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1966 IMO Problems/Problem 6

2012-05-16T17:01:01Z

Solution:

← Older revision		Revision as of 17:01, 16 May 2012
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	== Solution ==		== Solution ==
-	~~{{solution}}~~	+	Let the lengths of sides <math>BC</math>, <math>CA</math>, and <math>AB</math> be <math>a</math>, <math>b</math>, and <math>c</math>, respectively. Let <math>BK=d</math>, <math>CL=e</math>, and <math>AM=f</math>.

-	== See ~~also~~ ==	+	Now assume for the sake of contradiction that the areas of <math>\Delta AML</math>, <math>\Delta BKM</math>, and <math>\Delta CLK</math> are all at greater than one fourth of that of <math>\Delta ABC</math>. Therefore
		+
		+	<cmath>\frac{AM\cdot AL\sin{\angle BAC}}{2}>\frac{AB\cdot AC\sin{\angle BAC}}{8}</cmath>
		+
		+	In other words, <math>AM\cdot AL>\frac{1}{4}AB\cdot AC</math>, or <math>f(b-e)>\frac{bc}{4}</math>. Similarly, <math>d(c-f)>\frac{ac}{4}</math> and <math>e(a-d)>\frac{ab}{4}</math>. Multiplying these three inequalities together yields
		+
		+	<cmath>def(a-d)(b-e)(c-f)>\frac{a^2b^2c^2}{64}</cmath>
		+
		+	We also have that <math>d(a-d)\leq \frac{a^2}{4}</math>, <math>e(b-e)\leq \frac{b^2}{4}</math>, and <math>f(c-f)\leq \frac{c^2}{4}</math> from the [[Arithmetic Mean-Geometric Mean Inequality]]. Multiplying these three inequalities together yields
		+
		+	<cmath>def(a-d)(b-e)(c-f)\leq\frac{a^2b^2c^2}{64}</cmath>
		+
		+	This is a contradiction, which shows that our assumption must have been false in the first place. This proves the desired result.
		+
		+	== See Also ==
	{{IMO box\|year=1966\|num-b=5\|after=Last Problem}}		{{IMO box\|year=1966\|num-b=5\|after=Last Problem}}
	[[Category:Olympiad Geometry Problems]]		[[Category:Olympiad Geometry Problems]]

User talk:1=2

2012-05-16T16:46:49Z

added more incompletes

Show changes

Math books

2012-05-16T16:26:52Z

updated links to AoPS textbooks, added Prealgebra text

← Older revision		Revision as of 16:26, 16 May 2012
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	=== Algebra ===		=== Algebra ===
	====Getting Started====		====Getting Started====
		+	* [[AoPS]] publishes [[Richard Rusczyk]]'s, [[David Patrick]]'s, and [[Ravi Boppana]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=prealgebra Prealgebar] textbook, which is recommended for advanced elementary and middle school students.
	* [[AoPS]] publishes [[Richard Rusczyk]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:algebra Introduction to Algebra] textbook, which is recommended for advanced elementary, middle, and high school students.		* [[AoPS]] publishes [[Richard Rusczyk]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:algebra Introduction to Algebra] textbook, which is recommended for advanced elementary, middle, and high school students.

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	* [http://www.amazon.com/exec/obidos/ASIN/0817636773/artofproblems-20 Algebra] by I.M. Gelfand and Alexander Shen.		* [http://www.amazon.com/exec/obidos/ASIN/0817636773/artofproblems-20 Algebra] by I.M. Gelfand and Alexander Shen.
	* [http://www.amazon.com/Problems-Algebra-Training-Team-Enrichment/dp/187642012X/ref=sr_1_2?ie=UTF8&s=books&qid=1204029534&sr=8-2 101 Problems in Algebra from the Training of the US IMO Team] by Titu Andreescu and Zuming Feng		* [http://www.amazon.com/Problems-Algebra-Training-Team-Enrichment/dp/187642012X/ref=sr_1_2?ie=UTF8&s=books&qid=1204029534&sr=8-2 101 Problems in Algebra from the Training of the US IMO Team] by Titu Andreescu and Zuming Feng
-	* [[AoPS]] publishes [[Richard Rusczyk]]'s and [[Mathew Crawford]]'s [http://www.artofproblemsolving.com/~~Books~~/~~AoPS_B_Item~~.php?~~item_id~~=~~300~~ Intermediate Algebra] textbook, which is recommended for advanced middle and high school students.	+	* [[AoPS]] publishes [[Richard Rusczyk]]'s and [[Mathew Crawford]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=interm:algebra Intermediate Algebra] textbook, which is recommended for advanced middle and high school students.
	* [http://www.amazon.com/Complex-Numbers-Z-Titu-Andreescu/dp/0817643265/ref=sr_1_1?ie=UTF8&s=books&qid=1204029652&sr=1-1 Complex Numbers from A to... Z] by [[Titu Andreescu]]		* [http://www.amazon.com/Complex-Numbers-Z-Titu-Andreescu/dp/0817643265/ref=sr_1_1?ie=UTF8&s=books&qid=1204029652&sr=1-1 Complex Numbers from A to... Z] by [[Titu Andreescu]]

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	=== Calculus ===		=== Calculus ===
	==== High School ====		==== High School ====
		+	* [[AoPS]] publishes Dr. [[David Patrick]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=calculus Calculus] textbook, which is recommended for advanced middle and high school students.
	* [http://www.amazon.com/exec/obidos/ASIN/0914098896/artofproblems-20 Calculus] by [[Michael Spivak]]. Top students swear by this book.		* [http://www.amazon.com/exec/obidos/ASIN/0914098896/artofproblems-20 Calculus] by [[Michael Spivak]]. Top students swear by this book.
	* [http://www.amazon.com/exec/obidos/ASIN/0883858126/artofproblems-20 The Hitchhiker's Guide to Calculus] by [[Michael Spivak]].		* [http://www.amazon.com/exec/obidos/ASIN/0883858126/artofproblems-20 The Hitchhiker's Guide to Calculus] by [[Michael Spivak]].
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	=== Combinatorics ===		=== Combinatorics ===
	==== Getting Started ====		==== Getting Started ====
-	* [[AoPS]] publishes Dr. [[David Patrick]]'s [http://www.artofproblemsolving.com/~~Books~~/~~AoPS_B_Item~~.php?~~page_id~~=3 Introduction to Counting & Probability] textbook, which is recommended for advanced middle and high school students.	+	* [[AoPS]] publishes Dr. [[David Patrick]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:counting Introduction to Counting & Probability] textbook, which is recommended for advanced middle and high school students.

	==== Intermediate ====		==== Intermediate ====
-	* [[AoPS]] publishes Dr. [[David Patrick]]'s [http://www.artofproblemsolving.com/~~Books~~/~~AoPS_B_Item~~.php?~~item_id~~=~~302~~ Intermediate Counting & Probability] textbook, which is recommended for advanced middle and high school students.	+	* [[AoPS]] publishes Dr. [[David Patrick]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=interm:counting Intermediate Counting & Probability] textbook, which is recommended for advanced middle and high school students.
	* [http://www.amazon.com/exec/obidos/ASIN/0883856158/artofproblems-20 Mathematics of Choice] by Ivan Niven.		* [http://www.amazon.com/exec/obidos/ASIN/0883856158/artofproblems-20 Mathematics of Choice] by Ivan Niven.
	* [http://www.amazon.com/exec/obidos/ASIN/0817643176/artofproblems-20 102 Combinatorial Problems] by [[Titu Andreescu]] and [[Zuming Feng]].		* [http://www.amazon.com/exec/obidos/ASIN/0817643176/artofproblems-20 102 Combinatorial Problems] by [[Titu Andreescu]] and [[Zuming Feng]].
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	=== Geometry ===		=== Geometry ===
	==== Getting Started ====		==== Getting Started ====
-	* [[AoPS]] publishes [[Richard Rusczyk]]'s [http://www.artofproblemsolving.com/~~Books~~/~~AoPS_B_Item~~.php?~~page_id~~=9 Introduction to Geometry] textbook, which is recommended for advanced middle and high school students.	+	* [[AoPS]] publishes [[Richard Rusczyk]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:geometry Introduction to Geometry] textbook, which is recommended for advanced middle and high school students.

