grew up to be a math professor, I would try to be just like him. However, now I know, just as he realized, that helping others with my knowledge is just as important as pursuing it, and much more meaningful than trying to be better than everyone else.[/quote]

I was so sorry to hear that you lost your father at such an early age, and I think your thoughtful and insightful essay--and this statement in particular--would make a parent very proud and happy.  No matter what your future path in life, you can know joy in your life from using your knowledge to help others.

I hope you have a wonderful summer visiting your family in Taiwan this summer and great college experiences ahead at MIT.

with such a great work ethic im sure you will be a great mathematician. I think u should consider doing the Putnam just for the sake of the great problems

This is truly an odyssey. As unqualified as I am to say, I think you got more than what there is out of math olympiads. Best luck in your future endeavors.

I was literally brought to tears by this; I think everyone's suffered the agonizing feeling one gets after a math test where you were just off from advancing. This year, for example, I had a 129/8 index that prevented me from even making USAMO, and I made several very stupid mistakes on the AMC 10 that I would desperately love to fix. On the other hand, I don't think many of us can claim to have the same devotion to problem solving and the same work ethic that you have demonstrated: this skill, and this development of your affinity and intuition in mathematics in particular, is much, much more important than making MOP, I think.

And another thing: it made me happy to find out that you also want to go into mathematics. :D

I have to say, I could really feel the sequence of thoughts you put across in this essay.  From the time I really started taking the AMC competition seriously in my freshman year, through two years of missing the USAMO by an AIME point, I feel I had the same sort of thoughts as you initially had (except I wasn't training as hard).  I saw the competitions as actual competitions, and, as you said, didn't enjoy them as much as I could have.  There was always the nail-biting nervousness as time ran out, the anxiety afterwards of making mistakes you would not have made had you just looked that one more time...And I took as much free time as I could to do that one more AIME problem, one more problem from the Pre-Olympiad forum, one more USAMO problem...

And the sinking feeling afterwards...I'm sure I had it nowhere nearly as bad, since I had two more chances, but I remember it vividly.

[quote]I wanted to be the best, but there’s no point. Isn’t there room in the US for thousands of math professors? We should all work hard, but it’s not necessary to be better than everyone else: everyone working in the field contributes to it in some way, and often the work of hundreds of unknown people matters greatly. Who are we to judge?

If you anticipate fame and glory ahead, every failure will be that much more disappointing, but if you truly enjoy struggling with problems, it will only motivate you to continue. If you don’t look for competition, you will like what you do without needing to win, and because of the purity of your motivation, likely find success as well.
[/quote]

I think that these two paragraphs, in particular the very last sentence, resonates soundly with me.  I know this is the most important thing I learned from math competitions.  Before I found, in my sophomore year (2006), I'd missed the USAMO just barely for a second time (and the year I was hoping to make blue), I would always be measuring people's worth by what they did.  How good were they at math?  Music?  Other subjects?  Etc etc...the math competitions were a measure of my worth, they were what made me feel I had purpose, something I was good at.

After that, I think I re-evaluated things and the two paragraphs above were close to my thoughts then.  In general, I think I also really started to understand that in any competition of any kind, there are factors you cannot change, luck elements.  It's impossible to absolutely guarantee you'll end up at the top, even if you're Alex Zhai.  Now there are some extremely smart people I know who did not do so well at these competitions.  I kinda let go of the competition part, and simply tried to enjoy the mathematics in every competition I did.  And I started to try to let go of the competitive rush too.

(skip the next two years of high school)

I'm glad to know you took these lessons from your competitive experience going into college (and glad to know MIT's getting someone like you, I'll keep an eye out for you among math majors :D).  Once you get to college, I'll be honest, it's extremely collaborative, doing math because you love it with other like-minded people.  Even though math competitions seemed like the end-all be-all of high school, I look back at it now along with people who didn't do competitions and we laugh at it over a math problem.  And yeah, tbh people don't care all that much about Putnam in college...  What you've got ahead of you is much bigger, and it looks like you're perfectly prepared for it.

Thanks to everyone for your positive and supportive comments! I admit, I never thought this would resonate so much with everyone. I'll be sure to take your advice to heart.

What a touching and profound soliloquy! Look forward to see you in MIT this fall.

