OK, so here's what you do. Everyone says three words each, and we make a story! Let me start.

The first time

The first time AoPS failed due [b]to large amounts[/b]

The first time AoPS failed due to large amounts of  :D s was [b]one day when[/b]

The first time AoPS failed due to large amounts of  :Ds was one day when [b]evilhamster ate a[/b]

The first time AoPS failed due to large amounts of  :D s was one day when evilhamster ate a really big chili dog with  :D, [b]but then Arvind_sn[/b]

The first time AoPS failed due to large amounts of  :D s was one day when evilhamster ate a really big chili dog with  :D, but then Arvind_sn got pwned because

The first time AoPS failed due to large amounts of  :D s was one day when evilhamster ate a really big chili dog with  :D , but then Arvind_sn got pwned because it was too intense for qwertythecucumber. [b]Lotsofmath started playing[/b]

The first time AoPS failed due to large amounts of  :D s was one day when evilhamster ate a really big chili dog with  :D , but then Arvind_sn got pwned because it was too intense for qwertythecucumber. Lotsofmath started playing [b]basketball with his[/b]

The first time AoPS failed due to large amounts of :Ds was one day when evilhamster ate a really big chili dog with :D, but then Arvind_sn got pwned because it was too intense for qwertythecucumber. Lotsofmath started playing basketball with his pants off until

The first time AoPS failed due to large amounts of  Mr. Greens was one day when evilhamster ate a really big chili dog with Mr. Green, but then Arvind_sn got pwned because it was too intense for qwertythecucumber. Lotsofmath started playing basketball with his pants off until he was slapped by a cop. The cop arrested lotsofmath for indecent expose of the...

This was reported to aleph0 when Wickedestjr, another cop, let lotsofmath go. Arvind_sn turned into a huge pikachu and used thunder and incinerated louiethechewy123 and also had apple bottom in his power because of mutation. Wickedestjr commented on how crazy this huge pikachu was. But Arvind_sn turned ytrewq into an antagonistic meerkat because he felt like eating a meerkat. He ended up getting pwned by the last samurai without his shirt. Then Arvind_sn's monkey boisterously screeched and ate itself. Arvind_sn got very angry because he didn't get to eat his monkey, so he ate aleph0 in his dream. Eggylv999 won the Mathcounts National Failureship, and mousy won second place  :winner_second:  by beating eggs. Eggylv999 pwned mousy, but blucarneal came and kicked Godzilla, who then punched bluecarneal's nose and stepped on mousy. Then mousy ran into Godzilla's mouth and Godzilla turned and tripped over Dojo's miniscule brain. Mousy escaped and suicided. There was A BIG COW that punched tcjy in the face. A balrog said "Jesus revived mousy!" Then a dog ate Dojo's "brain". The dog then barfed up Dojo's brain. THE END.

Wait there's more- no there isn't. Actually mousy died and Eggylv999 said, "I'll go read Aops now." Then bluecarneal ran into him, so everyone died. THE FTW CHAMPIONSHIP started and lotsofmath won along with bluecarneal, his teammate. Lotsofmath gave his trophy to bluecarneal because he cheated and
was very sorry. Then, fooblah came and died. Gross monsters ransacked the whole world. Nike was very angry. Then bluecarneal pwned punctual precocious platypi. Then pianogirl's ravenous dinosaurs reincarnated fooblah then devoured 1=2 and pianogirl. Bluecarneal was very scared. pi didn't exist. Then came tcjy and tcjv to ytrewq said "You eat pie well." Then PowerOfPi said "mousy was revived!" And then suddenly, MM appeared from the thirteenth dimension along with PowerOfPi, and then they were pwnt. :cleaning:  Actually, they ate lots of pie filled with Leopluredons and then with poisonous mushrooms. Bluecarneal pwned everybody but algebra2000 tasered himself accidently. Then algebra2000 ate bluecarneal. :ninja:  pwned mousy. Mousy died again and 1=2 said, "ALGEBRA2000 PWNS ALL!" i mean YTREWQ. Myyellowducky82 rocks and only algebra2000 pwns. Then algebra2000 died in a right...ALGEBRA2000 STILL PWNS!

please include this to make the three word game thing very long :)  Error: no sense.

My name is ytrewq. I like pie. Myyellowducky82 is just plain weird. Actually myyellowducky82 rocks. And ytrewq pwns Myyellowducky82 and eats pianogirl's strange looking no adorable chihuahua. But then :bruce: decided to meet  :agent: at the algebra2000 pwnage store. Then they both got algebra2000-style pwned and drycleaned their algebra2000-style pwned hats. Ytrewq pwnd  :bruce: ; algebra2000 pwned ALL!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! except for ytrewq. HE WAS SUPER-PWNED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! No  :moose: was! Then ytrewq dicided to spell correctly.
NO HE DIDRN'T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
YES HE DID!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
THEN THEY DIED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
NO HE DIDN"T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
ONLY TCJV DIED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
AND ALGEBRA2000 TOO!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
NOT HE ISN"T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
EXCLAMATION POINTS=SPAM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
No exclamation points. DOWN WITH SPAM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

On June 12th a happy mushroom, an emo mushroom, cut itself. Badly. Then is said "What the  :censored:, HAHA IM EMO! " The happy mushroom was actually emo. He shot himself to death because he phailed USAMO. [b]The happy/emo mushroom[/b] jumped off an MOTHER****IN BOAT! YEAH!

WTH his mother WAS A BOAT!

THE END, REALLY. BE QUIET TCJV!!!

TCJV walked away to eat a something big, weird, purple, hairy acrylic; a 42-tuple burger.

TCJV walked away to eat a something big, weird, purple, hairy acrylic; a 42-tuple burger.
Then algebra2000 decided: travelling to  :D  Land. Algebra2000 then decided to go to
------------------------------------------------------
We are currently out of smilies. If you want to put another smilie in you will have to replace one of the first ones with something else.

The first time AoPS failed due to large amounts of  Mr. Greens was one day when evilhamster ate a really big chili dog with Mr. Green, but then Arvind_sn got pwned because it was too intense for qwertythecucumber. Lotsofmath started playing basketball with his pants off until he was slapped by a cop. The cop arrested lotsofmath for indecent expose of the...

This was reported to aleph0 when Wickedestjr, another cop, let lotsofmath go. Arvind_sn turned into a huge pikachu and used thunder and incinerated louiethechewy123 and also had apple bottom in his power because of mutation. Wickedestjr commented on how crazy this huge pikachu was. But Arvind_sn turned ytrewq into an antagonistic meerkat because he felt like eating a meerkat. He ended up getting pwned by the last samurai without his shirt. Then Arvind_sn's monkey boisterously screeched and ate itself. Arvind_sn got very angry because he didn't get to eat his monkey, so he ate aleph0 in his dream. Eggylv999 won the Mathcounts National Failureship, and mousy won second place  :winner_second:  by beating eggs. Eggylv999 pwned mousy, but blucarneal came and kicked Godzilla, who then punched bluecarneal's nose and stepped on mousy. Then mousy ran into Godzilla's mouth and Godzilla turned and tripped over Dojo's miniscule brain. Mousy escaped and suicided. There was A BIG COW that punched tcjy in the face. A balrog said "Jesus revived mousy!" Then a dog ate Dojo's "brain". The dog then barfed up Dojo's brain. THE END.

Wait there's more- no there isn't. Actually mousy died and Eggylv999 said, "I'll go read Aops now." Then bluecarneal ran into him, so everyone died. THE FTW CHAMPIONSHIP started and lotsofmath won along with bluecarneal, his teammate. Lotsofmath gave his trophy to bluecarneal because he cheated and
was very sorry. Then, fooblah came and died. Gross monsters ransacked the whole world. Nike was very angry. Then bluecarneal pwned punctual precocious platypi. Then pianogirl's ravenous dinosaurs reincarnated fooblah then devoured 1=2 and pianogirl. Bluecarneal was very scared. pi didn't exist. Then came tcjy and tcjv to ytrewq said "You eat pie well." Then PowerOfPi said "mousy was revived!" And then suddenly, MM appeared from the thirteenth dimension along with PowerOfPi, and then they were pwnt. :cleaning:  Actually, they ate lots of pie filled with Leopluredons and then with poisonous mushrooms. Bluecarneal pwned everybody but algebra2000 tasered himself accidently. Then algebra2000 ate bluecarneal. :ninja:  pwned mousy. Mousy died again and 1=2 said, "ALGEBRA2000 PWNS ALL!" i mean YTREWQ. Myyellowducky82 rocks and only algebra2000 pwns. Then algebra2000 died in a right...ALGEBRA2000 STILL PWNS!

please include this to make the three word game thing very long :)  Error: no sense.

My name is ytrewq. I like pie. Myyellowducky82 is just plain weird. Actually myyellowducky82 rocks. And ytrewq pwns Myyellowducky82 and eats pianogirl's strange looking no adorable chihuahua. But then :bruce: decided to meet  :agent: at the algebra2000 pwnage store. Then they both got algebra2000-style pwned and drycleaned their algebra2000-style pwned hats. Ytrewq pwnd  :bruce: ; algebra2000 pwned ALL!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! except for ytrewq. HE WAS SUPER-PWNED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! No  :moose: was! Then ytrewq dicided to spell correctly.
NO HE DIDRN'T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
YES HE DID!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
THEN THEY DIED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
NO HE DIDN"T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
ONLY TCJV DIED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
AND ALGEBRA2000 TOO!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
NOT HE ISN"T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
EXCLAMATION POINTS=SPAM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
No exclamation points. DOWN WITH SPAM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

On June 12th a happy mushroom, an emo mushroom, cut itself. Badly. Then is said "What the  :censored:, HAHA IM EMO! " The happy mushroom was actually emo. He shot himself to death because he phailed USAMO. [b]The happy/emo mushroom[/b] jumped off an MOTHER****IN BOAT! YEAH!

WTH his mother WAS A BOAT!

THE END, REALLY. BE QUIET TCJV!!!

TCJV walked away to eat a something big, weird, purple, hairy acrylic; a 42-tuple burger.

TCJV walked away to eat a something big, weird, purple, hairy acrylic; a 42-tuple burger.
Then algebra2000 decided: travelling to  :D  Land. Algebra2000 then decided to go to apologize to tcjv.
------------------------------------------------------
We are currently out of smilies. If you want to put another smilie in you will have to replace one of the first ones with something else.

The first time AoPS failed due to large amounts of  Mr. Greens was one day when evilhamster ate a really big chili dog with Mr. Green, but then Arvind_sn got pwned because it was too intense for qwertythecucumber. Lotsofmath started playing basketball with his pants off until he was slapped by a cop. The cop arrested lotsofmath for indecent expose of the...

This was reported to aleph0 when Wickedestjr, another cop, let lotsofmath go. Arvind_sn turned into a huge pikachu and used thunder and incinerated louiethechewy123 and also had apple bottom in his power because of mutation. Wickedestjr commented on how crazy this huge pikachu was. But Arvind_sn turned ytrewq into an antagonistic meerkat because he felt like eating a meerkat. He ended up getting pwned by the last samurai without his shirt. Then Arvind_sn's monkey boisterously screeched and ate itself. Arvind_sn got very angry because he didn't get to eat his monkey, so he ate aleph0 in his dream. Eggylv999 won the Mathcounts National Failureship, and mousy won second place  :winner_second:  by beating eggs. Eggylv999 pwned mousy, but blucarneal came and kicked Godzilla, who then punched bluecarneal's nose and stepped on mousy. Then mousy ran into Godzilla's mouth and Godzilla turned and tripped over Dojo's miniscule brain. Mousy escaped and suicided. There was A BIG COW that punched tcjy in the face. A balrog said "Jesus revived mousy!" Then a dog ate Dojo's "brain". The dog then barfed up Dojo's brain. THE END.

Wait there's more- no there isn't. Actually mousy died and Eggylv999 said, "I'll go read Aops now." Then bluecarneal ran into him, so everyone died. THE FTW CHAMPIONSHIP started and lotsofmath won along with bluecarneal, his teammate. Lotsofmath gave his trophy to bluecarneal because he cheated and
was very sorry. Then, fooblah came and died. Gross monsters ransacked the whole world. Nike was very angry. Then bluecarneal pwned punctual precocious platypi. Then pianogirl's ravenous dinosaurs reincarnated fooblah then devoured 1=2 and pianogirl. Bluecarneal was very scared. pi didn't exist. Then came tcjy and tcjv to ytrewq said "You eat pie well." Then PowerOfPi said "mousy was revived!" And then suddenly, MM appeared from the thirteenth dimension along with PowerOfPi, and then they were pwnt. :cleaning:  Actually, they ate lots of pie filled with Leopluredons and then with poisonous mushrooms. Bluecarneal pwned everybody but algebra2000 tasered himself accidently. Then algebra2000 ate bluecarneal. :ninja:  pwned mousy. Mousy died again and 1=2 said, "ALGEBRA2000 PWNS ALL!" i mean YTREWQ. Myyellowducky82 rocks and only algebra2000 pwns. Then algebra2000 died in a right...ALGEBRA2000 STILL PWNS!

please include this to make the three word game thing very long :)  Error: no sense.