	==== Intermediate ====		==== Intermediate ====
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	=== Number Theory ===		=== Number Theory ===
	==== Introductory ====		==== Introductory ====
-	* The AoPS [http://www.artofproblemsolving.com/~~Books~~/~~AoPS_B_Item~~.php?~~page_id~~=10 Introduction to Number Theory] by [[Mathew Crawford]].	+	* The AoPS [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:nt Introduction to Number Theory] by [[Mathew Crawford]].
	==== Olympiad ====		==== Olympiad ====
	* [http://www.amazon.com/exec/obidos/ASIN/081763245X/artofproblems-20 Number Theory: A Problem-Solving Approach] by [[Titu Andreescu]] and Dorin Andrica.		* [http://www.amazon.com/exec/obidos/ASIN/081763245X/artofproblems-20 Number Theory: A Problem-Solving Approach] by [[Titu Andreescu]] and Dorin Andrica.

2003 USAMO

2012-05-16T16:13:34Z

made it perty

← Older revision		Revision as of 16:13, 16 May 2012
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-	The '''32nd [[USAMO]]''' was held on April 29–30, 2003.	+	The '''32nd [[USAMO]]''' was held on April 29–30, 2003. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	* [[2003 USAMO Problems]]	+	* [[2003 USAMO Problems\|Entire Test]]
-	* [[2003 USAMO Problems/Problem 1 \| Problem 1]]	+	** [[2003 USAMO Problems/Problem 1 \| Problem 1]]
-	* [[2003 USAMO Problems/Problem 2 \| Problem 2]]	+	** [[2003 USAMO Problems/Problem 2 \| Problem 2]]
-	* [[2003 USAMO Problems/Problem 3 \| Problem 3]]	+	** [[2003 USAMO Problems/Problem 3 \| Problem 3]]
-	* [[2003 USAMO Problems/Problem 4 \| Problem 4]]	+	** [[2003 USAMO Problems/Problem 4 \| Problem 4]]
-	* [[2003 USAMO Problems/Problem 5 \| Problem 5]]	+	** [[2003 USAMO Problems/Problem 5 \| Problem 5]]
-	* [[2003 USAMO Problems/Problem 6 \| Problem 6]]	+	** [[2003 USAMO Problems/Problem 6 \| Problem 6]]

-	== ~~Resources~~ ==	+	== See Also ==
		+	* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-tex/usamo2003.tex USAMO 2003 (.TEX file)]
		+	* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-pdf/usamo2003.pdf USAMO 2003 (PDF)]
		+	* [[Mathematics competitions]]
		+	* [[Mathematics competition resources]]
		+	* [[Math books]]

-	* [[USAMO Problems and Solutions]]	+	{{USAMO box\|year=2003\|before=[[2002 USAMO]]\|after=[[2004 USAMO]]}}
-		+
-		+
-	{{USAMO ~~newbox~~\|year=2003\|before=[[2002 USAMO]]\|after=[[2004 USAMO]]}}	+

2002 USAMO

2012-05-16T16:12:48Z

made it perty

← Older revision		Revision as of 16:12, 16 May 2012
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-	The participants of the 2002 [[USAMO]] were invited to the [[MIT]] campus in Cambridge, Massachussetts, for the exam [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2002-ua/02usamolist.html]. For the first time, four and a half hours, instead of three, were allowed for each paper of three questions.	+	The participants of the 2002 [[USAMO]] were invited to the [[MIT]] campus in Cambridge, Massachussetts, for the exam [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2002-ua/02usamolist.html]. For the first time, four and a half hours, instead of three, were allowed for each paper of three questions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	== Problems ==	+	* [[2002 USAMO Problems\|Entire Test]]
		+	** [[2002 USAMO Problems/Problem 1 \| Problem 1]]
		+	** [[2002 USAMO Problems/Problem 2 \| Problem 2]]
		+	** [[2002 USAMO Problems/Problem 3 \| Problem 3]]
		+	** [[2002 USAMO Problems/Problem 4 \| Problem 4]]
		+	** [[2002 USAMO Problems/Problem 5 \| Problem 5]]
		+	** [[2002 USAMO Problems/Problem 6 \| Problem 6]]

-	* [~~[2002~~ USAMO ~~Problems]~~]	+	== See Also ==
-	* [~~[2002~~ USAMO ~~Problems~~/~~Problem 1 \| Problem 1]~~]	+	* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-tex/usamo2002.tex USAMO 2002 (.TEX file)]
-	* [[~~2002 USAMO Problems/Problem 2 \| Problem 2~~]]	+	* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-pdf/usamo2002.pdf USAMO 2002 (PDF)]
-	* [[~~2002 USAMO Problems/Problem 3 \| Problem 3~~]]	+	* [[Mathematics competitions]]
-	* [[~~2002 USAMO Problems/Problem 4 \| Problem 4~~]]	+	* [[Mathematics competition resources]]
-	* [[~~2002~~ USAMO ~~Problems/Problem 5 \| Problem 5~~]]	+	* [[Math books]]
-	* [[~~2002~~ USAMO ~~Problems/Problem 6 \| Problem 6~~]]	+
		+	{{USAMO box\|year=2002\|before=[[2001 USAMO]]\|after=[[2003 USAMO]]}}

2001 USAMO

2012-05-16T16:11:28Z

made it perty

← Older revision		Revision as of 16:11, 16 May 2012
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-	The '''2001 [[USAMO]]''' was held on May 1, 2001. The examination has six questions divided into Part 1 (9 a.m. to 12 p.m.) and Part 2 (1 p.m. to 4 p.m.)	+	The '''2001 [[USAMO]]''' was held on May 1, 2001. The examination has six questions divided into Part 1 (9 a.m. to 12 p.m.) and Part 2 (1 p.m. to 4 p.m.) The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	*[[2001 USAMO Problems/Problem 1]]	+	* [[2001 USAMO Problems\|Entire Test]]
-	*[[2001 USAMO Problems/Problem 2]]	+	** [[2001 USAMO Problems/Problem 1 \| Problem 1]]
-	*[[2001 USAMO Problems/Problem 3]]	+	** [[2001 USAMO Problems/Problem 2 \| Problem 2]]
-	*[[2001 USAMO Problems/Problem 4]]	+	** [[2001 USAMO Problems/Problem 3 \| Problem 3]]
-	*[[2001 USAMO Problems/Problem 5]]	+	** [[2001 USAMO Problems/Problem 4 \| Problem 4]]
-	*[[2001 USAMO Problems/Problem 6]]	+	** [[2001 USAMO Problems/Problem 5 \| Problem 5]]
		+	** [[2001 USAMO Problems/Problem 6 \| Problem 6]]
		+
		+	== See Also ==
		+	* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-tex/usamo2001.tex USAMO 2001 (.TEX file)]
		+	* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-pdf/usamo2001.pdf USAMO 2001 (PDF)]
		+	* [[Mathematics competitions]]
		+	* [[Mathematics competition resources]]
		+	* [[Math books]]
		+
		+	{{USAMO box\|year=2001\|before=[[2000 USAMO]]\|after=[[2002 USAMO]]}}