[quote="BOGTRO"]* Math competitions aren't that important. Until you do them.
[/quote]I'm gonna have to disagree with you here. While we may get caught up in the moment and think that they are, math competitions are not important. If that's what you are saying, then disregard this post.

[quote="BOGTRO"]This is a very nice piece of work. Some random thoughts I had were:

* I didn't think you should have spent every spare moment on difficult math problems. It might have been good to take an hour or so a day to just have fun. A computer game, a good book, anything probably would have helped. Although working is a good thing, too much of a good thing is... too much of a good thing.

* I was glad that you began working very hard and continued to do so even after your tragedy. I was also glad that you were not discouraged by the USAMO. You have great perseverence.

* Math competitions aren't that important. Until you do them.

* I felt that if you had heard of math competitions earlier, even just a year, you would have had a good chance at MOSP. If you had heard of math competitions at the age I did (4th grade), I have no doubt you would have been a USAMO winner.[/quote]
HM is still amazing and is something many MOPpers will never achieve!!

I don't mean to revive this thread too much but...wow. An emotional essay that really shows your personality. I wish I could have the same willpower as you. Amazing.

^^^ Thank you for sharing. Your opinion is probably right, as it is confirmed by the [b]over 100 above posts[/b] saying exactly the same thing.

:)

lol yeah :)
The author should feel proud

I will be downvoted for this but it was meh. This article isn't life-changing but it is touching while not as amazing as many aopsers are making it out to be. Well-written but a bit boring.

And I was right, aops is so predictable.

[quote=FlyingCucumber]I will be downvoted for this but it was meh. This article isn't life-changing but it is touching while not as amazing as many aopsers are making it out to be. Well-written but a bit boring.[/quote]

Umm...

[quote=First]A good honest article.[/quote]

Yup.

Very touching.  You seem extremely perseverant.

I read this every time I get discouraged. It helps. :)

Beautiful essay that was very well written.

I am in 7th grade and I recently got a pretty "meh" score on the AMC8. But what really pissed me off is that whenever I took the practice tests I would take near-perfect scores. So, this essay really changed my outlook and I am going to look to work harder because I don't think I deserve what I can do today. Most kids have to work to do what I can but the fact that I just sit around watching random YouTube videos. You really taught me a lot and I think I should start to look to the future. After all, who can judge me for my old AMC8 scores? So today, to the AoPS community, I have a promise. Although no one may ever read this comment, I am going to work harder than what I do right now. I am going to make sure I go to MathCounts States this year and get top 30. Many of my friends reading this will laugh and say I am a cocky idiot, thinking I can do that. But I promise I will try my very best. Thank You for the touching essay!

Same here, I'm in 5th grade and I took the AMC8 this year, and actually got a pretty good score! This also changes my outlook on my math life!

-Cubey

I've read this entire essay more than once and it has not failed to move me every time. Although not many AoPSers may be able to truly connect to the author, I think everyone has something to learn from someone who has journeyed through high school and perhaps experienced and overcame many more challenges and frustrations than many of us.

What I get from the essay most is the ardent belief that hard work will eventually turn into success, though failure is almost always inevitable along the way. Furthermore, this success is not totally materialistic, a "higher score" or a Fields medal. It can be meaningful in other ways as well. Glory, fame, getting into a good college, winning competitions, and other utilitarian things can be replaced by passion, inspiration, dedication, and much more.

On a related note, a good time to be thankful today...

First, I am thankful for my parents and family, knowing that I'm forever in their debt. I realize that many times I take them for granted, and then I am greatly saddened when I know there are many out there like me who have lost a parent or close family member. I am thankful for my friends, my school, my community (which includes AoPS), all of which give me an extremely vibrant and open opportunity for discussion and for a meaningful social life, with well-developed relationships. I am thankful for my teachers, who have taught me much more than in the classroom. I am thankful to those, whether supernatural or not, that have made it possible for me to be where I am today, breathing and typing here.

Sometimes it helps to look not at what you don't have, but what you do have.