My name is ytrewq. I like pie. Myyellowducky82 is just plain weird. Actually myyellowducky82 rocks. And ytrewq pwns Myyellowducky82 and eats pianogirl's strange looking no adorable chihuahua. But then :bruce: decided to meet  :agent: at the algebra2000 pwnage store. Then they both got algebra2000-style pwned and drycleaned their algebra2000-style pwned hats. Ytrewq pwnd  :bruce: ; algebra2000 pwned ALL!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! except for ytrewq. HE WAS SUPER-PWNED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! No  :moose: was! Then ytrewq dicided to spell correctly.
NO HE DIDRN'T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
YES HE DID!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
THEN THEY DIED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
NO HE DIDN"T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
ONLY TCJV DIED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
AND ALGEBRA2000 TOO!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
NOT HE ISN"T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
EXCLAMATION POINTS=SPAM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
No exclamation points. DOWN WITH SPAM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

On June 12th a happy mushroom, an emo mushroom, cut itself. Badly. Then is said "What the  :censored:, HAHA IM EMO! " The happy mushroom was actually emo. He shot himself to death because he phailed USAMO. [b]The happy/emo mushroom[/b] jumped off an MOTHER****IN BOAT! YEAH!

WTH his mother WAS A BOAT!

THE END, REALLY. BE QUIET TCJV!!!

TCJV walked away to eat a something big, weird, purple, hairy acrylic; a 42-tuple burger.

TCJV walked away to eat a something big, weird, purple, hairy acrylic; a 42-tuple burger.
Then algebra2000 decided: travelling to  :D  Land. Algebra2000 then decided to go to apologize to tcjv.

[b]But then Dragon96[/b] decided to lick a [quote] tag
------------------------------------------------------
We are currently out of smilies. If you want to put another smilie in you will have to replace one of the first ones with something else.

The first time AoPS failed due to large amounts of Mr. Greens was one day when evilhamster ate a really big chili dog with Mr. Green, but then Arvind_sn got pwned because it was too intense for qwertythecucumber. Lotsofmath started playing basketball with his pants off until he was slapped by a cop. The cop arrested lotsofmath for indecent expose of the...

This was reported to aleph0 when Wickedestjr, another cop, let lotsofmath go. Arvind_sn turned into a huge pikachu and used thunder and incinerated louiethechewy123 and also had apple bottom in his power because of mutation. Wickedestjr commented on how crazy this huge pikachu was. But Arvind_sn turned ytrewq into an antagonistic meerkat because he felt like eating a meerkat. He ended up getting pwned by the last samurai without his shirt. Then Arvind_sn's monkey boisterously screeched and ate itself. Arvind_sn got very angry because he didn't get to eat his monkey, so he ate aleph0 in his dream. Eggylv999 won the Mathcounts National Failureship, and mousy won second place Winner - Second by beating eggs. Eggylv999 pwned mousy, but blucarneal came and kicked Godzilla, who then punched bluecarneal's nose and stepped on mousy. Then mousy ran into Godzilla's mouth and Godzilla turned and tripped over Dojo's miniscule brain. Mousy escaped and suicided. There was A BIG COW that punched tcjy in the face. A balrog said "Jesus revived mousy!" Then a dog ate Dojo's "brain". The dog then barfed up Dojo's brain. THE END.

Wait there's more- no there isn't. Actually mousy died and Eggylv999 said, "I'll go read Aops now." Then bluecarneal ran into him, so everyone died. THE FTW CHAMPIONSHIP started and lotsofmath won along with bluecarneal, his teammate. Lotsofmath gave his trophy to bluecarneal because he cheated and
was very sorry. Then, fooblah came and died. Gross monsters ransacked the whole world. Nike was very angry. Then bluecarneal pwned punctual precocious platypi. Then pianogirl's ravenous dinosaurs reincarnated fooblah then devoured 1=2 and pianogirl. Bluecarneal was very scared. pi didn't exist. Then came tcjy and tcjv to ytrewq said "You eat pie well." Then PowerOfPi said "mousy was revived!" And then suddenly, MM appeared from the thirteenth dimension along with PowerOfPi, and then they were pwnt. Cleaning Actually, they ate lots of pie filled with Leopluredons and then with poisonous mushrooms. Bluecarneal pwned everybody but algebra2000 tasered himself accidently. Then algebra2000 ate bluecarneal. Ninja pwned mousy. Mousy died again and 1=2 said, "ALGEBRA2000 PWNS ALL!" i mean YTREWQ. Myyellowducky82 rocks and only algebra2000 pwns. Then algebra2000 died in a right...ALGEBRA2000 STILL PWNS!

please include this to make the three word game thing very long Smile Error: no sense.

My name is ytrewq. I like pie. Myyellowducky82 is just plain weird. Actually myyellowducky82 rocks. And ytrewq pwns Myyellowducky82 and eats pianogirl's strange looking no adorable chihuahua. But then Bruce Lee decided to meet Agent at the algebra2000 pwnage store. Then they both got algebra2000-style pwned and drycleaned their algebra2000-style pwned hats. Ytrewq pwnd Bruce Lee ; algebra2000 pwned ALL!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! except for ytrewq. HE WAS SUPER-PWNED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! No Moose was! Then ytrewq dicided to spell correctly.
NO HE DIDRN'T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
YES HE DID!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
THEN THEY DIED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
NO HE DIDN"T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
ONLY TCJV DIED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
AND ALGEBRA2000 TOO!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
NOT HE ISN"T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
EXCLAMATION POINTS=SPAM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
No exclamation points. DOWN WITH SPAM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

On June 12th a happy mushroom, an emo mushroom, cut itself. Badly. Then is said "What the Censored, HAHA IM EMO! " The happy mushroom was actually emo. He shot himself to death because he phailed USAMO. The happy/emo mushroom jumped off an MOTHER****IN BOAT! YEAH!

WTH his mother WAS A BOAT!

THE END, REALLY. BE QUIET TCJV!!!

TCJV walked away to eat a something big, weird, purple, hairy acrylic; a 42-tuple burger.

TCJV walked away to eat a something big, weird, purple, hairy acrylic; a 42-tuple burger.
Then algebra2000 decided: travelling to Mr. Green Land. Algebra2000 then decided to go to apologize to tcjv.

But then Dragon96 decided to lick a [quote] tag and farted ilovepink.

[quote="iheartchemistry2010"][quote="ytrewq"]The first time AoPS failed due to large amounts of  Mr. Greens was one day when evilhamster ate a really big chili dog with Mr. Green, but then Arvind_sn got pwned because it was too intense for qwertythecucumber. Lotsofmath started playing basketball with his pants off until he was slapped by a cop. The cop arrested lotsofmath for indecent expose of the...

This was reported to aleph0 when Wickedestjr, another cop, let lotsofmath go. Arvind_sn turned into a huge pikachu and used thunder and incinerated louiethechewy123 and also had apple bottom in his power because of mutation. Wickedestjr commented on how crazy this huge pikachu was. But Arvind_sn turned ytrewq into an antagonistic meerkat because he felt like eating a meerkat. He ended up getting pwned by the last samurai without his shirt. Then Arvind_sn's monkey boisterously screeched and ate itself. Arvind_sn got very angry because he didn't get to eat his monkey, so he ate aleph0 in his dream. Eggylv999 won the Mathcounts National Failureship, and mousy won second place  :winner_second:  by beating eggs. Eggylv999 pwned mousy, but blucarneal came and kicked Godzilla, who then punched bluecarneal's nose and stepped on mousy. Then mousy ran into Godzilla's mouth and Godzilla turned and tripped over Dojo's miniscule brain. Mousy escaped and suicided. There was A BIG COW that punched tcjy in the face. A balrog said "Jesus revived mousy!" Then a dog ate Dojo's "brain". The dog then barfed up Dojo's brain. THE END.

Wait there's more- no there isn't. Actually mousy died and Eggylv999 said, "I'll go read Aops now." Then bluecarneal ran into him, so everyone died. THE FTW CHAMPIONSHIP started and lotsofmath won along with bluecarneal, his teammate. Lotsofmath gave his trophy to bluecarneal because he cheated and
was very sorry. Then, fooblah came and died. Gross monsters ransacked the whole world. Nike was very angry. Then bluecarneal pwned punctual precocious platypi. Then pianogirl's ravenous dinosaurs reincarnated fooblah then devoured 1=2 and pianogirl. Bluecarneal was very scared. pi didn't exist. Then came tcjy and tcjv to ytrewq said "You eat pie well." Then PowerOfPi said "mousy was revived!" And then suddenly, MM appeared from the thirteenth dimension along with PowerOfPi, and then they were pwnt. :cleaning:  Actually, they ate lots of pie filled with Leopluredons and then with poisonous mushrooms. Bluecarneal pwned everybody but algebra2000 tasered himself accidently. Then algebra2000 ate bluecarneal. :ninja:  pwned mousy. Mousy died again and 1=2 said, "ALGEBRA2000 PWNS ALL!" i mean YTREWQ. Myyellowducky82 rocks and only algebra2000 pwns. Then algebra2000 died in a right...ALGEBRA2000 STILL PWNS!

please include this to make the three word game thing very long :)  Error: no sense.

My name is ytrewq. I like pie. Myyellowducky82 is just plain weird. Actually myyellowducky82 rocks. And ytrewq pwns Myyellowducky82 and eats pianogirl's strange looking no adorable chihuahua. But then :bruce: decided to meet  :agent: at the algebra2000 pwnage store. Then they both got algebra2000-style pwned and drycleaned their algebra2000-style pwned hats. Ytrewq pwnd  :bruce: ; algebra2000 pwned ALL!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! except for ytrewq. HE WAS SUPER-PWNED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! No  :moose: was! Then ytrewq dicided to spell correctly.
NO HE DIDRN'T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
YES HE DID!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
THEN THEY DIED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
NO HE DIDN"T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
ONLY TCJV DIED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
AND ALGEBRA2000 TOO!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
NOT HE ISN"T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
EXCLAMATION POINTS=SPAM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
No exclamation points. DOWN WITH SPAM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

On June 12th a happy mushroom, an emo mushroom, cut itself. Badly. Then is said "What the  :censored:, HAHA IM EMO! " The happy mushroom was actually emo. He shot himself to death because he phailed USAMO. [b]The happy/emo mushroom[/b] jumped off an MOTHER****IN BOAT! YEAH!

WTH his mother WAS A BOAT!

THE END, REALLY. BE QUIET TCJV!!!

TCJV walked away to eat a something big, weird, purple, hairy acrylic; a 42-tuple burger.

TCJV walked away to eat a something big, weird, purple, hairy acrylic; a 42-tuple burger.
Then algebra2000 decided: travelling to  :D  Land. Algebra2000 then decided to go to apologize to tcjv.

[b]But then Dragon96[/b] decided to lick a [quote] tag that loved meerkats
------------------------------------------------------
We are currently out of smilies. If you want to put another smilie in you will have to replace one of the first ones with something else.[/quote][/quote][/quote]

I'M SERIOUS! -----END-----

The first time AoPS failed due to large amounts of  Mr. Greens was one day when evilhamster ate a really big chili dog with Mr. Green, but then Arvind_sn got pwned because it was too intense for qwertythecucumber. Lotsofmath started playing basketball with his pants off until he was slapped by a cop. The cop arrested lotsofmath for indecent expose of the...

This was reported to aleph0 when Wickedestjr, another cop, let lotsofmath go. Arvind_sn turned into a huge pikachu and used thunder and incinerated louiethechewy123 and also had apple bottom in his power because of mutation. Wickedestjr commented on how crazy this huge pikachu was. But Arvind_sn turned ytrewq into an antagonistic meerkat because he felt like eating a meerkat. He ended up getting pwned by the last samurai without his shirt. Then Arvind_sn's monkey boisterously screeched and ate itself. Arvind_sn got very angry because he didn't get to eat his monkey, so he ate aleph0 in his dream. Eggylv999 won the Mathcounts National Failureship, and mousy won second place  :winner_second:  by beating eggs. Eggylv999 pwned mousy, but blucarneal came and kicked Godzilla, who then punched bluecarneal's nose and stepped on mousy. Then mousy ran into Godzilla's mouth and Godzilla turned and tripped over Dojo's miniscule brain. Mousy escaped and suicided. There was A BIG COW that punched tcjy in the face. A balrog said "Jesus revived mousy!" Then a dog ate Dojo's "brain". The dog then barfed up Dojo's brain. THE END.

