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• 14:58, 30 December 2008 (diff | hist) . . (-3)‎ . . m 1972 USAMO Problems ‎ (→‎See also)
• 14:57, 30 December 2008 (diff | hist) . . (+2,293)‎ . . N 1972 USAMO Problems ‎ (New page: Problems from the '''1972 USAMO'''. ==Problem 1== The symbols <math>(a,b,\ldots,g)</math> and <math>[a,b,\ldots, g]</math> denote the ...)
• 14:52, 30 December 2008 (diff | hist) . . (+4)‎ . . m 1972 USAMO Problems/Problem 5 ‎ (→‎See also)
• 14:52, 30 December 2008 (diff | hist) . . (+21)‎ . . m 1972 USAMO Problems/Problem 5 ‎ (→‎See also)
• 14:52, 30 December 2008 (diff | hist) . . (+805)‎ . . N 1972 USAMO Problems/Problem 5 ‎ (New page: ==Problem== A given convex pentagon <math>ABCDE</math> has the property that the area of each of the five triangles <math>ABC</math>, <math>BCD</math>, <math>CDE</math>, <math>DEA</math>, ...)
• 14:42, 30 December 2008 (diff | hist) . . (+404)‎ . . N 1972 USAMO Problems/Problem 4 ‎ (New page: ==Problem== Let <math>R</math> denote a non-negative rational number. Determine a fixed set of integers <math>a,b,c,d,e,f</math>, such that for ''every'' choice of <math>R</math>, <center...)
• 14:39, 30 December 2008 (diff | hist) . . (+530)‎ . . N 1972 USAMO Problems/Problem 1 ‎ (New page: ==Problem== The symbols <math>(a,b,\ldots,g)</math> and <math>[a,b,\ldots, g]</math> denote the greatest common divisor and least common multiple, respectively, of the positive integers <m...)
• 14:35, 30 December 2008 (diff | hist) . . (+296)‎ . . N 1972 USAMO Problems/Problem 2 ‎ (New page: ==Problem== A given tetrahedron <math>ABCD</math> is isosceles, that is, <math>AB=CD, AC=BD, AD=BC</math>. Show that the faces of the tetrahedron are acute-angled triangles. ==Solution== ...)
• 14:33, 30 December 2008 (diff | hist) . . (0)‎ . . m 1972 USAMO Problems/Problem 3 ‎ (→‎See also)
• 14:33, 30 December 2008 (diff | hist) . . (0)‎ . . m 1972 USAMO Problems/Problem 3 ‎ (→‎See also)
• 14:31, 30 December 2008 (diff | hist) . . (+1,897)‎ . . N 1973 USAMO Problems ‎ (New page: Problems from the '''1973 USAMO'''. ==Problem 1== Two points <math>P</math> and <math>Q</math> lie in the interior of a regular tetrah...)
• 14:26, 30 December 2008 (diff | hist) . . (+41)‎ . . m 1973 USAMO Problems/Problem 5 ‎ (→‎Problem)
• 14:24, 30 December 2008 (diff | hist) . . (+443)‎ . . N 1973 USAMO Problems/Problem 3 ‎ (New page: ==Problem== Three distinct vertices are chosen at random from the vertices of a given regular polygon of <math>(2n+1)</math> sides. If all such choices are equally likely, what is the prob...)
• 14:22, 30 December 2008 (diff | hist) . . (+13)‎ . . m 1973 USAMO Problems/Problem 1 ‎ (→‎Solution)
• 14:21, 30 December 2008 (diff | hist) . . (+284)‎ . . N 1973 USAMO Problems/Problem 1 ‎ (New page: ==Problem== Two points <math>P</math> and <math>Q</math> lie in the interior of a regular tetrahedron <math>ABCD</math>. Prove that angle <math>PAQ<60^o</math>. ==Solution== ==See also==...)
• 14:18, 30 December 2008 (diff | hist) . . (-152)‎ . . m 1973 USAMO Problems/Problem 2 ‎ (→‎Problem)
• 01:46, 21 December 2008 (diff | hist) . . (+2,829)‎ . . N 2001 USAMO Problems ‎ (New page: Problems of the 2001 USAMO. == Day 1 == === Problem 1 === Each of eight boxes contains six balls. Each ball has been colored with one of <math>n</math> colors, such...)