I cannot upvote this enough. It actually brought tears to my eyes. Thank you for writing such a beautiful piece of work and I hope you never lose sight of the raw, powerful, undeterred passion you have for math.


linear algebra
matrix
linear algebra unsolved

Let $ A,B \in \mathcal M_n \left( \mathbb R \right)$.
Prove that $ \text{rank} \, A \plus{} \text{rank} \, B \leq n$ if and only if there is an invertible matrix $ X \in \mathcal M_n \left( \mathbb R \right)$ such that $ AXB \equal{} O_n$.

Let $ L_{A},L_{B},L_{X}$ be linear transformations whose matrices in the standard basis are $ A,B,X$ respectively. Then the following conditions are obviously equivalent:
(a) there exists an invertible matrix $ X$ such that $ AXB \equal{} O_{n}$
(b) there exists an isomorphism $ L_{X}$ such that $ L_{A}\circ L_{X}\circ L_{B}$ is the zero transformation
(c) there exists an isomorphism $ L_{X}$ such that $ L_{X}\left(\mathrm{im}\, L_{B}\right)\subset\ker L_{A}$
(d) $ \dim \mathrm{im}\, L_{B}\leqslant \dim\ker A$
(e) $ \mathrm{rank}\, A \plus{} \mathrm{rank}\, B\leqslant n$.


analytic geometry
geometry proposed
geometry

Let $ ABC$ be a triangle and $ A'B'C'$ is pedal triangle of an arbitrary point $ P$, let $ PB',PC'$ intersect $ C'A',A'B'$ ay $ Y,Z$, resp. $ BY,CZ$ intersect $ CA, AB$ at $ E,F$, resp. $ PB$ intersects $ EF$ at $ D$. Prove that $ A',B',D$ are collinear.

Note that, this is generalization of the problem [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1638781#1638781]ABOUT collinear[/url].

Let $(u:v:w)$ be the barycentric coordinates of $P$ with respect to $\triangle ABC.$ Then

$A' \ (0:a^2v+uS_C:a^2w+uS_B)$

$B' \ (b^2u+vS_C:0:b^2w+vS_A)$

$C' \ (c^2u+wS_B:c^2v+wS_A:0)$

$A'B' \equiv \left[\begin{array}{ccc}0 & a^2v+uS_C & a^2w+uS_B\\  b^2u+vS_C & 0 & b^2w+vS_A\\ x & y & z\end{array}\right]=0$

$ E \ (u(c^2u+wS_B):0: w(a^2w+uS_B)) \ , \ F \ (u(b^2u+vS_C):v(a^2v+uS_C):0)$

$EF \equiv \left[\begin{array}{ccc} u(c^2u+wS_B) & 0 & w(a^2w+uS_B)\\  u(b^2u+vS_C) & v(a^2v+uS_C) & 0\\ x & y & z\end{array}\right]=0$

Therefore, $A'B',EF,BP$ concur at a point $D \ (x_0:y_0:z_0)$ with barycentric coordinates:

$x_0=u(a^2w+uS_B)(b^2u+vS_C)$

$y_0=v(a^2v+uS_C)(wS_C-uS_A)$

$z_0=w(a^2w+uS_B)(b^2u+vS_C)$

Actually the problem is merely projective. It can be generalized as follows: $P$ is arbitrary point on the plane and $A',B',C'$ are abitrary points on $BC,CA,AB,$ resp. $PB',PC'$ cut $A'C',A'B'$ at $Y,Z,$ resp and $BY,CZ$ cut $AC,AB$ at $E,F.$ Then the lines $A'B',EF,BP$ concur.

Proof: Let $PB,PC$ cut $A'B',A'C'$ at $M,N,$ resp and $MN$ cuts $AC,AB$ at $E',F'.$ By Pappus theorem for $A'NCE'BM,$ the intersections $Y' \equiv A'N \cap E'B,$ $P \equiv NC \cap BM$ and $B' \equiv CE' \cap MA'$ are collinear $\Longrightarrow$ $Y' \equiv Y$ $\Longrightarrow$ $E' \equiv E.$ By similar reasoning $F' \equiv F \equiv MN \cap AB \cap CZ$ $\Longrightarrow$ $A'B',EF,BP$ concur at $M.$


SuMAC
summer program
PROMYS
Ross Mathematics Program
Mathcamp
AMC
AIME

Hi. I'm currently a sophomore, and I got into SUMaC summer math program. But I am not sure if PROMYS, ROSS, or MathCamp would have been a better choice. If I wanted to major in fields of applied mathematics, what math camp would you recommend the most?