Wait there's more- no there isn't. Actually mousy died and Eggylv999 said, "I'll go read Aops now." Then bluecarneal ran into him, so everyone died. THE FTW CHAMPIONSHIP started and lotsofmath won along with bluecarneal, his teammate. Lotsofmath gave his trophy to bluecarneal because he cheated and
was very sorry. Then, fooblah came and died. Gross monsters ransacked the whole world. Nike was very angry. Then bluecarneal pwned punctual precocious platypi. Then pianogirl's ravenous dinosaurs reincarnated fooblah then devoured 1=2 and pianogirl. Bluecarneal was very scared. pi didn't exist. Then came tcjy and tcjv to ytrewq said "You eat pie well." Then PowerOfPi said "mousy was revived!" And then suddenly, MM appeared from the thirteenth dimension along with PowerOfPi, and then they were pwnt. :cleaning:  Actually, they ate lots of pie filled with Leopluredons and then with poisonous mushrooms. Bluecarneal pwned everybody but algebra2000 tasered himself accidently. Then algebra2000 ate bluecarneal. :ninja:  pwned mousy. Mousy died again and 1=2 said, "ALGEBRA2000 PWNS ALL!" i mean YTREWQ. Myyellowducky82 rocks and only algebra2000 pwns. Then algebra2000 died in a right...ALGEBRA2000 STILL PWNS!

please include this to make the three word game thing very long :)  Error: no sense.

My name is ytrewq. I like pie. Myyellowducky82 is just plain weird. Actually myyellowducky82 rocks. And ytrewq pwns Myyellowducky82 and eats pianogirl's strange looking no adorable chihuahua. But then :bruce: decided to meet  :agent: at the algebra2000 pwnage store. Then they both got algebra2000-style pwned and drycleaned their algebra2000-style pwned hats. Ytrewq pwnd  :bruce: ; algebra2000 pwned ALL!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! except for ytrewq. HE WAS SUPER-PWNED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! No  :moose: was! Then ytrewq dicided to spell correctly.
NO HE DIDRN'T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
YES HE DID!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
THEN THEY DIED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
NO HE DIDN"T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
ONLY TCJV DIED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
AND ALGEBRA2000 TOO!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
NOT HE ISN"T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
EXCLAMATION POINTS=SPAM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
No exclamation points. DOWN WITH SPAM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

On June 12th a happy mushroom, an emo mushroom, cut itself. Badly. Then is said "What the  :censored:, HAHA IM EMO! " The happy mushroom was actually emo. He shot himself to death because he phailed USAMO. [b]The happy/emo mushroom[/b] jumped off an MOTHER****IN BOAT! YEAH!

WTH his mother WAS A BOAT!

THE END, REALLY. BE QUIET TCJV!!!

TCJV walked away to eat a something big, weird, purple, hairy acrylic; a 42-tuple burger.

TCJV walked away to eat a something big, weird, purple, hairy acrylic; a 42-tuple burger.
Then algebra2000 decided: travelling to  :D  Land. Algebra2000 then decided to go to apologize to tcjv.

[b]But then Dragon96[/b] decided to lick a [quote ] tag that loved meerkats
------------------------------------------------------
We are currently out of smilies. If you want to put another smilie in you will have to replace one of the first ones with something else.

I'M SERIOUS! -----END-----
DUDE WTH NO!

The first time AoPS failed due to large amounts of Mr. Greens was one day when evilhamster ate a really big chili dog with Mr. Green, but then Arvind_sn got pwned because it was too intense for qwertythecucumber. Lotsofmath started playing basketball with his pants off until he was slapped by a cop. The cop arrested lotsofmath for indecent expose of the...

This was reported to aleph0 when Wickedestjr, another cop, let lotsofmath go. Arvind_sn turned into a huge pikachu and used thunder and incinerated louiethechewy123 and also had apple bottom in his power because of mutation. Wickedestjr commented on how crazy this huge pikachu was. But Arvind_sn turned ytrewq into an antagonistic meerkat because he felt like eating a meerkat. He ended up getting pwned by the last samurai without his shirt. Then Arvind_sn's monkey boisterously screeched and ate itself. Arvind_sn got very angry because he didn't get to eat his monkey, so he ate aleph0 in his dream. Eggylv999 won the Mathcounts National Failureship, and mousy won second place Winner - Second by beating eggs. Eggylv999 pwned mousy, but blucarneal came and kicked Godzilla, who then punched bluecarneal's nose and stepped on mousy. Then mousy ran into Godzilla's mouth and Godzilla turned and tripped over Dojo's miniscule brain. Mousy escaped and suicided. There was A BIG COW that punched tcjy in the face. A balrog said "Jesus revived mousy!" Then a dog ate Dojo's "brain". The dog then barfed up Dojo's brain. THE END.

Wait there's more- no there isn't. Actually mousy died and Eggylv999 said, "I'll go read Aops now." Then bluecarneal ran into him, so everyone died. THE FTW CHAMPIONSHIP started and lotsofmath won along with bluecarneal, his teammate. Lotsofmath gave his trophy to bluecarneal because he cheated and
was very sorry. Then, fooblah came and died. Gross monsters ransacked the whole world. Nike was very angry. Then bluecarneal pwned punctual precocious platypi. Then pianogirl's ravenous dinosaurs reincarnated fooblah then devoured 1=2 and pianogirl. Bluecarneal was very scared. pi didn't exist. Then came tcjy and tcjv to ytrewq said "You eat pie well." Then PowerOfPi said "mousy was revived!" And then suddenly, MM appeared from the thirteenth dimension along with PowerOfPi, and then they were pwnt. Cleaning Actually, they ate lots of pie filled with Leopluredons and then with poisonous mushrooms. Bluecarneal pwned everybody but algebra2000 tasered himself accidently. Then algebra2000 ate bluecarneal. Ninja pwned mousy. Mousy died again and 1=2 said, "ALGEBRA2000 PWNS ALL!" i mean YTREWQ. Myyellowducky82 rocks and only algebra2000 pwns. Then algebra2000 died in a right...ALGEBRA2000 STILL PWNS!

please include this to make the three word game thing very long Smile Error: no sense.

My name is ytrewq. I like pie. Myyellowducky82 is just plain weird. Actually myyellowducky82 rocks. And ytrewq pwns Myyellowducky82 and eats pianogirl's strange looking no adorable chihuahua. But then Bruce Lee decided to meet Agent at the algebra2000 pwnage store. Then they both got algebra2000-style pwned and drycleaned their algebra2000-style pwned hats. Ytrewq pwnd Bruce Lee ; algebra2000 pwned ALL!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! except for ytrewq. HE WAS SUPER-PWNED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! No Moose was! Then ytrewq dicided to spell correctly.
NO HE DIDRN'T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
YES HE DID!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
THEN THEY DIED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
NO HE DIDN"T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
ONLY TCJV DIED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
AND ALGEBRA2000 TOO!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
NOT HE ISN"T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
EXCLAMATION POINTS=SPAM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
No exclamation points. DOWN WITH SPAM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

On June 12th a happy mushroom, an emo mushroom, cut itself. Badly. Then is said "What the Censored, HAHA IM EMO! " The happy mushroom was actually emo. He shot himself to death because he phailed USAMO. The happy/emo mushroom jumped off an MOTHER****IN BOAT! YEAH!

WTH his mother WAS A BOAT!

THE END, REALLY. BE QUIET TCJV!!!

TCJV walked away to eat a something big, weird, purple, hairy acrylic; a 42-tuple burger.

TCJV walked away to eat a something big, weird, purple, hairy acrylic; a 42-tuple burger.
Then algebra2000 decided: travelling to Mr. Green Land. Algebra2000 then decided to go to apologize to tcjv.

[b]But then Dragon96[/b] decided to lick a [quote ] tag that loved meerkats AND tcj[size=200][b]V[/b][/size] Can't

The first time AoPS failed due to large amounts of Mr. Greens was one day when evilhamster ate a really big chili dog with Mr. Green, but then Arvind_sn got pwned because it was too intense for qwertythecucumber. Lotsofmath started playing basketball with his pants off until he was slapped by a cop. The cop arrested lotsofmath for indecent expose of the...

This was reported to aleph0 when Wickedestjr, another cop, let lotsofmath go. Arvind_sn turned into a huge pikachu and used thunder and incinerated louiethechewy123 and also had apple bottom in his power because of mutation. Wickedestjr commented on how crazy this huge pikachu was. But Arvind_sn turned ytrewq into an antagonistic meerkat because he felt like eating a meerkat. He ended up getting pwned by the last samurai without his shirt. Then Arvind_sn's monkey boisterously screeched and ate itself. Arvind_sn got very angry because he didn't get to eat his monkey, so he ate aleph0 in his dream. Eggylv999 won the Mathcounts National Failureship, and mousy won second place Winner - Second by beating eggs. Eggylv999 pwned mousy, but blucarneal came and kicked Godzilla, who then punched bluecarneal's nose and stepped on mousy. Then mousy ran into Godzilla's mouth and Godzilla turned and tripped over Dojo's miniscule brain. Mousy escaped and suicided. There was A BIG COW that punched tcjy in the face. A balrog said "Jesus revived mousy!" Then a dog ate Dojo's "brain". The dog then barfed up Dojo's brain. THE END.

Wait there's more- no there isn't. Actually mousy died and Eggylv999 said, "I'll go read Aops now." Then bluecarneal ran into him, so everyone died. THE FTW CHAMPIONSHIP started and lotsofmath won along with bluecarneal, his teammate. Lotsofmath gave his trophy to bluecarneal because he cheated and
was very sorry. Then, fooblah came and died. Gross monsters ransacked the whole world. Nike was very angry. Then bluecarneal pwned punctual precocious platypi. Then pianogirl's ravenous dinosaurs reincarnated fooblah then devoured 1=2 and pianogirl. Bluecarneal was very scared. pi didn't exist. Then came tcjy and tcjv to ytrewq said "You eat pie well." Then PowerOfPi said "mousy was revived!" And then suddenly, MM appeared from the thirteenth dimension along with PowerOfPi, and then they were pwnt. Cleaning Actually, they ate lots of pie filled with Leopluredons and then with poisonous mushrooms. Bluecarneal pwned everybody but algebra2000 tasered himself accidently. Then algebra2000 ate bluecarneal. Ninja pwned mousy. Mousy died again and 1=2 said, "ALGEBRA2000 PWNS ALL!" i mean YTREWQ. Myyellowducky82 rocks and only algebra2000 pwns. Then algebra2000 died in a right...ALGEBRA2000 STILL PWNS!

please include this to make the three word game thing very long Smile Error: no sense.

My name is ytrewq. I like pie. Myyellowducky82 is just plain weird. Actually myyellowducky82 rocks. And ytrewq pwns Myyellowducky82 and eats pianogirl's strange looking no adorable chihuahua. But then Bruce Lee decided to meet Agent at the algebra2000 pwnage store. Then they both got algebra2000-style pwned and drycleaned their algebra2000-style pwned hats. Ytrewq pwnd Bruce Lee ; algebra2000 pwned ALL!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! except for ytrewq. HE WAS SUPER-PWNED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! No Moose was! Then ytrewq dicided to spell correctly.
NO HE DIDRN'T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
YES HE DID!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
THEN THEY DIED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
NO HE DIDN"T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
ONLY TCJV DIED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
AND ALGEBRA2000 TOO!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
NOT HE ISN"T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
EXCLAMATION POINTS=SPAM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
No exclamation points. DOWN WITH SPAM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

On June 12th a happy mushroom, an emo mushroom, cut itself. Badly. Then is said "What the Censored, HAHA IM EMO! " The happy mushroom was actually emo. He shot himself to death because he phailed USAMO. The happy/emo mushroom jumped off an MOTHER****IN BOAT! YEAH!

WTH his mother WAS A BOAT!

THE END, REALLY. BE QUIET TCJV!!!

TCJV walked away to eat a something big, weird, purple, hairy acrylic; a 42-tuple burger.

TCJV walked away to eat a something big, weird, purple, hairy acrylic; a 42-tuple burger.
Then algebra2000 decided: travelling to Mr. Green Land. Algebra2000 then decided to go to apologize to tcjv.