2000 USAMO

2012-05-16T16:09:23Z

made it perty

← Older revision		Revision as of 16:09, 16 May 2012
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-	The '''30th [[USAMO]]''' was held in April 2000. The first part of the exam occurred in the morning; the second part occurred in the afternoon. Three hours were permitted for each part.	+	The '''30th [[USAMO]]''' was held April, 2000. The first part of the exam occurred in the morning; the second part occurred in the afternoon. Three hours were permitted for each part. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	* [[2000 USAMO Problems]]	+	* [[2000 USAMO Problems\|Entire Test]]
-	* [[2000 USAMO Problems/Problem 1 \| Problem 1]]	+	** [[2000 USAMO Problems/Problem 1 \| Problem 1]]
-	* [[2000 USAMO Problems/Problem 2 \| Problem 2]]	+	** [[2000 USAMO Problems/Problem 2 \| Problem 2]]
-	* [[2000 USAMO Problems/Problem 3 \| Problem 3]]	+	** [[2000 USAMO Problems/Problem 3 \| Problem 3]]
-	* [[2000 USAMO Problems/Problem 4 \| Problem 4]]	+	** [[2000 USAMO Problems/Problem 4 \| Problem 4]]
-	* [[2000 USAMO Problems/Problem 5 \| Problem 5]]	+	** [[2000 USAMO Problems/Problem 5 \| Problem 5]]
-	* [[2000 USAMO Problems/Problem 6 \| Problem 6]]	+	** [[2000 USAMO Problems/Problem 6 \| Problem 6]]

-	== ~~Resources~~ ==	+	== See Also ==
-		+
-	* [[USAMO Problems and Solutions]]	+
-	* <url>resources.php?c=182&cid=27&year=2000 Resources</url>	+
	* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-tex/usamo2000.tex USAMO 2000 (.TEX file)]		* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-tex/usamo2000.tex USAMO 2000 (.TEX file)]
	* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-pdf/usamo2000.pdf USAMO 2000 (PDF)]		* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-pdf/usamo2000.pdf USAMO 2000 (PDF)]
		+	* [[Mathematics competitions]]
		+	* [[Mathematics competition resources]]
		+	* [[Math books]]
		+
		+	{{USAMO box\|year=2000\|before=[[1999 USAMO]]\|after=[[2001 USAMO]]}}

1999 USAMO

2012-05-16T16:08:12Z

made it pertty

← Older revision		Revision as of 16:08, 16 May 2012
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-	The '''29th [[USAMO]]''' was held April 27, 1999. The first part of the exam occurred in the morning; the second part occurred in the afternoon. Three hours were permitted for each part.	+	The '''29th [[USAMO]]''' was held April 27, 1999. The first part of the exam occurred in the morning; the second part occurred in the afternoon. Three hours were permitted for each part. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	* [[1999 USAMO Problems]]	+	* [[1999 USAMO Problems\|Entire Test]]
-	* [[1999 USAMO Problems/Problem 1 \| Problem 1]]	+	** [[1999 USAMO Problems/Problem 1 \| Problem 1]]
-	* [[1999 USAMO Problems/Problem 2 \| Problem 2]]	+	** [[1999 USAMO Problems/Problem 2 \| Problem 2]]
-	* [[1999 USAMO Problems/Problem 3 \| Problem 3]]	+	** [[1999 USAMO Problems/Problem 3 \| Problem 3]]
-	* [[1999 USAMO Problems/Problem 4 \| Problem 4]]	+	** [[1999 USAMO Problems/Problem 4 \| Problem 4]]
-	* [[1999 USAMO Problems/Problem 5 \| Problem 5]]	+	** [[1999 USAMO Problems/Problem 5 \| Problem 5]]
-	* [[1999 USAMO Problems/Problem 6 \| Problem 6]]	+	** [[1999 USAMO Problems/Problem 6 \| Problem 6]]

-	== ~~Resources~~ ==	+	== See Also ==
-		+
-	* [[USAMO Problems and Solutions]]	+
	* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-tex/usamo1999.tex USAMO 1999 (.TEX file)]		* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-tex/usamo1999.tex USAMO 1999 (.TEX file)]
	* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-pdf/usamo1999.pdf USAMO 1999 (PDF)]		* [http://www.unl.edu/amc/a-activities/a7-problems/USAMO-IMO/q-usamo/-pdf/usamo1999.pdf USAMO 1999 (PDF)]
		+	* [[Mathematics competitions]]
		+	* [[Mathematics competition resources]]
		+	* [[Math books]]
		+
		+	{{USAMO box\|year=1999\|before=[[1998 USAMO]]\|after=[[2000 USAMO]]}}

1998 USAMO

2012-05-16T16:05:52Z

made it pretty

← Older revision		Revision as of 16:05, 16 May 2012
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-	~~The~~ 1998 USAMO problems. There were 6 problems total, to be completed over a period of 2 days, 3 problems per day, four and a half hours per day.	+	'''1998 USAMO''' problems and solutions. There were 6 problems total, to be completed over a period of 2 days, 3 problems per day, four and a half hours per day. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	*[[1998 USAMO Problems]]	+	* [[1998 USAMO Problems\|Entire Test]]
-	*[[1998 USAMO Problems/Problem 1]]	+	** [[1998 USAMO Problems/Problem 1\|Problem 1]]
-	*[[1998 USAMO Problems/Problem 2]]	+	** [[1998 USAMO Problems/Problem 2\|Problem 2]]
-	*[[1998 USAMO Problems/Problem 3]]	+	** [[1998 USAMO Problems/Problem 3\|Problem 3]]
-	*[[1998 USAMO Problems/Problem 4]]	+	** [[1998 USAMO Problems/Problem 4\|Problem 4]]
-	*[[1998 USAMO Problems/Problem 5]]	+	** [[1998 USAMO Problems/Problem 5\|Problem 5]]
-	*[[1998 USAMO Problems/Problem 6]]	+	** [[1998 USAMO Problems/Problem 6\|Problem 6]]
		+
		+	== See Also ==
		+	* [[Mathematics competitions]]
		+	* [[Mathematics competition resources]]
		+	* [[Math books]]
		+	* [[USAMO]]
		+
		+	{{USAMO box\|year=1998\|before=[[1997 USAMO]]\|after=[[1999 USAMO]]}}

1995 USAMO

2012-05-16T16:04:53Z

made it pretty

← Older revision		Revision as of 16:04, 16 May 2012
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-	~~The~~ '''1995 [[USAMO]]''' ~~was~~ the ~~twenty-fourth USAMO~~.	+	'''1995 USAMO''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	*[[1995 USAMO Problems]]	+	* [[1995 USAMO Problems\|Entire Test]]
-	**[[1995 USAMO Problems/Problem 1\|Problem 1]]	+	** [[1995 USAMO Problems/Problem 1\|Problem 1]]
-	**[[1995 USAMO Problems/Problem 2\|Problem 2]]	+	** [[1995 USAMO Problems/Problem 2\|Problem 2]]
-	**[[1995 USAMO Problems/Problem 3\|Problem 3]]	+	** [[1995 USAMO Problems/Problem 3\|Problem 3]]
-	**[[1995 USAMO Problems/Problem 4\|Problem 4]]	+	** [[1995 USAMO Problems/Problem 4\|Problem 4]]
-	**[[1995 USAMO Problems/Problem 5\|Problem 5]]	+	** [[1995 USAMO Problems/Problem 5\|Problem 5]]
-	**[[~~1995~~ USAMO ~~Problems/Problem 6\|Problem 6~~]]	+
		+	== See Also ==
		+	* [[Mathematics competitions]]
		+	* [[Mathematics competition resources]]
		+	* [[Math books]]
		+	* [[USAMO]]

	{{USAMO box\|year=1995\|before=[[1994 USAMO]]\|after=[[1996 USAMO]]}}		{{USAMO box\|year=1995\|before=[[1994 USAMO]]\|after=[[1996 USAMO]]}}