A little bit about me..
AMC 12: 114/150        2008 (not so good)
AIME: 7/15                2008 (not so good either)
AMC 10: 150/150        2009
Intel ISEF Finalsit (mathematical science)
Some University-competitions (HMMT, PUMaC, etc.) with no notable awards

Thank you.

what was your aime score? i dont think attending pumac/hmmt will help. what was your intel project?

My aime score last year was 7 as i wrote up there, I usually get 9-11 on practice ones tho.

The Intel project was about a new algorithm that I wrote for computing the size of certain spherical fullerene molecules, which also proved applicable to distorted spherical objects or spherical polyhedrons...

Ross/Promys focus a great deal on number theory; that may help you.

Is intel isef as prestigious as just being an Intel Finalist?


MATHCOUNTS
probability
geometry
rectangle
ratio
inequalities
number theory

1.
Darts are thrown randomly at a board showing three concentric circles of radii 2, 3, and 4.  If a dart hits inside one of the circles, what is the probability, expressed as a common fraction, that it is in the interior of the circle of radius 3 but not in the interior of the circle of radius 2?

2.
Three tennis balls are tightly sealed in a right cylindrical can, so that the balls cannot move inside the can.  If the radius of each ball is 4 cm, find the number of cubic centimeters in the volume of the space within the cylinder, not taken up by the three tennis balls.  Express your answer in terms if $\pi$.

3.
A custom made window is a single pane of glass consisting of a semicircular sheet of glass on top of the narrow end of a rectangular sheet of glass.  The rectangle has dimensions of 4 feet by 3.5 feet.  How many feet of weather-stripping will be needed to go around the glass?  Express your answer to the nearest integer.

4.
The numbers from 1 through 8 are separated into two sets, A and B.  The numbers in A are multiplied together to get $a$.  The numbers in B are multiplied together to get $b$.  The larger of the two numbers $a$ and $b$ is written down.  What is the smallest possible number that can be written down using this procedure?

5.
Eric's Florist Shop sells roses for 20 dollars per dozen, daises for 5 dollars per dozen, and chrysanthemums for 10 dollars per dozen.  At these rates how much would an arrangement of 9 roses, 6 daises, and 18 chrysanthemums cost in dollars?

6.
Given the equation $\displaystyle \frac{1}{m} + \frac{1}{n} = \frac{2}{15}$, where $m$ and $n$ are positive integers, find the value of $m+n$ such that $m>n$ and $m$ is not a multiple of $n$.

7.
Use the pattern below to express $100^2$ in the form $a^2+b^2-c^2$.  Then find $a+b+c$.
\begin{eqnarray*}
12^2 = 8^2 + 9^2 - 1^2\\
14^2 = 10^2 + 10^2 - 2^2\\
16^2 = 12^2 + 11^2 - 3^2\\
18^2 = 14^2 + 12^2 - 4^2
\end{eqnarray*}

8.
The ratio of the length of a rectangle to its width is the same as that of the diagonal to the length.  If the width is 2, how many units are in the length of the diagonal?

9.
Suppose a number $x$ satisfies $x^2-4x-1=0$.  How much greater is $x$ than its reciprocal?

10.
What is the difference between the largest and the smallest values of $x$ satisfying the following system of inequalities?
Express your answer as a mixed number.
\begin{eqnarray*}
2x-6 \le 6\\
4x+3 \ge 13
\end{eqnarray*}

I'll do one
1.  The circle with 3 has an area of 9 pi and the total circle is 16 pi.  The circle's bull'seye is 4 pi so 5/16 is the total chance.

Looking at number 7, start listing
12 a+b+c=18
14 a+b+c=22
26
30 and so on.