But then Dragon96 decided to lick a [quote ] tag that loved meerkats AND tcjV Can't eat a 42-tuple

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

hummm..... where'd all the smilies go?

The first time AoPS failed due to large amounts of Mr. Greens was one day when evilhamster ate a really big chili dog with Mr. Green, but then Arvind_sn got pwned because it was too intense for qwertythecucumber. Lotsofmath started playing basketball with his pants off until he was slapped by a cop. The cop arrested lotsofmath for indecent expose of the...

This was reported to aleph0 when Wickedestjr, another cop, let lotsofmath go. Arvind_sn turned into a huge pikachu and used thunder and incinerated louiethechewy123 and also had apple bottom in his power because of mutation. Wickedestjr commented on how crazy this huge pikachu was. But Arvind_sn turned ytrewq into an antagonistic meerkat because he felt like eating a meerkat. He ended up getting pwned by the last samurai without his shirt. Then Arvind_sn's monkey boisterously screeched and ate itself. Arvind_sn got very angry because he didn't get to eat his monkey, so he ate aleph0 in his dream. Eggylv999 won the Mathcounts National Failureship, and mousy won second place Winner - Second by beating eggs. Eggylv999 pwned mousy, but blucarneal came and kicked Godzilla, who then punched bluecarneal's nose and stepped on mousy. Then mousy ran into Godzilla's mouth and Godzilla turned and tripped over Dojo's miniscule brain. Mousy escaped and suicided. There was A BIG COW that punched tcjy in the face. A balrog said "Jesus revived mousy!" Then a dog ate Dojo's "brain". The dog then barfed up Dojo's brain. THE END.

Wait there's more- no there isn't. Actually mousy died and Eggylv999 said, "I'll go read Aops now." Then bluecarneal ran into him, so everyone died. THE FTW CHAMPIONSHIP started and lotsofmath won along with bluecarneal, his teammate. Lotsofmath gave his trophy to bluecarneal because he cheated and
was very sorry. Then, fooblah came and died. Gross monsters ransacked the whole world. Nike was very angry. Then bluecarneal pwned punctual precocious platypi. Then pianogirl's ravenous dinosaurs reincarnated fooblah then devoured 1=2 and pianogirl. Bluecarneal was very scared. pi didn't exist. Then came tcjy and tcjv to ytrewq said "You eat pie well." Then PowerOfPi said "mousy was revived!" And then suddenly, MM appeared from the thirteenth dimension along with PowerOfPi, and then they were pwnt. Cleaning Actually, they ate lots of pie filled with Leopluredons and then with poisonous mushrooms. Bluecarneal pwned everybody but algebra2000 tasered himself accidently. Then algebra2000 ate bluecarneal. Ninja pwned mousy. Mousy died again and 1=2 said, "ALGEBRA2000 PWNS ALL!" i mean YTREWQ. Myyellowducky82 rocks and only algebra2000 pwns. Then algebra2000 died in a right...ALGEBRA2000 STILL PWNS!

please include this to make the three word game thing very long Smile Error: no sense.

My name is ytrewq. I like pie. Myyellowducky82 is just plain weird. Actually myyellowducky82 rocks. And ytrewq pwns Myyellowducky82 and eats pianogirl's strange looking no adorable chihuahua. But then Bruce Lee decided to meet Agent at the algebra2000 pwnage store. Then they both got algebra2000-style pwned and drycleaned their algebra2000-style pwned hats. Ytrewq pwnd Bruce Lee ; algebra2000 pwned ALL!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! except for ytrewq. HE WAS SUPER-PWNED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! No Moose was! Then ytrewq dicided to spell correctly.
NO HE DIDRN'T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
YES HE DID!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
THEN THEY DIED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
NO HE DIDN"T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
ONLY TCJV DIED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
AND ALGEBRA2000 TOO!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
NOT HE ISN"T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
EXCLAMATION POINTS=SPAM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
No exclamation points. DOWN WITH SPAM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

On June 12th a happy mushroom, an emo mushroom, cut itself. Badly. Then is said "What the Censored, HAHA IM EMO! " The happy mushroom was actually emo. He shot himself to death because he phailed USAMO. The happy/emo mushroom jumped off an MOTHER****IN BOAT! YEAH!

WTH his mother WAS A BOAT!

THE END, REALLY. BE QUIET TCJV!!!

TCJV walked away to eat a something big, weird, purple, hairy acrylic; a 42-tuple burger.

TCJV walked away to eat a something big, weird, purple, hairy acrylic; a 42-tuple burger.
Then algebra2000 decided: travelling to Mr. Green Land. Algebra2000 then decided to go to apologize to tcjv.

But then Dragon96 decided to lick a [quote ] tag that loved meerkats AND tcjV Can't eat a 42-tuple burger.

THE END

+ IT'S NOT THE END

------------------------
the story ^^^^^

(used up tcj[size=200]V[/size] extra word he didn't use  :rotfl:  :D  :lol:  :wink:

The first time AoPS failed due to large amounts of Mr. Greens was one day when evilhamster ate a really big chili dog with Mr. Green, but then Arvind_sn got pwned because it was too intense for qwertythecucumber. Lotsofmath started playing basketball with his pants off until he was slapped by a cop. The cop arrested lotsofmath for indecent expose of the...

This was reported to aleph0 when Wickedestjr, another cop, let lotsofmath go. Arvind_sn turned into a huge pikachu and used thunder and incinerated louiethechewy123 and also had apple bottom in his power because of mutation. Wickedestjr commented on how crazy this huge pikachu was. But Arvind_sn turned ytrewq into an antagonistic meerkat because he felt like eating a meerkat. He ended up getting pwned by the last samurai without his shirt. Then Arvind_sn's monkey boisterously screeched and ate itself. Arvind_sn got very angry because he didn't get to eat his monkey, so he ate aleph0 in his dream. Eggylv999 won the Mathcounts National Failureship, and mousy won second place Winner - Second by beating eggs. Eggylv999 pwned mousy, but blucarneal came and kicked Godzilla, who then punched bluecarneal's nose and stepped on mousy. Then mousy ran into Godzilla's mouth and Godzilla turned and tripped over Dojo's miniscule brain. Mousy escaped and suicided. There was A BIG COW that punched tcjy in the face. A balrog said "Jesus revived mousy!" Then a dog ate Dojo's "brain". The dog then barfed up Dojo's brain. THE END.

Wait there's more- no there isn't. Actually mousy died and Eggylv999 said, "I'll go read Aops now." Then bluecarneal ran into him, so everyone died. THE FTW CHAMPIONSHIP started and lotsofmath won along with bluecarneal, his teammate. Lotsofmath gave his trophy to bluecarneal because he cheated and
was very sorry. Then, fooblah came and died. Gross monsters ransacked the whole world. Nike was very angry. Then bluecarneal pwned punctual precocious platypi. Then pianogirl's ravenous dinosaurs reincarnated fooblah then devoured 1=2 and pianogirl. Bluecarneal was very scared. pi didn't exist. Then came tcjy and tcjv to ytrewq said "You eat pie well." Then PowerOfPi said "mousy was revived!" And then suddenly, MM appeared from the thirteenth dimension along with PowerOfPi, and then they were pwnt. Cleaning Actually, they ate lots of pie filled with Leopluredons and then with poisonous mushrooms. Bluecarneal pwned everybody but algebra2000 tasered himself accidently. Then algebra2000 ate bluecarneal. Ninja pwned mousy. Mousy died again and 1=2 said, "ALGEBRA2000 PWNS ALL!" i mean YTREWQ. Myyellowducky82 rocks and only algebra2000 pwns. Then algebra2000 died in a right...ALGEBRA2000 STILL PWNS!

please include this to make the three word game thing very long Smile Error: no sense.

My name is ytrewq. I like pie. Myyellowducky82 is just plain weird. Actually myyellowducky82 rocks. And ytrewq pwns Myyellowducky82 and eats pianogirl's strange looking no adorable chihuahua. But then Bruce Lee decided to meet Agent at the algebra2000 pwnage store. Then they both got algebra2000-style pwned and drycleaned their algebra2000-style pwned hats. Ytrewq pwnd Bruce Lee ; algebra2000 pwned ALL!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! except for ytrewq. HE WAS SUPER-PWNED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! No Moose was! Then ytrewq dicided to spell correctly.
NO HE DIDRN'T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
YES HE DID!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
THEN THEY DIED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
NO HE DIDN"T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
ONLY TCJV DIED!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
AND ALGEBRA2000 TOO!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
NOT HE ISN"T!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
EXCLAMATION POINTS=SPAM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
No exclamation points. DOWN WITH SPAM!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

On June 12th a happy mushroom, an emo mushroom, cut itself. Badly. Then is said "What the Censored, HAHA IM EMO! " The happy mushroom was actually emo. He shot himself to death because he phailed USAMO. The happy/emo mushroom jumped off an MOTHER****IN BOAT! YEAH!

WTH his mother WAS A BOAT!

THE END, REALLY. BE QUIET TCJV!!!

TCJV walked away to eat a something big, weird, purple, hairy acrylic; a 42-tuple burger.

TCJV walked away to eat a something big, weird, purple, hairy acrylic; a 42-tuple burger.
Then algebra2000 decided: travelling to Mr. Green Land. Algebra2000 then decided to go to apologize to tcjv.

But then Dragon96 decided to lick a [quote ] tag that loved meerkats AND tcjV Can't eat a 42-tuple burger.

THE END

+ IT'S NOT THE END

ytrewq returned from the land of


modular arithmetic
algorithm
Euler
function
search
number theory
algebra

Is this statement true - If a (insert here three lined equal sign - congruent modulo) c (mod n), then a^b (insert here three lined equal sign - congruent modulo) c^b (mod n)?
Morover, is it's converse true?
Can you prove it?
Can anyone recommend a good place to learn about mods?
Thanks!
All help is greatly appreciated.

One can prove the more general statement that if $ a \equiv b \bmod n$ and $ c \equiv d \bmod n$ then $ ac \equiv bd \bmod n$ - in other words, congruence is compatible with multiplication (it is clearly compatible with addition).

Its converse is very false.  For example, modulo a prime $ p$ every integer $ a$ not divisible by $ p$ satisfies $ a^{p\minus{}1} \equiv 1 \bmod p$, so here $ a^{p\minus{}1} \equiv b^{p\minus{}1} \bmod p$ certainly does not imply $ a \equiv b \bmod p$.  (This is not even true in the integers - one cannot say that $ a^2 \equal{} b^2$ implies $ a \equal{} b$.)

thanks...
so the property - If a (insert here three lined equal sign - congruent modulo) c (mod n), then a^b (insert here three lined equal sign - congruent modulo) c^b (mod n) - is true?
And, how would you solve something like what is the remainder when (17)^(35) is divided by 5?
I read the other posts on remainders but I still don't understand it...
Can anyone help me and give me a step by step reasoning?
This is because I am just learning mods  :(
Thanks in advance!  :lol:

yes, it is true (but the converse is not true).

Given: $ a \equiv c \pmod{n}$

Then there exists some integer $ k$ such that $ a\plus{}nk \equal{} c$.  Raising both sides to the $ b$th power, we get

$ (a\plus{}nk)^b \equal{} a^b \plus{} \binom{b}{1}a^{b\minus{}1}nk \plus{} \ldots \plus{} \binom{b}{b\minus{}1}a(nk)^{b\minus{}1} \plus{} (nk)^b \equal{} c^b$ by binomial theorem.  Since every term in the binomial expansion except the first term is a multiple of $ n$, we have $ a^c \cong b^c \pmod{n}$, as desired.

The simplest counterexample for the converse is that for $ n>2$, $ \minus{}1$ is not congruent to $ 1$ but $ (\minus{}1)^2 \cong 1^2 \cong 1 \pmod{n}$.

As for a place to look up mods... I have yet to find a good one.  I think it may be better to look up some key theorems and concepts

Euclid's algorithm
Exponentiation by squaring
multiplicative inverse
Chinese remainder theorem

these next three are all the same topic:
Fermat's little theorem
Euler Totient Theorem
Carmichael's theorem

Modular arithmetic normally falls under the category of abstract algebra.

Anyway, here are the few, perhaps not so helpful bookmarks I have:
not too hard intro: [url]http://www.cut-the-knot.org/blue/Modulo.shtml[/url]
abstract algebra notes, university of Minnesota (gets difficult at times)[url]http://www.math.umn.edu/~garrett/m/intro_algebra/notes.pdf[/url]

Thanks!
But how would you solve something like what is the remainder when (17)^(35) is divided by 5?  I read the other posts on remainders but I still don't understand it...
Can anyone help me and give me a step by step reasoning?
This is because I am just learning mods
Thanks in advance!