1994 USAMO

2012-05-16T16:03:55Z

made it pretty

← Older revision		Revision as of 16:03, 16 May 2012
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-	The '''23<sup>rd</sup> [[USAMO]]''' occurred April 28<sup>th</sup>, 1994. There were five problems given in one session of three and a half hours.	+	The '''23<sup>rd</sup> [[USAMO]]''' occurred April 28<sup>th</sup>, 1994. There were five problems given in one session of three and a half hours. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	*[[1994 USAMO Problems]]	+	* [[1994 USAMO Problems\|Entire Test]]
-	**[[1994 USAMO Problems/Problem 1\|Problem 1]]	+	** [[1994 USAMO Problems/Problem 1\|Problem 1]]
-	**[[1994 USAMO Problems/Problem 2\|Problem 2]]	+	** [[1994 USAMO Problems/Problem 2\|Problem 2]]
-	**[[1994 USAMO Problems/Problem 3\|Problem 3]]	+	** [[1994 USAMO Problems/Problem 3\|Problem 3]]
-	**[[1994 USAMO Problems/Problem 4\|Problem 4]]	+	** [[1994 USAMO Problems/Problem 4\|Problem 4]]
-	**[[1994 USAMO Problems/Problem 5\|Problem 5]]	+	** [[1994 USAMO Problems/Problem 5\|Problem 5]]
		+
		+	== See Also ==
		+	* [[Mathematics competitions]]
		+	* [[Mathematics competition resources]]
		+	* [[Math books]]
		+	* [[USAMO]]

	{{USAMO box\|year=1994\|before=[[1993 USAMO]]\|after=[[1995 USAMO]]}}		{{USAMO box\|year=1994\|before=[[1993 USAMO]]\|after=[[1995 USAMO]]}}

1993 USAMO

2012-05-16T16:03:28Z

made it pretty

← Older revision		Revision as of 16:03, 16 May 2012
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-	The '''22<sup>nd</sup> [[USAMO]]''' occurred April 29<sup>th</sup>, 1993. There were five problems given in one session of three and a half hours.	+	The '''22<sup>nd</sup> [[USAMO]]''' occurred April 29<sup>th</sup>, 1993. There were five problems given in one session of three and a half hours. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	* [[1993 USAMO Problems]]	+	* [[1993 USAMO Problems\|Entire Test]]
-	* [[1993 USAMO Problems/Problem 1 \| Problem 1]]	+	** [[1993 USAMO Problems/Problem 1\|Problem 1]]
-	* [[1993 USAMO Problems/Problem 2 \| Problem 2]]	+	** [[1993 USAMO Problems/Problem 2\|Problem 2]]
-	* [[1993 USAMO Problems/Problem 3 \| Problem 3]]	+	** [[1993 USAMO Problems/Problem 3\|Problem 3]]
-	* [[1993 USAMO Problems/Problem 4 \| Problem 4]]	+	** [[1993 USAMO Problems/Problem 4\|Problem 4]]
-	* [[1993 USAMO Problems/Problem 5 \| Problem 5]]	+	** [[1993 USAMO Problems/Problem 5\|Problem 5]]

-	== ~~Resources~~ ==	+	== See Also ==
		+	* [[Mathematics competitions]]
		+	* [[Mathematics competition resources]]
		+	* [[Math books]]
		+	* [[USAMO]]

	{{USAMO box\|year=1993\|before=[[1992 USAMO]]\|after=[[1994 USAMO]]}}		{{USAMO box\|year=1993\|before=[[1992 USAMO]]\|after=[[1994 USAMO]]}}

1992 USAMO

2012-05-16T16:02:58Z

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-	The '''21st [[USAMO]]''' occurred April 30, 1992. There were five problems given in one session of three and a half hours.	+	The '''21st [[USAMO]]''' occurred April 30, 1992. There were five problems given in one session of three and a half hours. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	* [[1992 USAMO Problems]]	+	* [[1992 USAMO Problems\|Entire Test]]
-	* [[1992 USAMO Problems/Problem 1 \| Problem 1]]	+	** [[1992 USAMO Problems/Problem 1\|Problem 1]]
-	* [[1992 USAMO Problems/Problem 2 \| Problem 2]]	+	** [[1992 USAMO Problems/Problem 2\|Problem 2]]
-	* [[1992 USAMO Problems/Problem 3 \| Problem 3]]	+	** [[1992 USAMO Problems/Problem 3\|Problem 3]]
-	* [[1992 USAMO Problems/Problem 4 \| Problem 4]]	+	** [[1992 USAMO Problems/Problem 4\|Problem 4]]
-	* [[1992 USAMO Problems/Problem 5 \| Problem 5]]	+	** [[1992 USAMO Problems/Problem 5\|Problem 5]]

-	== ~~Resources~~ ==	+	== See Also ==
		+	* [[Mathematics competitions]]
		+	* [[Mathematics competition resources]]
		+	* [[Math books]]
		+	* [[USAMO]]

	{{USAMO box\|year=1992\|before=[[1991 USAMO]]\|after=[[1993 USAMO]]}}		{{USAMO box\|year=1992\|before=[[1991 USAMO]]\|after=[[1993 USAMO]]}}

1991 USAMO

2012-05-16T16:02:31Z

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← Older revision		Revision as of 16:02, 16 May 2012
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-	The '''20th [[USAMO]]''' occurred April 23, 1991. There were five problems, and the time limit was three and a half hours.	+	The '''20th [[USAMO]]''' occurred April 23, 1991. There were five problems, and the time limit was three and a half hours. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	* [[1991 USAMO Problems]]	+	* [[1991 USAMO Problems\|Entire Test]]
-	* [[1991 USAMO Problems/Problem 1 \| Problem 1]]	+	** [[1991 USAMO Problems/Problem 1\|Problem 1]]
-	* [[1991 USAMO Problems/Problem 2 \| Problem 2]]	+	** [[1991 USAMO Problems/Problem 2\|Problem 2]]
-	* [[1991 USAMO Problems/Problem 3 \| Problem 3]]	+	** [[1991 USAMO Problems/Problem 3\|Problem 3]]
-	* [[1991 USAMO Problems/Problem 4 \| Problem 4]]	+	** [[1991 USAMO Problems/Problem 4\|Problem 4]]
-	* [[1991 USAMO Problems/Problem 5 \| Problem 5]]	+	** [[1991 USAMO Problems/Problem 5\|Problem 5]]

-	== ~~Resources~~ ==	+	== See Also ==
		+	* [[Mathematics competitions]]
		+	* [[Mathematics competition resources]]
		+	* [[Math books]]
		+	* [[USAMO]]

	{{USAMO box\|year=1991\|before=[[1990 USAMO]]\|after=[[1992 USAMO]]}}		{{USAMO box\|year=1991\|before=[[1990 USAMO]]\|after=[[1992 USAMO]]}}

1990 USAMO

2012-05-16T16:01:50Z

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← Older revision		Revision as of 16:01, 16 May 2012
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-	The '''19th [[USAMO]]''' took place April 24, 1990. The time limit was three and a half hours, and total scores were out of 100 points.	+	The '''19th [[USAMO]]''' took place April 24, 1990. The time limit was three and a half hours, and total scores were out of 100 points. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	* [[1990 USAMO Problems]]	+	* [[1990 USAMO Problems\|Entire Test]]
-	* [[1990 USAMO Problems/Problem 1 \| Problem 1]]	+	** [[1990 USAMO Problems/Problem 1\|Problem 1]]
-	* [[1990 USAMO Problems/Problem 2 \| Problem 2]]	+	** [[1990 USAMO Problems/Problem 2\|Problem 2]]
-	* [[1990 USAMO Problems/Problem 3 \| Problem 3]]	+	** [[1990 USAMO Problems/Problem 3\|Problem 3]]
-	* [[1990 USAMO Problems/Problem 4 \| Problem 4]]	+	** [[1990 USAMO Problems/Problem 4\|Problem 4]]
-	* [[1990 USAMO Problems/Problem 5 \| Problem 5]]	+	** [[1990 USAMO Problems/Problem 5\|Problem 5]]