• 01:42, 21 December 2008 (diff | hist) . . (-45)‎ . . m 2002 USAMO Problems/Problem 3 ‎ (→‎Problem)
• 01:42, 21 December 2008 (diff | hist) . . (0)‎ . . m 2002 USAMO Problems ‎ (→‎Problem 3)
• 01:40, 21 December 2008 (diff | hist) . . (+423)‎ . . N 2001 USAMO Problems/Problem 6 ‎ (New page: == Problem == Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three ...)
• 01:38, 21 December 2008 (diff | hist) . . (+523)‎ . . N 2001 USAMO Problems/Problem 5 ‎ (New page: == Problem == Let <math>S</math> be a set of integers (not necessarily positive) such that (a) there exist <math>a,b \in S</math> with <math>\gcd(a,b) = \gcd(a - 2,b - 2) = 1</math>; (b...)
• 01:37, 21 December 2008 (diff | hist) . . (+4)‎ . . 2001 USAMO Problems/Problem 4 ‎
• 01:36, 21 December 2008 (diff | hist) . . (+463)‎ . . N 2001 USAMO Problems/Problem 4 ‎ (New page: == Problem == Let <math>P</math> be a point in the plane of triangle <math>ABC</math> such that the segments <math>PA</math>, <math>PB</math>, and <math>PC</math> are the sides of an obtu...)
• 01:35, 21 December 2008 (diff | hist) . . (+312)‎ . . N 2001 USAMO Problems/Problem 3 ‎ (New page: == Problem == Let <math>a, b, c \geq 0</math> and satisfy <center><math>a^2 + b^2 + c^2 + abc = 4.</math></center> Show that <center><math>ab + bc + ca - abc \leq 2.</math></center> == So...)
• 01:29, 21 December 2008 (diff | hist) . . (+3,035)‎ . . N 2002 USAMO Problems ‎ (New page: Problems of the 2002 USAMO. == Day 1 == === Problem 1 === Let <math>S </math> be a set with 2002 elements, and let <math>N </math> be an integer with <math> 0 \le N...)
• 01:19, 21 December 2008 (diff | hist) . . (+6)‎ . . m 2002 USAMO Problems/Problem 6 ‎
• 01:18, 21 December 2008 (diff | hist) . . (+809)‎ . . N 2002 USAMO Problems/Problem 6 ‎ (New page: == Problem == I have an <math>n \times n</math> sheet of stamps, from which I've been asked to tear out blocks of three adjacent stamps in a single row or column. (I can only tear along t...)
• 18:58, 18 December 2008 (diff | hist) . . (+801)‎ . . 2003 USAMO Problems/Problem 5 ‎
• 18:54, 18 December 2008 (diff | hist) . . (+598)‎ . . 2003 USAMO Problems/Problem 4 ‎
• 18:40, 18 December 2008 (diff | hist) . . (+4,499)‎ . . 2003 USAMO Problems/Problem 6 ‎
• 17:43, 18 December 2008 (diff | hist) . . (+495)‎ . . m Cauchy-Schwarz Inequality ‎ (→‎Introductory)
• 18:01, 20 August 2008 (diff | hist) . . (+354)‎ . . N 2001 IMO Shortlist Problems/N6 ‎ (New page: == Problem == Is it possible to find 100 positive integers not exceeding 25,000, such that all pairwise sums of them are different? == Solution == {{solution}} == Resources == * [[2001 ...)
• 17:59, 20 August 2008 (diff | hist) . . (+423)‎ . . N 2001 IMO Shortlist Problems/N5 ‎ (New page: == Problem == Let <math>a > b > c > d</math> be positive integers and suppose that <center><math>ac + bd = (b + d + a - c)(b + d - a + c).</math></center> Prove that <math>ab + cd</math> i...)
• 17:57, 20 August 2008 (diff | hist) . . (+485)‎ . . N 2001 IMO Shortlist Problems/N4 ‎ (New page: == Problem == Let <math>p \geq 5</math> be a prime number. Prove that there exists an integer <math>a</math> with <math>1 \leq a \leq p - 2</math> such that neither <math>a^{p - 1} - 1</ma...)