P.S. Can anyone give me a couple of good modular exponential theorems, because I just don't understand it  :(

$ 17 \equiv 2 \pmod{5}$ so
$ 17^{35} \equiv 2^{35} \pmod{5}$
$ 2^{35} \equiv 2 \cdot 4^{17} \pmod{5}$
$ 2\cdot4^{17} \equiv 2\cdot( \minus{} 1)^{17} \pmod{5}$
$ 2\cdot( \minus{} 1)^{17} \equiv \minus{} 2 \equiv 3 \pmod{5}$

So the remainder when $ 17^{35}$ is divided by $ 5$ is $ 3$.

Alternatively, we can use fermat's little theorem to first say that $ 35 \equiv 3 \pmod{4}$ (5 is prime so $ a^{4} \equiv 1 \pmod{5}$ as long as $ a$ is not divisible by $ 5$) so $ 17^{35} \equiv 17^3 \pmod{5}$.  Then we can apply $ 17 \equiv 2 \pmod{5}$ so our answer is $ 2^3 \equiv 3 \pmod{5}$.

But how can you say -  2^{35} \equiv 2 \cdot 4^{17} \pmod{5} and then
2\cdot4^{17} \equiv 2\cdot(-1)^17 \pmod{5}?????
How can you go from steps 3 to 4?
AND can someone give me the properties of exponents for mods (I just can't seem to find them online)?

I hope you realize $ 2^{35} \equal{} 2\cdot 4^{17}$... thats just arithmetic, nothing fancy.

Then, since $ 4 \equiv\minus{} 1 \pmod{5}$, $ 4^{17} \equiv ( \minus{} 1)^{17} \pmod{5}$.

The property is if $ a \equiv b \pmod{p}$, then $ a^n \equiv b^n \pmod{p}$, which follows from binomial theorem, or considering that if $ a \equiv b \pmod{p}$ and $ c \equiv d \pmod{p}$ then $ ac \equiv bd \pmod{p}$.

The first property allows us to change the base of an exponent.  The second allows us to subsitute any equivalent numbers in a multiplication comptutation.  For instance, we can say $ 8*11 \equiv 8*1 \pmod{10}$ because $ 11 \equiv 1 \pmod{10}$

To change the exponent instead, we can break it down with simple arithmetic or we need to use fermat's little theorem, euler's totient theorem, or Carmichael's theorem.  Carmichael is an improvement on Euler's totient theorem; fermat's little theorem is the case of the prime modulo, which gives the same result as the other two theorems.

sorry  :blush: I meant the expressions on the left of the congruent modulo sign and on the right (ohh... thanks! I think I get it now - is it because the exponents merely how many numbers you multiply together, so you can do the mod stuff even with the exponent signs on?)
I apologize for any confusion...
Thanks for the theorems!

But how do they help?
Or maybe I phrased my question wrong  :oops:
Sorry again...
I meant like in the 125^625 (mod 6) = [125^(104 + 1)] (mod 6) - step 1 of #1 Is this a correct solution
modular arithmetic problem Post
How do you know that 625 = 104 + 1 (mod 6) - I mean, how do you know that and how can you say that (a theorem, maybe)?

Thanks again for all your help!  :D

Anyone? Or should this go to intermediate or classroom?

I don't know where you got the notion that $ 625 \equiv 104 \plus{} 1 \pmod{6}$; this simply isn't true.  However, it is true in mod $ 5$ - but this is not what you want.

I think the 1st theorem you need is fermat's little theorem:
[quote="Fermat's little theorem"]For any $ p$ prime and integer $ a$ which is not a multiple of $ p$, $ a^{p \minus{} 1} \equiv 1 \pmod{p}$.

Or equivalently, $ a^{p} \equiv a \pmod{p}$  for prime $ p$.
[/quote]

Using the first version, we see that in mod $ p$ we can reduce exponents to their equivalents in mod $ p \minus{} 1$.  For example

$ 3^{92}\equiv 3^{2} \pmod{7}$ because $ 92 \equiv 2 \pmod{6}$.  Thus, $ 3^{92} \equiv 2 \pmod{7}$.

Now in your example, you asked for $ 125^{625} \pmod{6}$.  This is actually rather easy to compute since $ 125 \equiv \minus{} 1 \pmod{6}$, so $ 125^{625} \equiv ( \minus{} 1)^{625} \equiv \minus{} 1 \pmod{6}$, so our answer is $ 5$.

But I you probably want to know what to do when a problem like that doesn't work so nicely.  So lets consider $ 129^{625} \equiv 3^{625} \pmod{6}$.  Now since $ 3^2 \equiv 3 \pmod{6}$, $ 3^{625} \equiv 3 \pmod{6}$.  One way to think of this is that all multiples of three fall under the residue classes $ 0$ or $ 3$ in mod $ 6$, but to get in the $ 0$ class we must also be a multiple of $ 2$, which cannot happen when we are only multiplying $ 3$'s together.

The big issue here is that $ 129$ is not relatively prime to $ 6$.  In fact, the only relatively prime numbers to $ 6$ are $ 1$ and $ 5 \equiv \minus{} 1$, which are both easy to compute.  In some situations like this we may want one of the "division" rules:

[quote="Division Rule"]If $ na \equiv nb \pmod{np}$, then $ a \equiv b \pmod{p}$.[/quote]

For instance, we can solve $ 3x\equiv 6 \pmod {9}$ by dividing to get $ x \equiv 2 \pmod{3}$.

Anyway, onward.  We want a rule for reducing the exponents in composite moduli.  There are actually two theorems: Euler's totient theorem, and Carmichael theorem.

[quote="Euler's Totient Theorem"]
For $ a$ relatively prime to $ n$, $ a^{\phi(n)} \equiv 1 \pmod{n}$ where $ \phi(n)$ is Euler's totient function.
[/quote]

Euler's totient function is the number of positive integers less than $ n$ relatively prime to $ n$.  This can be computed by $ \phi(n) \equal{} n(1 \minus{} 1/p_1)(1 \minus{} 1/p_2)(1 \minus{} 1/p_3) \cdots$ where the $ p$'s are unique prime factors of $ n$.  Note that the exponents in the prime factorization are irrelevant with the exception of the value of $ n$.  For instance, $ \phi(36) \equal{} 36(1 \minus{} 1/2)(1 \minus{} 1/3) \equal{} 12$.  The exponents from the prime factorization $ 36 \equal{} 2^2 \cdot 3^2$ do not go into the formula anywhere.

The other theorem is an improvement on euler's theorem:

[quote="Carmichael's theorem"]
The carmichael function, $ \lambda(n)$ is the least number that for all $ a$ relatively prime to $ n$, $ a^{\lambda(n)} \equiv 1 \pmod{n}$.  This function is defined as follows:

1) for $ p \ge 3$ prime, $ \lambda(p^k) \equal{} p^{k \minus{} 1}(p \minus{} 1)$.  Note that this is the same as Euler's totient theorem.
2) for $ n \equal{} p_1^{e_1}p_2^{e_2}p_3^{e_3}\cdots$ where the $ p$'s are prime numbers, $ \lambda(n) \equal{} lcm(\lambda(p_1^{e_1}), \lambda(p_2^{e_2}), \lambda(p_3^{e_3}), \ldots)$.  Euler's totient theorem is to take the product of all these numbers instead of their lcm.
3) $ \lambda(2) \equal{} 1$, $ \lambda(4) \equal{} 2$, $ \lambda(2^k) \equal{} 2^{k \minus{} 2}$ for $ k \ge 3$
[/quote]

Hope this helps.  Oh, and I should probably mention the "other" division rule:

[quote="Division Rule 2"]
For $ a$ relatively prime to $ p$, $ ab \equiv ac \pmod{p}$ implies $ b \equiv c \pmod{p}$.
[/quote]

One example that we really can't deal with with division is $ 2x \equiv 1 \pmod{10}$.  This equation has no solutions: multiplying both sides by $ 5$ we get $ 0 \equiv 2 \pmod{10}$ which is clearly incorrect!

anyway, I don't have time to make up for examples right now... try a few problems and post any questions.

Thanks a lot, faciis!!!
It really helped, even though I still have a couple of questions and am still absorbing the information.
I really appreciate your effort and dedication

[b][u]And how do you get from (-1)^625 (mod 6) to -1 (mod 6)?[/u][/b]

I know you raise -1 to the 625th power, but how can you do that (I mean, (-1)^625 (mod 6) means -1 (mod 6) * (-1) (mod 6) * so on and so forth - so don't you have to multiply all of them [i]first[/i] before you do the mods - or [u]is there a rule/theorem that allows you to do the exponentiation, because we already know that we can break down 125^625 (mod 6) to (-1)^625 (mod 6) - this is in your post - which illustrates that an exponents merely means how much you multiply something, so 125^625 (mod 6) would mean 125 (mod 6) * 125 (mod 6) * so on and so forth - this reasoning is right, right?)[/u]

This is the first post on the post titled Is this a correct solution
modular arithmetic problem - step 1. Whereupon it is agreed to be correct - so if it is,

--------------------------------------------------------------------------------
Okay I just learned this cool modular arithmetic trick from ernie on the post titled remainder, and I want to see if I'm doing this correctly. If you know how to solve the problem, then please tell me if I did it right.

Problem;
What is the remainder when [125^(625)]/6

My Solution;

125^625 (mod 6) = [125^(104 + 1)] (mod 6)
125^105 (mod 6) = [125^(17 + 3)] (mod 6)
125^20 (mod 6) = [125^(3 + 2)](mod 6)
125^5 (mod 6) = 5^15 (mod 6) = [5^(2 + 3)] (mod 6)
5^5 (mod 6) = 3125 (mod 6)
5 (mod 6)

So the remainder is 5.what is the rule/theorem that you use to make that step - any help is greatly appreciated

Thanks again facis!!!!!!!!!

I see what you are doing but I don't think it is valid - if it is, I'll be shocked.  What happens though is that $ 125 \equiv - 1 \pmod{6}$, so $ 125$ raised to any power will alternate between $ - 1 \pmod{6}$ and $ 1 \pmod{6}$, so we have a fifty-fifty chance that any random integer exponent of $ 125$ will give a remainder of $ - 1$ ($ 5$) and not $ 1$.  So essentially you have a 50-50 chance that your answer is right no matter the validity of what you did to the exponent.

So we could go the route that $ 125^{625} \equiv ( - 1)^{625} \equiv - 1 \pmod{6}$ - which I think is simplest.  I'll proceed with carmichael's theorem instead.

$ \lambda(6) = \text{lcm}(\lambda(2), \lambda(3)) = \text{lcm}(1,2) = 2$.  So we reduce the exponent in mod $ 2$, which is $ 625 \equiv 1 \pmod{2}$.  So $ 125^{625} \equiv 125 \equiv 5 \pmod{6}$.

Note that we could have also used Euler's totient theorem.  $ \phi(6) = 6(1 - 1/2)(1 - 1/3) = 2$, so the solution is nearly the same either way.

I'm just posting this again because you replied when I was editing...
Thanks again, facis!  :D

#12Thanks a lot, faciis!!!
It really helped, even though I still have a couple of questions and am still absorbing the information.
I really appreciate your effort and dedication

And how do you get from (-1)^625 (mod 6) to -1 (mod 6)?

I know you raise -1 to the 625th power, but how can you do that (I mean, (-1)^625 (mod 6) means -1 (mod 6) * (-1) (mod 6) * so on and so forth - so don't you have to multiply all of them first before you do the mods - or is there a rule/theorem that allows you to do the exponentiation, because we already know that we can break down 125^625 (mod 6) to (-1)^625 (mod 6) - this is in your post - which illustrates that an exponents merely means how much you multiply something, so 125^625 (mod 6) would mean 125 (mod 6) * 125 (mod 6) * so on and so forth - this reasoning is right, right?)

This is the first post on the post titled Is this a correct solution
modular arithmetic problem - step 1. Whereupon it is agreed to be correct - so if it is,

--------------------------------------------------------------------------------
Okay I just learned this cool modular arithmetic trick from ernie on the post titled remainder, and I want to see if I'm doing this correctly. If you know how to solve the problem, then please tell me if I did it right.

Problem;
What is the remainder when [125^(625)]/6

My Solution;

125^625 (mod 6) = [125^(104 + 1)] (mod 6)
125^105 (mod 6) = [125^(17 + 3)] (mod 6)
125^20 (mod 6) = [125^(3 + 2)](mod 6)
125^5 (mod 6) = 5^15 (mod 6) = [5^(2 + 3)] (mod 6)
5^5 (mod 6) = 3125 (mod 6)
5 (mod 6)

So the remainder is 5.what is the rule/theorem that you use to make that step - any help is greatly appreciated

Thanks again facis!!!!!!!!!