-	== ~~Resources~~ ==	+	== See Also ==
	* [[Mathematics competitions]]		* [[Mathematics competitions]]
	* [[Mathematics competition resources]]		* [[Mathematics competition resources]]
	* [[Math books]]		* [[Math books]]
		+	* [[USAMO]]

	{{USAMO box\|year=1990\|before=[[1989 USAMO]]\|after=[[1991 USAMO]]}}		{{USAMO box\|year=1990\|before=[[1989 USAMO]]\|after=[[1991 USAMO]]}}

1989 USAMO

2012-05-16T16:01:16Z

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← Older revision		Revision as of 16:01, 16 May 2012
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	'''1989 USAMO''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.		'''1989 USAMO''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	* [[1989 USAMO Problems]]	+	* [[1989 USAMO Problems\|Entire Test]]
-	* [[1989 USAMO Problems/Problem 1]]	+	** [[1989 USAMO Problems/Problem 1\|Problem 1]]
-	* [[1989 USAMO Problems/Problem 2]]	+	** [[1989 USAMO Problems/Problem 2\|Problem 2]]
-	* [[1989 USAMO Problems/Problem 3]]	+	** [[1989 USAMO Problems/Problem 3\|Problem 3]]
-	* [[1989 USAMO Problems/Problem 4]]	+	** [[1989 USAMO Problems/Problem 4\|Problem 4]]
-	* [[1989 USAMO Problems/Problem 5]]	+	** [[1989 USAMO Problems/Problem 5\|Problem 5]]
-	== See ~~also~~ ==	+
		+	== See Also ==
	* [[Mathematics competitions]]		* [[Mathematics competitions]]
	* [[Mathematics competition resources]]		* [[Mathematics competition resources]]
	* [[Math books]]		* [[Math books]]
		+	* [[USAMO]]

	{{USAMO box\|year=1989\|before=[[1988 USAMO]]\|after=[[1990 USAMO]]}}		{{USAMO box\|year=1989\|before=[[1988 USAMO]]\|after=[[1990 USAMO]]}}

1972 USAMO

2012-05-16T16:00:26Z

nitpicking

← Older revision		Revision as of 16:00, 16 May 2012
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-	The '''First [[USAMO]]''' was held May 9, 1972. There were five problems.	+	The '''First [[USAMO]]''' was held May 9, 1972. There were five problems. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

	* [[1972 USAMO Problems\|Entire Test]]		* [[1972 USAMO Problems\|Entire Test]]

1973 USAMO

2012-05-16T15:59:52Z

nitpicking

← Older revision		Revision as of 15:59, 16 May 2012
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-	The '''Second [[USAMO]]''' was held May 1, 1973. There were five problems.	+	The '''Second [[USAMO]]''' was held May 1, 1973. There were five problems. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

	* [[1973 USAMO Problems \| Entire Test]]		* [[1973 USAMO Problems \| Entire Test]]

1974 USAMO

2012-05-16T15:59:32Z

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← Older revision		Revision as of 15:59, 16 May 2012
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-	The '''Third [[USAMO]]''' was held May 7, 1974. There were five problems.	+	The '''Third [[USAMO]]''' was held May 7, 1974. There were five problems. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	* [[1974 USAMO Problems]]	+	* [[1974 USAMO Problems\|Entire Test]]
-	* [[1974 USAMO Problems/Problem 1 \| Problem 1]]	+	** [[1974 USAMO Problems/Problem 1\|Problem 1]]
-	* [[1974 USAMO Problems/Problem 2 \| Problem 2]]	+	** [[1974 USAMO Problems/Problem 2\|Problem 2]]
-	* [[1974 USAMO Problems/Problem 3 \| Problem 3]]	+	** [[1974 USAMO Problems/Problem 3\|Problem 3]]
-	* [[1974 USAMO Problems/Problem 4 \| Problem 4]]	+	** [[1974 USAMO Problems/Problem 4\|Problem 4]]
-	* [[1974 USAMO Problems/Problem 5 \| Problem 5]]	+	** [[1974 USAMO Problems/Problem 5\|Problem 5]]

-	== ~~Resources~~ ==	+	== See Also ==
		+	* [[Mathematics competitions]]
		+	* [[Mathematics competition resources]]
		+	* [[Math books]]
		+	* [[USAMO]]

	{{USAMO box\|year=1974\|before=[[1973 USAMO]]\|after=[[1975 USAMO]]}}		{{USAMO box\|year=1974\|before=[[1973 USAMO]]\|after=[[1975 USAMO]]}}

1975 USAMO

2012-05-16T15:59:05Z

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← Older revision		Revision as of 15:59, 16 May 2012
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-	The '''Fourth [[USAMO]]''' was held May 6, 1975. There were five problems.	+	The '''Fourth [[USAMO]]''' was held May 6, 1975. There were five problems. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	* [[1975 USAMO Problems]]	+	* [[1975 USAMO Problems\|Entire Test]]
-	* [[1975 USAMO Problems/Problem 1 \| Problem 1]]	+	** [[1975 USAMO Problems/Problem 1\|Problem 1]]
-	* [[1975 USAMO Problems/Problem 2 \| Problem 2]]	+	** [[1975 USAMO Problems/Problem 2\|Problem 2]]
-	* [[1975 USAMO Problems/Problem 3 \| Problem 3]]	+	** [[1975 USAMO Problems/Problem 3\|Problem 3]]
-	* [[1975 USAMO Problems/Problem 4 \| Problem 4]]	+	** [[1975 USAMO Problems/Problem 4\|Problem 4]]
-	* [[1975 USAMO Problems/Problem 5 \| Problem 5]]	+	** [[1975 USAMO Problems/Problem 5\|Problem 5]]

-	== ~~Resources~~ ==	+	== See Also ==
		+	* [[Mathematics competitions]]
		+	* [[Mathematics competition resources]]
		+	* [[Math books]]
		+	* [[USAMO]]

	{{USAMO box\|year=1975\|before=[[1974 USAMO]]\|after=[[1976 USAMO]]}}		{{USAMO box\|year=1975\|before=[[1974 USAMO]]\|after=[[1976 USAMO]]}}

1976 USAMO

2012-05-16T15:58:34Z

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← Older revision		Revision as of 15:58, 16 May 2012
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-	The '''Fifth [[USAMO]]''' was held on May 4, 1976. There were five problems.	+	The '''Fifth [[USAMO]]''' was held on May 4, 1976. There were five problems. The first link contains the full set of test problems. The rest contain each individual problem and its solution.
-	* [[1976 USAMO Problems]]	+
-	* [[1976 USAMO Problems/Problem 1 \| Problem 1]]	+
-	* [[1976 USAMO Problems/Problem 2 \| Problem 2]]	+
-	* [[1976 USAMO Problems/Problem 3 \| Problem 3]]	+
-	* [[1976 USAMO Problems/Problem 4 \| Problem 4]]	+
-	* [[1976 USAMO Problems/Problem 5 \| Problem 5]]	+

-	== ~~Resources~~ ==	+	* [[1976 USAMO Problems\|Entire Test]]
		+	** [[1976 USAMO Problems/Problem 1\|Problem 1]]
		+	** [[1976 USAMO Problems/Problem 2\|Problem 2]]
		+	** [[1976 USAMO Problems/Problem 3\|Problem 3]]
		+	** [[1976 USAMO Problems/Problem 4\|Problem 4]]
		+	** [[1976 USAMO Problems/Problem 5\|Problem 5]]
		+
		+	== See Also ==
		+	* [[Mathematics competitions]]
		+	* [[Mathematics competition resources]]
		+	* [[Math books]]
		+	* [[USAMO]]