For x, a+b+c=2x-6
For 100, a+b+c=194

[hide="1"]$\frac{5}{16}$[/hide]
[hide="2"]$(24\times 16\pi)-(3\times\frac{4}{3}\times 64\pi)=$volume of cylinder-area of balls$=256\pi$[/hide]
[hide="3"]$11.5+1.75\pi\approx 17$[/hide]

thanks ragnarok  :lol:

[quote="daermon"][hide="1"]$\frac{5}{16}$[/hide]
[hide="2"]$11.5+1.75\pi\approx 17$[/hide]

is the area of a sphere $\pi r^{3}?$[/quote]
$\frac{4}{3}\pi r^{3}$

[hide="4"]prime factorization together is $2^{7}\times 3^{2}\times 5 \times 7.$ To minimize larger number, they should be as close as possible.  closest are $240$ and $168$, so $240$[/hide]
[hide="5"]flowers cost $15,2.50,$and$15$dollars, respectively, for a total of $32.50$[/hide]
[hide="7"]the pattern is $(12+x)^{2}=(8+x)^{2}+(\frac{x}{2}+9)^{2}-(\frac{x}{2}+1)^{2}$ Plugging in $x=88$ gives $100^{2}=96^{2}+54^{2}-45^{2}$, so the answer is $96+54-45=105$[/hide]


analytic geometry
absolute value
complex numbers

Why does   $\sqrt{5-12i}= a+bi$ for some number a and some number b.  Why not $\sqrt{5-12i}= a-bi$ ?

Does $\sqrt{5+12i}= a-bi$?

Thanks :P

I guess its something to do with convention.

You can get $b$ as negative while solving,so no harm.There's nothing wrong in taking it the other way round as well.Both work but in each,the sign of $b$ would be different.

It's not just convention. Say you were taking the general $n$th root. You would much rather write it as $a +bi$ because you don't have to worry about the whether $n$ was odd or even.

It's because of two things:
1) In this case, if you use $a+bi$, you would end up finding that $b$ is negative. If you use $a-bi$, you would end up finding that $b$ is positive. Putting in the numbers, with $a+bi$, you would have found that one of the solutions is $a=3$ and $b=-2$, while with $a-bi$, you would have found the same solution as $a=3$ and $b=2$ instead. Either way is fine in this case.
2) As Centy said, in more complicated versions of this question, $a+bi$ is often simpler to use.

[quote="mathgeniuse^ln(x)"]This is sort of random but since we are talking about complex numbers, what does magnitude mean?[/quote]

The easy way to think about it is plotting a complex number on a coordinate grid.  The x-axis is the real component and the y-axis is the imaginary component.

Then the magnitude (or absolute value) of a complex number is its distance from the origin.

(This fits the definition of absolute value of a real number perfectly, by the way, which is why it's so nice and intuitive.)

Useful fact:  When multiplying two complex numbers, the product of the magnitudes is equal to the magnitude of the product.  :)


linear algebra
matrix
algebra
polynomial
superior algebra
superior algebra unsolved

Let K be an extension of the field F.
Suppose that a,b in K such that [F(a):F] =n , and [F(b):F]=m
and (n,m)=1

Show that F(a,b)=F(a+b)

--------------------------------
F(a+b)< F(a,b) is obvious...
but I don't know how to show a and b can be expressed as a linear combination in F(a+b)

It's enough to show that the two extensions have the same degree over $ F$.  The linear transformation $ x \mapsto (a \plus{} b)x$ in $ F(a, b)$ has a matrix and a characteristic polynomial.  What is its degree?

i) Prove that $ [F(a,b): F] \equal{} mn$.

ii) Let $ p_b(x)$ be the minimum polynomial of $ b$ over $ F$ and define $ p_{a \plus{} b}(x) \equal{} p_b(x \minus{} a) \in F(a)[x]$. Then $ p_{a \plus{} b}$ is irreducible (prove this!) and of degree $ m$, so $ [F(a \plus{} b): F] \equal{} [F(a \plus{} b): F(a)][F(a): F] \equal{} mn$.

iii) On the other hand is $ F(a \plus{} b) \subset F(a,b)$ such that by i) it follows that $ F(a,b) \equal{} F(a \plus{} b)$.

@yannick: This only shows $ m\equal{}[F(a,a \plus{} b): F(a)]\le [F(a \plus{} b): F]$, but it doesn't divide in general.
This problem is probably wrong, but it's very hard for me to find a counterexample.

it should be true in the case of char(F)=0, but there's only a proof using representation theory,
http://www.mathlinks.ro/Forum/viewtopic.php?t=5828
@t0rajir0u: could you provide the details? ;)