[b]And also, I know you tried to explain this concept at first (thanks for that  :lol: ) but could you show me what to do in a situation like (17)^35 = ? (mod 5).

And can you also explain why [code]129^{625} \equiv 3^{625} \pmod{6}[/code] Now since 3^2 (mod 6), 3^625 = 3 (mod 6) - how do you know that/how can you make that generalization???. One way to think of this is that all multiples of three fall under the residue classes  or  in mod , but to get in the  class we must also be a multiple of , which cannot happen when we are only multiplying 's together.[/b]

your ideas about exponents are correct.  But modular arithmetic follows all normal rules of multiplication (and thus many of exponentiation), subtraction, and addition.  While doing arithmetic, we can ignore the mods until the end.  All the mod really says is to remember we are working in a circle with only integers; and this only has strange consequences for roots and division.  All other calculations can be done the same way you've done them since elementary school; The only difference is that at the end, we usually want to move them back into our residue classes $ 0,1,2,\ldots p \minus{} 1$, and some properties applied in the middle make computations easier.  It would be impractical to compute $ 125^{625}$ and then find the remainder of that when divided by $ 6$; the switch to $ ( \minus{} 1)^{625}$ makes computation rather straightforward: [have you ever noticed that $ \minus{} 1$ to any odd power is always $ \minus{} 1$?]

perhaps it may help if we define equivalence more rigorously.  $ a \equiv b \pmod{p}$ means that $ p|a \minus{} b$, or more precisely, that there exists an integer $ k$ such that $ kp \equal{} a \minus{} b$.  This then implies that $ a \equal{} b \plus{} kp$.  So we can think of equivalence as we can add or subtract some multiple of $ p$ to one number to get to the other numbers.  From here, we can show that multiplication holds:

$ a\equiv b \pmod{p}$ and $ c \equiv d \pmod{p}$ means that there exist $ m$ and $ n$ such that $ a \plus{} np \equal{} b$ and $ c \plus{} mp \equal{} d$.  Thus, $ cd \equal{} (a \plus{} np)(b \plus{} mp) \equal{} ab \plus{} nbp \plus{} map \plus{} mnp^2 \equal{} ab \plus{} p(nb \plus{} ma \plus{} mnp)$ and therefore there exists $ k$ such that $ ab \plus{} kp \equal{} cd$, namely $ k \equal{} nb \plus{} ma \plus{} mnp$.  This implies that $ ab \equiv cd \pmod{p}$.

Sorry...
I'm just posting this again because you replied when I was editing...
Thanks again, facis!

#12Thanks a lot, faciis!!!
It really helped, even though I still have a couple of questions and am still absorbing the information.
I really appreciate your effort and dedication

And how do you get from (-1)^625 (mod 6) to -1 (mod 6)?

I know you raise -1 to the 625th power, but how can you do that (I mean, (-1)^625 (mod 6) means -1 (mod 6) * (-1) (mod 6) * so on and so forth - so don't you have to multiply all of them first before you do the mods - or is there a rule/theorem that allows you to do the exponentiation, because we already know that we can break down 125^625 (mod 6) to (-1)^625 (mod 6) - this is in your post - which illustrates that an exponents merely means how much you multiply something, so 125^625 (mod 6) would mean 125 (mod 6) * 125 (mod 6) * so on and so forth - this reasoning is right, right?)

This is the first post on the post titled Is this a correct solution
modular arithmetic problem - step 1. Whereupon it is agreed to be correct - so if it is,

--------------------------------------------------------------------------------
Okay I just learned this cool modular arithmetic trick from ernie on the post titled remainder, and I want to see if I'm doing this correctly. If you know how to solve the problem, then please tell me if I did it right.

Problem;
What is the remainder when [125^(625)]/6

My Solution;

125^625 (mod 6) = [125^(104 + 1)] (mod 6)
125^105 (mod 6) = [125^(17 + 3)] (mod 6)
125^20 (mod 6) = [125^(3 + 2)](mod 6)
125^5 (mod 6) = 5^15 (mod 6) = [5^(2 + 3)] (mod 6)
5^5 (mod 6) = 3125 (mod 6)
5 (mod 6)

So the remainder is 5.what is the rule/theorem that you use to make that step - any help is greatly appreciated

Thanks again facis!!!!!!!!!

And also, I know you tried to explain this concept at first (thanks for that  ) but could you show me what to do in a situation like (17)^35 = ? (mod 5).

And can you also explain why Code:
129^{625} \equiv 3^{625} \pmod{6}
Now since 3^2 (mod 6), 3^625 = 3 (mod 6) - how do you know that/how can you make that generalization???. One way to think of this is that all multiples of three fall under the residue classes or in mod , but to get in the class we must also be a multiple of , which cannot happen when we are only multiplying 's together.

Is this restatement true - "a (mod m) * b (mod m) = (a*b) (mod m)?

Anyways, how would you solve this (Sorry, but I really don't get it - when you multiply something other than 1, how would you do it? - THANKS A LOT FOR YOUR DEDICATION) - (17)^(35) = ? (mod 5)??????

[quote]Is this restatement true - "a (mod m) * b (mod m) = (a*b) (mod m)? [/quote]
yes, though I am not quite sure why you want to attach the mod m's to individual numbers.

$ 129^{625} \equiv 3^{625} \pmod{6}$ because $ 129 \equiv 3 \pmod{6}$.
Now $ 3^2 \equiv 3 \pmod{6}$.  By multiplication, this means that $ 3^{k \plus{} 1} \equiv 3^k$ for all $ k \ge 1$.  Now proceed by induction.  The base case is $ 3^2 \equiv 3 \pmod{6}$, which is certainly true.  Now if $ 3^k \equiv 3 \pmod{6}$, then $ 3^{k \plus{} 1} \equiv 3^2 \equiv 3 \pmod{6}$.  Thus, $ 3^{k} \equiv 3 \pmod{6}$ implies $ 3^{k\plus{}1} \equiv 3 \pmod{6}$, which completes the induction.

$ 17^{35} \equiv 2^{35} \pmod{5}$ since $ 17 \equiv 2 \pmod{5}$.
Fermat's little theorem tells us that $ 2^4 \equiv 1 \pmod{5}$.  $ 2^{35} \equal{} 2^{4*8 \plus{} 3} \equal{} (2^4)^8 \cdot 2^3$ so we have that $ (2^4)^8 \cdot 2^3 \equiv 1^8 \cdot 2^3 \equiv 8 \equiv 3 \pmod{5}$.  So $ 17^{35} \equiv 3 \pmod{5}$

Thanks again, facis!!!!!  :lol:
I think I understand it now, although new questions might crop up as I continue to study mods.
But just a question question - "a (mod m) * b (mod m) = (a*b) (mod m)?" is basically the same as "a * b = (a*b) (mod m)", right? But to might it look nicer, we write a * b = (a*b) (mod m), right?
Thanks a lot, facis!  :D

Will you explain CRT and propose some problems. As i understand it it states that all the solutions to a system of modular congruences mod $ n_1, n_2 ...nk$ are congruent mod $ n_1*n_2*...n_k$

And also, can you, instead of applying the theorems and stuff (although they're really cool- - thanks again facis!), just look for mods of smaller exponents and same bases - also known as finding a pattern - and then apply that rule to bigger exponents? (This refers problems like 125^625 (mod 6) - really big exponents)

yes, I suppose you could, but the theorems basically tell you what the pattern is instead of finding it yourself.  I think that the theorems are easier, but they do require a little memorization.  As for $ 125^{625}$, you could look at $ 125^x \pmod{6}$ and find that when $ x$ is odd the remainder is $ 5$ and when $ x$ is even the remainder is $ 1$.  Or you could take $ 125 \equiv \minus{}1 \pmod{6}$ and use that to change the base - and then the result becomes obvious.

If you really want to search for patterns, I would suggest that you look into exponentiation by squaring, as it will speed up the computations you may need to make.  The basic idea is instead of computing $ n^k$ by $ n\cdot n^{k\minus{}1}$ is you would instead compute $ n^2$, $ (n^2)^2 \equal{} n^4$, $ (n^4)^2 \equal{} n^8$, etc., and then use these results to compute $ n^k$, choosing which ones to multiply together based on the binary representation of $ k$.  Perhaps an example would help illustrate it:

Lets compute $ 3^{29}$.  First, we find $ 3^2\equal{}9$, $ (3^2)^2 \equal{} 3^4 \equal{} 9^2 \equal{} 81$, $ (3^{4})^2 \equal{} 3^8 \equal{} 81^2 \equal{} 6561$, $ (3^8)^2 \equal{} 3^{16} \equal{} 6561^2 \equal{} 43046721$.  Now $ 29 \equal{} 11101_2$, so $ 3^{29} \equal{} 3 \cdot 3^4 \cdot 3^8 \cdot 3^{16} \equal{} 3 \cdot 81 \cdot 6561 \cdot 43049721$ which comes out to some big number.  This is not easy to compute, but notice that we would do a total of $ 7$ multiplications here.  If we instead tried to compute it using only $ n^k \equal{} n^{k\minus{}1} \cdot n$, it would take us $ 28$ multiplications to obtain the same result.

Using this algorithm in modular arithmetic, we can reduce the amount of work much further, because at any intermediate result, we can reduce our number to be less than our modulo (because of our multiplication rule).  For instance, lets try the same problem again, but in mod $ 50$:
$ 3^2 \equiv 9 \pmod{50}$
$ 3^4 \equiv 9^2 \equiv 81 \equiv 31 \pmod{50}$
$ 3^8 \equiv 31^2 \equiv 900\plus{}60\plus{}1 \equiv 961 \equiv 11 \pmod{50}$
$ 3^{16} \equiv 11^2 \equiv 121 \equiv 21 \pmod{50}$

So to compute $ 3^{29} \pmod{50}$, we multiply like before, but substitute in much smaller numbers: $ 3^{29} \equiv 3 \cdot 3^4 \cdot 3^8 \cdot 3^{16} \equiv 3\cdot 31 \cdot 11 \cdot 21 \pmod{50}$ which is $ 33 \pmod{50}$.

THANKS!!!!!  :D
Is that method also called the "Russian Peasant Method" or something like that?
So basically, it's okay to try to find patterns for smaller exponents (and you know that there will be a pattern because the way mods are defined) and apply that to bigger exponents, right?
And lastly, could it be said that the finding pattern method works for specific numbers best and theorems help prove general facts the best (even though they can also be used to solve equations wiht specific numbers), right?
Thanks again, facis!

no, this is not russian peasant method, its exponentiation by squaring.  I am not aware of any other names for it.

pattern finding can work.  I'm not entirely sure what you intend to do though.

In your problem $ 125^{625}$ you have two options: to look for a pattern with $ x^{625}$ or to look for a pattern with $ 125^x$. inevitably, both should give the same results, but both require huge numerical calculations.  Without any complex theorems in hand, I think it is still pretty clear that we can reduce $ 125$ to $ 5$ or $ \minus{}1$.  If you get down to $ 5$, then you can analyze $ 5^x$ instead of $ 125^x$, since they are the same.  And we find:
$ 5^1 \equiv 5 \pmod{6}$
$ 5^2 \equiv 25 \equiv 1 \pmod{6}$
$ 5^3 \equiv 125 \equiv 5 \pmod{6}$

And we immediately see that $ 5^x \pmod{6}$ alternates between $ 5 \pmod{6}$ and $ 1 \pmod{6}$.  And we could prove inductively, using $ 5^3 \equiv 5 \pmod{6}$ as our base case, that $ 5^{k\plus{}2} \equiv 5^k \pmod{6}$ for all integers $ k \ge 1$ - and this proves that the pattern we observe must hold.

Now if you intend to analyze $ x^{625} \pmod{6}$ instead, the work is more difficult, but with exponentiation by squaring, still possible.
$ 1^{625} \equiv 1 \pmod{6}$
$ 2^{625} \equiv  2 \pmod{6}$
$ 3^{625} \equiv 3 \pmod{6}$
etc.  To find each of these we could also analyze $ 2^x$, $ 3^x$, etc. in mod 6.

in any case, I think theorems are easier, if you can remember them.  Anyway, I suppose that the only way pattern finding fails is if you want to change your mod.  You could prove almost anything with enough work in a specific mod, but nearly nothing about other modulos.