	{{USAMO box\|year=1976\|before=[[1975 USAMO]]\|after=[[1977 USAMO]]}}		{{USAMO box\|year=1976\|before=[[1975 USAMO]]\|after=[[1977 USAMO]]}}

1977 USAMO

2012-05-16T15:58:04Z

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← Older revision		Revision as of 15:58, 16 May 2012
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-	~~The~~ '''~~Sixth [[~~USAMO]]''' ~~was held on *****, 1977~~. ~~There were five~~ problems.	+	'''1977 USAMO''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.
-	* [[1977 USAMO Problems]]	+
-	* [[1977 USAMO Problems/Problem 1 \| Problem 1]]	+
-	* [[1977 USAMO Problems/Problem 2 \| Problem 2]]	+
-	* [[1977 USAMO Problems/Problem 3 \| Problem 3]]	+
-	* [[1977 USAMO Problems/Problem 4 \| Problem 4]]	+
-	* [[1977 USAMO Problems/Problem 5 \| Problem 5]]	+

-	== ~~Resources~~ ==	+	* [[1977 USAMO Problems\|Entire Test]]
		+	** [[1977 USAMO Problems/Problem 1\|Problem 1]]
		+	** [[1977 USAMO Problems/Problem 2\|Problem 2]]
		+	** [[1977 USAMO Problems/Problem 3\|Problem 3]]
		+	** [[1977 USAMO Problems/Problem 4\|Problem 4]]
		+	** [[1977 USAMO Problems/Problem 5\|Problem 5]]
		+
		+	== See Also ==
		+	* [[Mathematics competitions]]
		+	* [[Mathematics competition resources]]
		+	* [[Math books]]
		+	* [[USAMO]]

	{{USAMO box\|year=1977\|before=[[1976 USAMO]]\|after=[[1978 USAMO]]}}		{{USAMO box\|year=1977\|before=[[1976 USAMO]]\|after=[[1978 USAMO]]}}

1978 USAMO

2012-05-16T15:57:34Z

redirects are ugly. I made the page pretty.

New page

'''1978 USAMO''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

* [[1978 USAMO Problems|Entire Test]]
** [[1978 USAMO Problems/Problem 1|Problem 1]]
** [[1978 USAMO Problems/Problem 2|Problem 2]]
** [[1978 USAMO Problems/Problem 3|Problem 3]]
** [[1978 USAMO Problems/Problem 4|Problem 4]]
** [[1978 USAMO Problems/Problem 5|Problem 5]]

== See Also ==
* [[Mathematics competitions]]
* [[Mathematics competition resources]]
* [[Math books]]
* [[USAMO]]

{{USAMO box|year=1978|before=[[1977 USAMO]]|after=[[1979 USAMO]]}}

1979 USAMO

2012-05-16T15:56:37Z

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← Older revision		Revision as of 15:56, 16 May 2012
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-	The '''Eighth [[USAMO]]''' was held on May 1, 1979. There were five problems.	+	The '''Eighth [[USAMO]]''' was held on May 1, 1979. There were five problems. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	* [[1979 USAMO Problems]]	+	* [[1979 USAMO Problems\|Entire Test]]
-	* [[1979 USAMO Problems/Problem 1 \| Problem 1]]	+	** [[1979 USAMO Problems/Problem 1\|Problem 1]]
-	* [[1979 USAMO Problems/Problem 2 \| Problem 2]]	+	** [[1979 USAMO Problems/Problem 2\|Problem 2]]
-	* [[1979 USAMO Problems/Problem 3 \| Problem 3]]	+	** [[1979 USAMO Problems/Problem 3\|Problem 3]]
-	* [[1979 USAMO Problems/Problem 4 \| Problem 4]]	+	** [[1979 USAMO Problems/Problem 4\|Problem 4]]
-	* [[1979 USAMO Problems/Problem 5 \| Problem 5]]	+	** [[1979 USAMO Problems/Problem 5\|Problem 5]]

-	== ~~Resources~~ ==	+	== See Also ==
		+	* [[Mathematics competitions]]
		+	* [[Mathematics competition resources]]
		+	* [[Math books]]
		+	* [[USAMO]]

-	*[[USAMO ~~Problems and Solutions~~]]	+	{{USAMO box\|year=1979\|before=[[1978 USAMO]]\|after=[[1980 USAMO]]}}

1980 USAMO

2012-05-16T15:56:03Z

redirects are ugly. I made the page pretty.

New page

'''1980 USAMO''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

* [[1980 USAMO Problems|Entire Test]]
** [[1980 USAMO Problems/Problem 1|Problem 1]]
** [[1980 USAMO Problems/Problem 2|Problem 2]]
** [[1980 USAMO Problems/Problem 3|Problem 3]]
** [[1980 USAMO Problems/Problem 4|Problem 4]]
** [[1980 USAMO Problems/Problem 5|Problem 5]]

== See Also ==
* [[Mathematics competitions]]
* [[Mathematics competition resources]]
* [[Math books]]
* [[USAMO]]

{{USAMO box|year=1980|before=[[1979 USAMO]]|after=[[1981 USAMO]]}}

1981 USAMO

2012-05-16T15:55:23Z

redirects are ugly. I made the page pretty.

← Older revision		Revision as of 15:55, 16 May 2012
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-	~~#REDIRECT~~ [[1981 USAMO Problems]]	+	'''1981 USAMO''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.
		+
		+	* [[1981 USAMO Problems\|Entire Test]]
		+	** [[1981 USAMO Problems/Problem 1\|Problem 1]]
		+	** [[1981 USAMO Problems/Problem 2\|Problem 2]]
		+	** [[1981 USAMO Problems/Problem 3\|Problem 3]]
		+	** [[1981 USAMO Problems/Problem 4\|Problem 4]]
		+	** [[1981 USAMO Problems/Problem 5\|Problem 5]]
		+
		+	== See Also ==
		+	* [[Mathematics competitions]]
		+	* [[Mathematics competition resources]]
		+	* [[Math books]]
		+	* [[USAMO]]
		+
		+	{{USAMO box\|year=1981\|before=[[1980 USAMO]]\|after=[[1982 USAMO]]}}

1982 USAMO

2012-05-16T15:54:54Z

redirects are ugly. I made the page pretty.

← Older revision		Revision as of 15:54, 16 May 2012
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-	~~#REDIRECT~~ [[1982 USAMO Problems]]	+	'''1982 USAMO''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.
		+
		+	* [[1982 USAMO Problems\|Entire Test]]
		+	** [[1982 USAMO Problems/Problem 1\|Problem 1]]
		+	** [[1982 USAMO Problems/Problem 2\|Problem 2]]
		+	** [[1982 USAMO Problems/Problem 3\|Problem 3]]
		+	** [[1982 USAMO Problems/Problem 4\|Problem 4]]
		+	** [[1982 USAMO Problems/Problem 5\|Problem 5]]
		+
		+	== See Also ==
		+	* [[Mathematics competitions]]
		+	* [[Mathematics competition resources]]
		+	* [[Math books]]
		+	* [[USAMO]]
		+
		+	{{USAMO box\|year=1982\|before=[[1981 USAMO]]\|after=[[1983 USAMO]]}}

1983 USAMO

2012-05-16T15:54:18Z

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← Older revision		Revision as of 15:54, 16 May 2012
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-	==1983 USAMO ~~Problems==~~	+	'''1983 USAMO''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	1983 USAMO ~~problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.~~	+	* [[1983 USAMO Problems\|Entire Test]]
		+	** [[1983 USAMO Problems/Problem 1\|Problem 1]]
		+	** [[1983 USAMO Problems/Problem 2\|Problem 2]]
		+	** [[1983 USAMO Problems/Problem 3\|Problem 3]]
		+	** [[1983 USAMO Problems/Problem 4\|Problem 4]]
		+	** [[1983 USAMO Problems/Problem 5\|Problem 5]]

-	* [[~~1983 USAMO Problems~~]]	+	== See Also ==
-	* [[~~1983 USAMO Problems/Problem 1 \| Problem 1~~]]	+	* [[Mathematics competitions]]
-	* [[~~1983 USAMO Problems/Problem 2 \| Problem 2~~]]	+	* [[Mathematics competition resources]]
-	* [[~~1983~~ USAMO ~~Problems/Problem 3 \| Problem 3~~]]	+	* [[Math books]]
-	* [[~~1983~~ USAMO ~~Problems/Problem 4 \| Problem 4~~]]	+	* [[USAMO]]
-	* [[~~1983~~ USAMO ~~Problems/Problem 5 \| Problem 5~~]]	+
		+	{{USAMO box\|year=1983\|before=[[1982 USAMO]]\|after=[[1984 USAMO]]}}

1984 USAMO

2012-05-16T15:53:44Z

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-	1984 USAMO ~~Problems~~. ~~There were 5~~ problems~~, with three~~ and ~~a half hours to solve them~~.	+	'''1984 USAMO''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	*[[1984 USAMO Problems]]	+	* [[1984 USAMO Problems\|Entire Test]]
-	*[[1984 USAMO Problems/Problem 1]]	+	** [[1984 USAMO Problems/Problem 1\|Problem 1]]
-	*[[1984 USAMO Problems/Problem 2]]	+	** [[1984 USAMO Problems/Problem 2\|Problem 2]]
-	*[[1984 USAMO Problems/Problem 3]]	+	** [[1984 USAMO Problems/Problem 3\|Problem 3]]
-	*[[1984 USAMO Problems/Problem 4]]	+	** [[1984 USAMO Problems/Problem 4\|Problem 4]]
-	*[[1984 USAMO Problems/Problem 5]]	+	** [[1984 USAMO Problems/Problem 5\|Problem 5]]
		+
		+	== See Also ==
		+	* [[Mathematics competitions]]
		+	* [[Mathematics competition resources]]
		+	* [[Math books]]
		+	* [[USAMO]]
		+
		+	{{USAMO box\|year=1984\|before=[[1983 USAMO]]\|after=[[1985 USAMO]]}}

1985 USAMO

2012-05-16T15:53:11Z

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	'''1985 USAMO''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.		'''1985 USAMO''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	* [[1985 USAMO Problems]]	+	* [[1985 USAMO Problems\|Entire Test]]
-	* [[1985 USAMO Problems/Problem 1]]	+	** [[1985 USAMO Problems/Problem 1\|Problem 1]]
-	* [[1985 USAMO Problems/Problem 2]]	+	** [[1985 USAMO Problems/Problem 2\|Problem 2]]
-	* [[1985 USAMO Problems/Problem 3]]	+	** [[1985 USAMO Problems/Problem 3\|Problem 3]]
-	* [[1985 USAMO Problems/Problem 4]]	+	** [[1985 USAMO Problems/Problem 4\|Problem 4]]
-	* [[1985 USAMO Problems/Problem 5]]	+	** [[1985 USAMO Problems/Problem 5\|Problem 5]]
		+
	== See Also ==		== See Also ==
	* [[Mathematics competitions]]		* [[Mathematics competitions]]
	* [[Mathematics competition resources]]		* [[Mathematics competition resources]]
	* [[Math books]]		* [[Math books]]
		+	* [[USAMO]]

	{{USAMO box\|year=1985\|before=[[1984 USAMO]]\|after=[[1986 USAMO]]}}		{{USAMO box\|year=1985\|before=[[1984 USAMO]]\|after=[[1986 USAMO]]}}

1986 USAMO

2012-05-16T15:52:41Z

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← Older revision		Revision as of 15:52, 16 May 2012
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	'''1986 USAMO''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.		'''1986 USAMO''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	* [[1986 USAMO Problems]]	+	* [[1986 USAMO Problems\|Entire Test]]
-	* [[1986 USAMO Problems/Problem 1]]	+	** [[1986 USAMO Problems/Problem 1\|Problem 1]]
-	* [[1986 USAMO Problems/Problem 2]]	+	** [[1986 USAMO Problems/Problem 2\|Problem 2]]
-	* [[1986 USAMO Problems/Problem 3]]	+	** [[1986 USAMO Problems/Problem 3\|Problem 3]]
-	* [[1986 USAMO Problems/Problem 4]]	+	** [[1986 USAMO Problems/Problem 4\|Problem 4]]
-	* [[1986 USAMO Problems/Problem 5]]	+	** [[1986 USAMO Problems/Problem 5\|Problem 5]]
		+
	== See Also ==		== See Also ==
	* [[Mathematics competitions]]		* [[Mathematics competitions]]
	* [[Mathematics competition resources]]		* [[Mathematics competition resources]]
	* [[Math books]]		* [[Math books]]
		+	* [[USAMO]]

	{{USAMO box\|year=1986\|before=[[1985 USAMO]]\|after=[[1987 USAMO]]}}		{{USAMO box\|year=1986\|before=[[1985 USAMO]]\|after=[[1987 USAMO]]}}

1987 USAMO

2012-05-16T15:51:55Z

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	'''1987 USAMO''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.		'''1987 USAMO''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	* [[1987 USAMO Problems]]	+	* [[1987 USAMO Problems\|Entire Test]]
-	* [[1987 USAMO Problems/Problem 1]]	+	** [[1987 USAMO Problems/Problem 1\|Problem 1]]
-	* [[1987 USAMO Problems/Problem 2]]	+	** [[1987 USAMO Problems/Problem 2\|Problem 2]]
-	* [[1987 USAMO Problems/Problem 3]]	+	** [[1987 USAMO Problems/Problem 3\|Problem 3]]
-	* [[1987 USAMO Problems/Problem 4]]	+	** [[1987 USAMO Problems/Problem 4\|Problem 4]]
-	* [[1987 USAMO Problems/Problem 5]]	+	** [[1987 USAMO Problems/Problem 5\|Problem 5]]
-	== See ~~also~~ ==	+
		+	== See Also ==
	* [[Mathematics competitions]]		* [[Mathematics competitions]]
	* [[Mathematics competition resources]]		* [[Mathematics competition resources]]
	* [[Math books]]		* [[Math books]]
		+	* [[USAMO]]

	{{USAMO box\|year=1987\|before=[[1986 USAMO]]\|after=[[1988 USAMO]]}}		{{USAMO box\|year=1987\|before=[[1986 USAMO]]\|after=[[1988 USAMO]]}}

1988 USAMO Problems/Problem 4

2012-05-16T15:45:13Z

added solution

← Older revision		Revision as of 15:45, 16 May 2012
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	==Solution==		==Solution==
-	{~~{solution~~}}	+	Let the circumcenters of <math>\Delta IAB</math>, <math>\Delta IBC</math>, and <math>\Delta ICA</math> be <math>O_c</math>, <math>O_a</math>, and <math>O_b</math>, respectively. It then suffices to show that <math>A</math>, <math>B</math>, <math>C</math>, <math>O_a</math>, <math>O_b</math>, and <math>O_c</math> are concyclic.
		+
		+	We shall prove that quadrilateral <math>ABO_aC</math> is cyclic first. Let <math>\angle BAC=\alpha</math>, <math>\angle CBA=\beta</math>, and <math>\angle ACB=\gamma</math>. Then <math>\angle ICB=\gamma/2</math> and <math>\angle IBC=\beta/2</math>. Therefore minor arc <math>\arc{BIC}</math> in the circumcircle of <math>IBC</math> has a degree measure of <math>\beta+\gamma</math>. This shows that <math>\angle CO_aB=\beta+\gamma</math>, implying that <math>\angle BAC+\angle BO_aC=\alpha+\beta+\gamma=180^{\circ}</math>. Therefore quadrilateral <math>ABO_aC</math> is cyclic.
		+
		+	This shows that point <math>O_a</math> is on the circumcircle of <math>\Delta ABC</math>. Analagous proofs show that <math>O_b</math> and <math>O_c</math> are also on the circumcircle of <math>ABC</math>, which completes the proof. <math>\blacksquare</math>

	==See Also==		==See Also==

User:Chaotic iak/ChaoTickets

2012-05-16T13:21:17Z

← Older revision		Revision as of 13:21, 16 May 2012
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-	'''ChaoTicket''' is the ~~new~~ currency that I use in my games and blog. ChaoTicket is abbreviated CT.	+	'''ChaoTicket''' is the now-obsolete currency that I used to use in my games and blog. ChaoTicket is abbreviated CT. This page is correct as of the last update of this page (23 September 2011).