THANKS FACIS!   :lol:
If I want to prepare for AMC/AIME/USAMO what books do you guys recommend the most (keeping in mind the amount of time before the actual test)...
I greatly appreciated any help...


Solve for r, w, b, and g where n = r+w+b+g. This comes from the 1996 AMC12 #26 http://www.mathlinks.ro/viewtopic.php?p=419893#419893

$ \frac{\binom r4}{\binom n4}\equal{}\frac{\binom{r}{3}\binom{w}{1}}{\binom n4}\equal{}\frac{\binom{r}{2}\binom{w}{1}\binom{b}{1}}{\binom n4}\equal{}\frac{\binom{r}{1}\binom{w}{1}\binom{b}{1}\binom{g}{1}}{\binom n4}$

This would be more appropriate in the intermediate forum.

I think you should look at [[Combination]]s.

[quote="displayname00"]Solve for r, w, b, and g where n = r+w+b+g. This comes from the 1996 AMC12 #26 http://www.mathlinks.ro/viewtopic.php?p=419893#419893

$ \frac {\binom r4}{\binom n4} = \frac {\binom{r}{3}\binom{w}{1}}{\binom n4} = \frac {\binom{r}{2}\binom{w}{1}\binom{b}{1}}{\binom n4} = \frac {\binom{r}{1}\binom{w}{1}\binom{b}{1}\binom{g}{1}}{\binom n4}$[/quote]

[hide]First, you can get rid of the denominator.

Next, expand the expressions. You get:

\[ \frac 1{24}r(r-1)(r-2)(r-3)=\frac 16 r(r-1)(r-2)w = \frac 12 r(r-1)wb = rwbg\]

Getting rid of the fractions:

\[ r(r-1)(r-2)(r-3)=4r(r-1)(r-2)w = 12r(r-1)wb = 24rwbg\]

From this, you can conclude the following:

\begin{align*}
r-3&=4w \\
r-2&=3b \\
r-1&=2g\end{align*}

And therefore, $ r=2g+1=3b+2=4w+3$. In other words, $ r\equiv 1\pmod 2\equiv 2\pmod 3\equiv 3\pmod 4$, and the smallest possible value for $ r$ that satisfies the above is $ 11$.

Working backwards, $ g=5$, $ b=3$, and $ w=2$. So the sum is $ 11+5+3+2=\boxed{21}$.[/hide]


a, b, c >0
Prove that
$ \frac{a^3}{a^2\plus{}ab\plus{}b^2}\plus{}\frac{b^3}{b^2\plus{}bc\plus{}c^2}\plus{} \frac{c^3}{c^2\plus{}ca\plus{}a^2} \geq \frac{a\plus{}b\plus{}c}{3}$

See here [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=1629879#1629879]here[/url] posts #52 , #53 and #54 .

My solution was using reverse technique  :roll:

By Am-Gm

$ \frac{a^3}{a^2 \plus{}ab \plus{}b^2} \equal{}a \minus{}\frac{a^2b \plus{}ab^2 }{a^2 \plus{}ab \plus{}b^2} \geq a\minus{} \frac{ab(a\plus{}b)}{3ab} \equal{}a \minus{}\frac{a\plus{}b}{3}$

Summing similars we have :

$ \frac{a^3}{a^2 \plus{}ab\plus{}b^2 } \plus{}\frac{b^3}{b^2 \plus{}bc \plus{}c^2 } \plus{}\frac{c^3}{c^2 \plus{}ca \plus{}a^2} \geq a\plus{}b\plus{}c \minus{}\frac{2(a\plus{}b\plus{}c)}{3} \equal{}\frac{a\plus{}b\plus{}c}{3}$

[quote="aktyw19"]a, b, c >0
Prove that
$ \frac {a^3}{a^2 \plus{} ab \plus{} b^2} \plus{} \frac {b^3}{b^2 \plus{} bc \plus{} c^2} \plus{} \frac {c^3}{c^2 \plus{} ca \plus{} a^2} \geq \frac {a \plus{} b \plus{} c}{3}$[/quote]

Let $ A \equal{} \sum {\frac{{a^3 }}{{a^2  \plus{} ab \plus{} b^2 }}} ,B \equal{} \sum {\frac{{b^3 }}{{a^2  \plus{} ab \plus{} b^2 }}}$, then\[ A \minus{} B \equal{} \sum {\frac{{a^3  \minus{} b^3 }}{{a^2  \plus{} ab \plus{} b^2 }}}  \equal{} \sum {\left( {a \minus{} b} \right)}  \equal{} 0\]so \[ \sum {\frac{{a^3 }}{{a^2  \plus{} ab \plus{} b^2 }}}  \ge \frac{{a \plus{} b \plus{} c}}{3}\]\[ \Leftrightarrow \sum {\frac{{a^3  \plus{} b^3 }}{{a^2  \plus{} ab \plus{} b^2 }}}  \ge \frac{{2\left( {a \plus{} b \plus{} c} \right)}}{3}\]\[ \Leftrightarrow \sum {\left( {\frac{{a^3  \plus{} b^3 }}{{a^2  \plus{} ab \plus{} b^2 }} \minus{} \frac{{a \plus{} b}}{3}} \right)}  \ge 0\]\[ \Leftrightarrow \sum {\frac{{2\left( {a \plus{} b} \right)}}{{3\left( {a^2  \plus{} ab \plus{} b^2 } \right)}}\left( {a \minus{} b} \right)^2 }  \ge 0.\]

another solution:
with loss of generality, we can assume that $ a \ge b \ge c$ or $ a \le b \le c$.
$ \sum {\frac {{a^3 }}{{a^2 \plus{} ab \plus{} b^2 }}} \ge \frac {{a \plus{} b \plus{} c}}{3}$
$ \Leftrightarrow \sum {\left( {\frac {{a^3 }}{{a^2 \plus{} ab \plus{} b^2 }} \minus{} \frac {{a \plus{} b}}{6}} \right)} \ge 0$
$ \Leftrightarrow \sum {\frac {{\left( {a \minus{} b} \right)\left( {5a^2 \plus{} 3ab \plus{} b^2 } \right)}}{{6\left( {a^2 \plus{} ab \plus{} b^2 } \right)}}} \ge 0$
$ \Leftrightarrow \frac {{\left( {a \minus{} b} \right)\left( {5a^2 \plus{} 3ab \plus{} b^2 } \right)}}{{6\left( {a^2 \plus{} ab \plus{} b^2 } \right)}} \plus{} \frac {{\left( {b \minus{} c} \right)\left( {5b^2 \plus{} 3bc \plus{} c^2 } \right)}}{{6\left( {b^2 \plus{} bc \plus{} c^2 } \right)}}$ $ \minus{} \frac {{\left( {a \minus{} b \plus{} b \minus{} c} \right)\left( {5c^2 \plus{} 3ca \plus{} a^2 } \right)}}{{6\left( {c^2 \plus{} ca \plus{} a^2 } \right)}} \ge 0$
$ \Leftrightarrow \left( {a \minus{} b} \right)\left( {\frac {{5a^2 \plus{} 3ab \plus{} b^2 }}{{6\left( {a^2 \plus{} ab \plus{} b^2 } \right)}} \minus{} \frac {{5c^2 \plus{} 3ca \plus{} a^2 }}{{6\left( {c^2 \plus{} ca \plus{} a^2 } \right)}}} \right)$ $ \plus{} \left( {b \minus{} c} \right)\left( {\frac {{5b^2 \plus{} 3bc \plus{} c^2 }}{{6\left( {b^2 \plus{} bc \plus{} c^2 } \right)}} \minus{} \frac {{5c^2 \plus{} 3ca \plus{} a^2 }}{{6\left( {c^2 \plus{} ca \plus{} a^2 } \right)}}} \right) \ge 0$
$ \Leftrightarrow \frac {{\left( {a \minus{} b} \right)\left( {a^2 \minus{} bc} \right)\left( {2a^2 \plus{} ab \plus{} ac \plus{} 2bc} \right)}}{{3\left( {a^2 \plus{} ab \plus{} b^2 } \right)\left( {c^2 \plus{} ca \plus{} a^2 } \right)}}$ $ \plus{} \frac {{\left( {b \minus{} c} \right)\left( {ab \minus{} c^2 } \right)\left( {2ab \plus{} ac \plus{} bc \plus{} 2c^2 } \right)}}{{3\left( {b^2 \plus{} bc \plus{} c^2 } \right)\left( {c^2 \plus{} ca \plus{} a^2 } \right)}} \ge 0$
when $ a \ge b \ge c$ or $ a \le b \le c$, we have $ {\left( {a \minus{} b} \right)\left( {a^2 \minus{} bc} \right)}\ge 0$ and $ {\left( {b \minus{} c} \right)\left( {ab \minus{} c^2 } \right)}\ge 0$, so the inequality holds.

Alternatively,  observe that the inequality is homogenious on both sides, it is free to make a constraint on the inequality, I use $ a\plus{}b\plus{}c\equal{}1$, thus we are going to show a equivalent inequality:
$ \quad\sum_{cyc} \frac{a^3}{a^2\plus{}b^2\plus{}ab}\ge\frac{1}{3}$.

On the other hand, it can be seen that $ (a^2\plus{}b^2\plus{}ab)(a\plus{}b\plus{}c)\leq 3(a^3\plus{}b^3\plus{}c^3)$ (this one I used the fact that $ ab^2\plus{}ba^2\leq a^3\plus{}b^3)$,
thus we have $ \sum_{cyc}\frac{a^3}{a^2\plus{}b^2\plus{}ab}\ge \frac{a^3\plus{}b^3\plus{}c^3}{3(a^3\plus{}b^3\plus{}c^3)}\equal{}\frac{1}{3}$.

[quote="Aile"]Alternatively,  observe that the inequality is homogenious on both sides, it is free to make a constraint on the inequality, I use $ a \plus{} b \plus{} c \equal{} 1$, thus we are going to show a equivalent inequality:
$ \quad\sum_{cyc} \frac {a^3}{a^2 \plus{} b^2 \plus{} ab}\ge\frac {1}{3}$.

On the other hand, it can be seen that $ (a^2 \plus{} b^2 \plus{} ab)(a \plus{} b \plus{} c)\leq 3(a^3 \plus{} b^3 \plus{} c^3)$ (this one I used the fact that $ ab^2 \plus{} ba^2\leq a^3 \plus{} b^3)$,
thus we have $ \sum_{cyc}\frac {a^3}{a^2 \plus{} b^2 \plus{} ab}\ge \frac {a^3 \plus{} b^3 \plus{} c^3}{3(a^3 \plus{} b^3 \plus{} c^3)} \equal{} \frac {1}{3}$.[/quote]
I think that $ (a^2 \plus{} b^2 \plus{} ab)(a \plus{} b \plus{} c)\leq 3(a^3 \plus{} b^3 \plus{} c^3)$ is not true...
try $ a\equal{}c\equal{}0.3, b\equal{}0.4$ ...


[quote="SuperMath"]Let $\large a_1,a_2,. . .,a_{2003} \in [-1;1]$ satisfying: $\large \sum a^3_j = 0$. Prove that:
$\large -2003 < 3 \sum a_j < 2003$[/quote]

This inequality isn't quite correct.
The parts could be equal  :!:

Let $\alpha_{i}=arcsin(a_{i})$ as $a_{i}$ satisfy the conditions $\Rightarrow$
$3\sum_{i=1}^{2003} a_{i}=3\sum_{i=1}^{2003} sin(\alpha_{i})-4\sum_{i=1}^{2003} sin^3(\alpha_{i}) = \sum_{i=1}^{2003} sin(3\alpha_{i})=S \Rightarrow \\ \\ 2003 \geq S \geq -2003$
The first relation is equality for $a_{1}=$...$=a_{1005}=\frac{1}{2}$ , $a_{1006}=-1$ , $a_{1007}=$...$a_{2003}=-\frac{1}{2}$
Also we can make the second relation be equaity (with another example)
 :)


Prove that the irrationals have usual neighborhoods and the rationals are isolated is metrizable.

This is the flip-flop of the Michael Line.  I do not see how to prove this.  I know we want to use Urysohn Metrization Theorem, but I do not see how to use it now.  Thanks in advance.

[quote="xboxlive89128"]Prove that the irrationals have usual neighborhoods and the rationals are isolated is metrizable. [/quote]
I don't understand that sentence.


is there an easy way to factor problems that have a number other than 1 in front of the quadratic term?

ex:  $ 4x^2 \plus{} 14x \plus{} 12$

(in this case, it would be 4)

$ 4x^2\plus{}14x\plus{}12 \rightarrow 4x^2\plus{}8x\plus{}6x\plus{}12$

$ 4x(x\plus{}2)\plus{}6(x\plus{}2) \rightarrow (4x\plus{}6)(x\plus{}2)$

[hide="Method"]
Here are the steps to what I did above:

Given $ ax^2\plus{}bx\plus{}c$

(i.) Multiply the coefficient of the quadratic term $ a$ with the constant term $ c$.