-	In this article, all mentions of singular first-person pronoun ("I", "me", "my") not in header refer to chaotic_iak, and all mentions of singular ~~first~~-person pronoun in headers refer to anyone reading this ("you").	+	In this article, all mentions of singular first-person pronoun ("I", "me", "my") not in header refer to chaotic_iak, and all mentions of singular second-person pronoun in headers refer to anyone reading this ("you").

	== What are the differences of ChaoTicket and all other currencies in AoPS? ==		== What are the differences of ChaoTicket and all other currencies in AoPS? ==

User:Thkim1011

2012-05-16T06:16:59Z

← Older revision		Revision as of 06:16, 16 May 2012
(2 intermediate revisions not shown)
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-	~~<div id="body" style="font-family:calibri;">~~	+	<div id="header" style="border:2px solid black;border-radius:10px;background-color:dimgrey;">
-	<div id="header" style="border:2px solid black;border-radius:10px;background-color:dimgrey;">	+	<p style="font-size:50px;color:white;">Welcome to my wiki page</p>
-	<p style="font-size:50px;color:white;">Welcome to my wiki page</p>	+	</div>
-	</div>	+	<div id="inner" style="border:2px solid black;border-radius:10px;background-color:dimgrey;position:relative;top:5px;">
-	<div id="~~content~~" style="~~background-color~~:~~grey~~;border-radius:10px;~~border~~:~~2px solid black~~;">	+	<p style="font-size:15px;color:white;">I am an AoPS member that joined last year.</p>
-	~~<div id="info1" style="">~~	+
-	<p style="~~color:white;~~font-size:15px;">I am an AoPS member that joined last year.</p>	+
-	~~</div>~~	+
-	~~</div~~>	+
	</div>		</div>

User talk:Thkim1011

2012-05-16T06:02:25Z

Created page with "Wait, what do we do here??"

New page

Wait, what do we do here??

User:Thkim1011

2012-05-16T05:58:35Z

← Older revision		Revision as of 05:58, 16 May 2012
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-	I am an AoPS member.	+	<div id="body" style="font-family:calibri;">
		+	<div id="header" style="border:2px solid black;border-radius:10px;background-color:dimgrey;">
		+	<p style="font-size:50px;color:white;">Welcome to my wiki page</p>
		+	</div>
		+	<div id="content" style="background-color:grey;border-radius:10px;border:2px solid black;">
		+	<div id="info1" style="">
		+	<p style="color:white;font-size:15px;">I am an AoPS member that joined last year.</p>
		+	</div>
		+	</div>
		+	</div>

File:A.png

2012-05-16T03:40:49Z

uploaded "[[File:A.png]]"

2010 AMC 8 Problems/Problem 18

2012-05-16T03:32:26Z

Created page with "The ratio of AD to AB is 3:2 and AB=30 inches. <math>If 30=2, then x=3. x=45. </math> The area of the rectangle is <math>45*30= 1350</math>. The semicircles, when combined, have..."

New page

The ratio of AD to AB is 3:2 and AB=30 inches.
<math>If 30=2, then x=3. x=45.
</math>
The area of the rectangle is <math>45*30= 1350</math>.

The semicircles, when combined, have the area of a regular circle.
Radius is 15 (<math>30/2=15</math>, for the record). Area of a circle is <math>\pi r^2</math> , so it's equal to 225<math>\pi</math>

<math>1350/225 \pi</math> is equivalent to <math>6/ \pi </math>
[http://i.imgur.com/5qEb2.png]
The image given in the test is given above for reference.

Answer is <math> \box{C: 6:\pi} </math>

Massachusetts MathCounts

2012-05-16T03:14:23Z

← Older revision		Revision as of 03:14, 16 May 2012
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	Massachusetts placed fourth in 2010.		Massachusetts placed fourth in 2010.
		+
		+	Massachusetts placed first in 2012.

← Older revision		Revision as of 16:04, 16 May 2012
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-	~~The~~ '''1995 [[USAMO]]''' ~~was~~ the ~~twenty-fourth USAMO~~.	+	'''1995 USAMO''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	*[[1995 USAMO Problems]]	+	* [[1995 USAMO Problems\|Entire Test]]
-	**[[1995 USAMO Problems/Problem 1\|Problem 1]]	+	** [[1995 USAMO Problems/Problem 1\|Problem 1]]
-	**[[1995 USAMO Problems/Problem 2\|Problem 2]]	+	** [[1995 USAMO Problems/Problem 2\|Problem 2]]
-	**[[1995 USAMO Problems/Problem 3\|Problem 3]]	+	** [[1995 USAMO Problems/Problem 3\|Problem 3]]
-	**[[1995 USAMO Problems/Problem 4\|Problem 4]]	+	** [[1995 USAMO Problems/Problem 4\|Problem 4]]
-	**[[1995 USAMO Problems/Problem 5\|Problem 5]]	+	** [[1995 USAMO Problems/Problem 5\|Problem 5]]
-	**[[~~1995~~ USAMO ~~Problems/Problem 6\|Problem 6~~]]	+
		+	== See Also ==
		+	* [[Mathematics competitions]]
		+	* [[Mathematics competition resources]]
		+	* [[Math books]]
		+	* [[USAMO]]

	{{USAMO box\|year=1995\|before=[[1994 USAMO]]\|after=[[1996 USAMO]]}}		{{USAMO box\|year=1995\|before=[[1994 USAMO]]\|after=[[1996 USAMO]]}}

← Older revision		Revision as of 16:03, 16 May 2012
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-	The '''23<sup>rd</sup> [[USAMO]]''' occurred April 28<sup>th</sup>, 1994. There were five problems given in one session of three and a half hours.	+	The '''23<sup>rd</sup> [[USAMO]]''' occurred April 28<sup>th</sup>, 1994. There were five problems given in one session of three and a half hours. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

-	*[[1994 USAMO Problems]]	+	* [[1994 USAMO Problems\|Entire Test]]
-	**[[1994 USAMO Problems/Problem 1\|Problem 1]]	+	** [[1994 USAMO Problems/Problem 1\|Problem 1]]
-	**[[1994 USAMO Problems/Problem 2\|Problem 2]]	+	** [[1994 USAMO Problems/Problem 2\|Problem 2]]
-	**[[1994 USAMO Problems/Problem 3\|Problem 3]]	+	** [[1994 USAMO Problems/Problem 3\|Problem 3]]
-	**[[1994 USAMO Problems/Problem 4\|Problem 4]]	+	** [[1994 USAMO Problems/Problem 4\|Problem 4]]
-	**[[1994 USAMO Problems/Problem 5\|Problem 5]]	+	** [[1994 USAMO Problems/Problem 5\|Problem 5]]
		+
		+	== See Also ==
		+	* [[Mathematics competitions]]
		+	* [[Mathematics competition resources]]
		+	* [[Math books]]
		+	* [[USAMO]]

	{{USAMO box\|year=1994\|before=[[1993 USAMO]]\|after=[[1995 USAMO]]}}		{{USAMO box\|year=1994\|before=[[1993 USAMO]]\|after=[[1995 USAMO]]}}