In this case, we multiply $ 4*12\equal{}48$.

(ii.) Factor the product into two numbers which add to the coefficient of the linear term $ b$.

In this case, the coefficient of the linear term is $ 14$. $ 48$ can be factored as $ 8*6$ and $ 8\plus{}6\equal{}14$.

(iii.) Rewrite the problem with the two factors as linear terms.

In this case, $ 4x^2\plus{}8x\plus{}6x\plus{}12$.

(iv.) Group and factor the problem.

In this case, $ 4x(x\plus{}2)\plus{}6(x\plus{}2)$

From here we can break this down into, $ (4x\plus{}6)(x\plus{}2)$

[/hide]

I believe that is known as the British method (could be mistaken, though). Also, you forgot to factor out 2.

[color=darkblue]Let $ f(x) \equal{} ax^2 \plus{} bx \plus{} c, a\neq 0 , \Delta \equal{} b^2 \minus{} 4ac \ge 0$.

$ f(x) \equal{} a\left(x^2 \plus{} \frac {b}{a}x \plus{} \frac {c}{a} \right)$

$ \equal{} a\left[ (x^2 \plus{} 2\frac {b}{2a}x \plus{} \frac {b^2}{4a^2}) \plus{} \frac {c}{a} \minus{} \frac {b^2}{4a^2} \right]$

$ \equal{} a\left[ (x \plus{} \frac {b}{2a})^2 \minus{} \frac {b^2 \minus{} 4ac}{4a^2} \right]$

$ \equal{} a\left[ (x\plus{} \frac {b}{2a})^2 \minus{} \frac {\Delta}{4a^2} \right]$

$ \equal{} a\left(x\plus{} \frac {b}{2a} \plus{} \frac {\sqrt {\Delta}}{2a} \right) \left(x \plus{} \frac {b}{2a} \minus{} \frac {\sqrt {\Delta}}{2a} \right)$

$ \equal{} a\left(x \plus{} \frac {b \plus{} \sqrt {\Delta}}{2a} \right) \left(x\plus{} \frac {b \minus{} \sqrt {\Delta}}{2a} \right)$[/color]


trigonometry
LaTeX
geometry
trig identities
Law of Sines
AMC

[quote="Sunday Math-er"]Please help, thanks in advance:

In triangle ABC, angle C is a right angle and CB > CA. Point D is located on BC so that angle CAD is twice angle DAB. If AC/AD =2/3, then CD/BD=m/n. Find m+n.[/quote]

use law of sines if DAB is x, then CAD is 2x, and angle B is 90-3x,

so

sin(90-3x)/3=sin(x)/DB,

you can solve for sin(x) and sin(90-3x) in terms of the sides (since $ sin2x \equal{} \sqrt {5}/3$) so your only leftr with DB (I hope you can find CD on your own without a problem).

Sorry I would type up the entire solution but its getting late and I dont feel like using latex with all the square roots and fractions..

[quote="Sunday Math-er"]thank you so much for the hint! I'll work on it :)[/quote]

In 1998 you were allowed to use calculator so it would only take a sec with that method (you can do sin^{-1} on your calc to find angles and just bash form there)..

However now, it is still easily do able but there is a little more calculation, like finding triple angle formulas the old fasioned way..

[quote="sponge008"]Use the angle bisector theorem (http://en.wikipedia.org/wiki/Angle_bisector_theorem)
Much simpler.[/quote]

which angle did you exactly bisect?

I know you can use stewarts but too many square roots (especailly without a calculator), but laws of sinces on ABD takes like 3 mins max.

Wait, does the title make sense? Which AMC? I thought AMCs only had 25 questions each...

It was the old AMC, the AHSME. That had 30 questions.

I thought the AHSME was around only in the 1980s... maybe I was mistaken.

They went all the way up to and including 1999.


Hi,

Let n an integer >2 ,  and denote
$ E\equal{}\{ (a_1 X_1\plus{}\cdots\plus{}a_n X_n)^n  / (a_1,\dots,a_n)\in \mathbb{C}^n \}$

Is it true that

$ X_1 X_2 \dots X_n \in \mbox{Vect}(E)$
or
$ \{  P\in \mathbb{C}[X_1,\dots,X_n]/ \quad \forall \lambda \in \mathbb{C} \quad P(\lambda X_1,\dots,\lambda X_n)\equal{}\lambda^n P      \} \equal{} \mbox{Vect}(E)$

thanks

Yes.  One way to see this is just to take the $ a_i$ to be roots of unity and apply the discrete Fourier transform.

[b]Lemma:[/b]  Let $ \zeta$ be a primitive $ m^{th}$ root of unity and let $ P(x) \equal{} \sum_{i \equal{} 0}^{d} a_i x^i$ be a polynomial.  Then

$ \sum_{i \equiv a \bmod m}^{d} a_i x^i \equal{} \frac {1}{m} \sum_{j \equal{} 0}^{m \minus{} 1} \zeta^{ \minus{} aj} P(\zeta^j x)$.

This implies that letting the $ a_i$ be appropriate $ (n \plus{} 1)^{th}$ roots of unity it is possible to extract an arbitrary monomial of degree $ n$.

hi,

i'm sorry i don't see the link with the initial question.

We can apply the lemma to each variable $ x_1, x_2, ... x_n$ to extract some monomial $ x_1^{k_1} x_2^{k_2} ... x_n^{k_n}$.


geometry
circumcircle
trigonometry
ratio
projective geometry
geometry proposed

Let triangle $ ABC$ inscribe in circumcircle $ (O)$. Let $ M_1,M_2$ be the point inside the triangle. Denote $ A_1,B_1,C_1$ the intersections of $ AM_1,BM_1,CM_1$ with $ (O)$, respectively; $ A_2,B_2,C_2$ the intersections of $ AM_2,BM_2,CM_2$ with $ (O)$, respectively. Let $ X\equiv A_1A_2 \cap BC$. Define the same for $ Y,Z$. Prove that: $ X,Y,Z$ are collinear. :D

[b]Hint[/b]: What do you think about using [b]Pascal[/b]

$ \frac{XB}{XC}\equal{}\frac{XB^2}{XB.XC}\equal{}\frac{XB^2}{XA_1.XA_2}\equal{}\frac{XB}{XA_1}.\frac{XB}{XA_2}\equal{}\frac{A_2B}{A_1C}.\frac{A_1B}{A_2C}$
Similar with $ \frac{XC}{XA},\frac{XA}{XB}$ and $ \frac{A_1B.B_1C.C_1A}{A_1C.B_1A.C_1B}\equal{}\frac{A_2B.B_2C.C_2A}{A_2C.B_2A.C_2B}\equal{}1$ then OK!

Another way of calculating:

$ \prod \frac{XB}{XC} \equal{} \prod \frac{S_{A_1XB}}{S_{A_1XC}}\equal{}\prod \frac{A_1B.\sin{\widehat{A_2AB}}}{A_1C\sin{\widehat{A_2AC}}}\equal{}\prod \frac{\sin{\widehat{A_1AB}}.\sin{\widehat{A_2AB}}}{\sin{\sin{\widehat{A_1AC}}.\widehat{A_2AC}}} \equal{} 1$

So by Ceva's theorem, we got what desired.

[b]Comment:[/b] Pascal's theorem as well as cross ratio may still work with this problem.


LaTeX
inequalities
blogs
inequalities unsolved

Is it true that

a^3 b + b^3 c + c^3 a  >= a^2 b^2 + b^2 c^2 +c^2 a^2

Left side is cyc sum a*a*a*b and right cyc sum a*a*b*b
and if it is true is it some special case of some well known theorem about cyclic sums ?

No, just take $ a\equal{}0, b\equal{}1, c\equal{}2$. :wink:

And for using LaTeX, just put one dollar sign at the beginning and one dollar sign at the end of the code.

your inequality is not true for every non-negative numbers, but it's true if $ a,b,c$ are lengthes of side of triangle, it's IMO 1983 problem 3, it's already posted, but if you wan a solution see here in my blog: http://mathsmaster.wordpress.com/2009/01/12/imo-1983-probleme-3/

It's true end easy to see that:
$ a^4 \plus{} b^4\plus{} c^4 \geq a^2 b^2 \plus{} b^2 c^2 \plus{}a^2 c^2\geq abc(a\plus{}b\plus{}c)$

[u]But try to prove:[/u]
$ a^3 b \plus{} b^3 c \plus{} b^3 a \geq abc(a\plus{}b\plus{}c)$

That's just Am-Gm:
$ \sum_{cyc} \frac{4a^2}{7c}\plus{} \frac{b^2}{7a}\plus{} \frac{2c^2}{7b} \ge \sum_{cyc} a$.

[quote="fen"]It's true end easy to see that:
$ a^4 \plus{} b^4 \plus{} c^4 \geq a^2 b^2 \plus{} b^2 c^2 \plus{} a^2 c^2\geq abc(a \plus{} b \plus{} c)$

[u]But try to prove:[/u]
$ a^3 b \plus{} b^3 c \plus{} b^3 a \geq abc(a \plus{} b \plus{} c)$[/quote]
your inequality is equivalent as,
$ \sum \frac{a^2}{c} \geq a\plus{}b\plus{}c$
wich is very easy only by Cauchy-Schwarz.

[quote="Smilley"]Ok then my ineq isn't true for all a,b,c positive

But is it true that

$ 2 a^3 b \plus{} 2 b^3 c \plus{} 2 b^3 a \geq a^2 b^2 \plus{} b^2 c^2 \plus{} c^2 a^2 \plus{} a^2 bc \plus{} b^2 ac \plus{} c^2 ab$[/quote]
Try $ b\equal{}0, a\equal{}c>0$...
Did you mean:
$ 2 a^3 b \plus{} 2 b^3 c \plus{} 2 c^3 a \geq a^2 b^2 \plus{} b^2 c^2 \plus{} c^2 a^2 \plus{} a^2 bc \plus{} b^2 ac \plus{} c^2 ab$ ?

[quote="Mathias_DK"][quote="Smilley"]Ok then my ineq isn't true for all a,b,c positive

But is it true that

$ 2 a^3 b \plus{} 2 b^3 c \plus{} 2 b^3 a \geq a^2 b^2 \plus{} b^2 c^2 \plus{} c^2 a^2 \plus{} a^2 bc \plus{} b^2 ac \plus{} c^2 ab$[/quote]
Try $ b \equal{} 0, a \equal{} c > 0$...
Did you mean:
$ 2 a^3 b \plus{} 2 b^3 c \plus{} 2 c^3 a \geq a^2 b^2 \plus{} b^2 c^2 \plus{} c^2 a^2 \plus{} a^2 bc \plus{} b^2 ac \plus{} c^2 ab (1)$ ?[/quote]

The last one is also wrong. Try a = 100, b = 300, c = 1. (Actually we can put a = 1, b = 3, c = 0, but it take long time to explain).

In addition, note that (1) becomes true if a, b, c is sides of a triangle.


I am currently studying at UC Davis as a sophmore. I applied to UC San Diege, UC LA, and UC Berkeley as a tranfer student and got into UCSD (in the John Muir college) and UCLA. My major is electrical engineering and I love math and physics.

I am kinda undecided on which school should I pick or should I just stay at UC Davis. Do you guys have any suggestions?

Thanks,
-Swapnil

They say that you should be carefull about what college you get into in UCSD. Is John Muir a good college?

UCLA has terrence tao, who many suspect will win the fields medal this summer.

er...I don't know if his awesomeness at math generalizes to the whole department.

[quote="agolsme"]UCLA has terrence tao, who many suspect will win the fields medal this summer.
[/quote]

UCSD has Efim Zelmanov, who already won a Fields Medal ;)  (He also taught at our Math Circle - more points for him!)

I have a friend who is a prof in the EE/CS department at UCSD who is, I think, one of the 5-10 best problem solvers I know.  And I know a lot of good problem solvers.

Going to a given university for your undergrad because of the presence of one or two very bright profs is generally a very poor idea.

UCSD and UCLA are both very good schools for math from what I know, with UCSD having more of an algebra/discrete math focus and UCLA having more of an analysis/applied focus.  Neither of these focuses will make much of a difference if you are an undergrad.

UCLA, better sports games, hehe, just a thought :D