• 17:56, 20 August 2008 (diff | hist) . . (+424)‎ . . N 2001 IMO Shortlist Problems/N3 ‎ (New page: == Problem == Let <math>a_1 = 11^{11}, \, a_2 = 12^{12}, \, a_3 = 13^{13}</math>, and <math>a_n = |a_{n - 1} - a_{n - 2}| + |a_{n - 2} - a_{n - 3}|, n \geq 4.</math> Determine <math>a_{14^...) (current)
• 17:55, 20 August 2008 (diff | hist) . . (+503)‎ . . N 2001 IMO Shortlist Problems/N2 ‎ (New page: == Problem == Consider the system <math>x + y = z + u,</math> <math>2xy & = zu.</math> Find the greatest value of the real constant <math>m</math> such that <math>m \leq x/y</math> for any...)
• 17:54, 20 August 2008 (diff | hist) . . (+426)‎ . . N 2001 IMO Shortlist Problems/N1 ‎ (New page: == Problem == Prove that there is no positive integer <math>n</math> such that, for <math>k = 1,2,\ldots,9</math>, the leftmost digit (in decimal notation) of <math>(n + k)!</math> equals ...)
• 17:51, 20 August 2008 (diff | hist) . . (+567)‎ . . N 2001 IMO Shortlist Problems/G8 ‎ (New page: == Problem == Let <math>ABC</math> be a triangle with <math>\angle BAC = 60^{\circ}</math>. Let <math>AP</math> bisect <math>\angle BAC</math> and let <math>BQ</math> bisect <math>\angle ...)
• 17:50, 20 August 2008 (diff | hist) . . (+726)‎ . . N 2001 IMO Shortlist Problems/G7 ‎ (New page: == Problem == Let <math>O</math> be an interior point of acute triangle <math>ABC</math>. Let <math>A_1</math> lie on <math>BC</math> with <math>OA_1</math> perpendicular to <math>BC</math...) (current)
• 17:49, 20 August 2008 (diff | hist) . . (+756)‎ . . N 2001 IMO Shortlist Problems/G6 ‎ (New page: == Problem == Let <math>ABC</math> be a triangle and <math>P</math> an exterior point in the plane of the triangle. Suppose the lines <math>AP</math>, <math>BP</math>, <math>CP</math> meet...)
• 17:47, 20 August 2008 (diff | hist) . . (+944)‎ . . N 2001 IMO Shortlist Problems/G5 ‎ (New page: == Problem == Let <math>ABC</math> be an acute triangle. Let <math>DAC,EAB</math>, and <math>FBC</math> be isosceles triangles exterior to <math>ABC</math>, with <math>DA = DC, EA = EB</m...) (current)
• 17:46, 20 August 2008 (diff | hist) . . (+831)‎ . . N 2001 IMO Shortlist Problems/G4 ‎ (New page: == Problem == Let <math>M</math> be a point in the interior of triangle <math>ABC</math>. Let <math>A'</math> lie on <math>BC</math> with <math>MA'</math> perpendicular to <math>BC</math>....) (current)
• 17:44, 20 August 2008 (diff | hist) . . (+551)‎ . . N 2001 IMO Shortlist Problems/G3 ‎ (New page: == Problem == Let <math>ABC</math> be a triangle with centroid <math>G</math>. Determine, with proof, the position of the point <math>P</math> in the plane of <math>ABC</math> such that <m...)
• 17:43, 20 August 2008 (diff | hist) . . (+594)‎ . . N 2001 IMO Shortlist Problems/G2 ‎ (New page: == Problem == Consider an acute-angled triangle <math>ABC</math>. Let <math>P</math> be the foot of the altitude of triangle <math>ABC</math> issuing from the vertex <math>A</math>, and le...)
• 17:39, 20 August 2008 (diff | hist) . . (+724)‎ . . N 2001 IMO Shortlist Problems/G1 ‎ (New page: == Problem == Let <math>A_1</math> be the center of the square inscribed in acute triangle <math>ABC</math> with two vertices of the square on side <math>BC</math>. Thus one of the two re...)
• 17:36, 20 August 2008 (diff | hist) . . (+588)‎ . . N 2001 IMO Shortlist Problems/C8 ‎ (New page: == Problem == Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and ...) (current)
• 17:36, 20 August 2008 (diff | hist) . . (+991)‎ . . N 2001 IMO Shortlist Problems/C7 ‎ (New page: == Problem == A pile of <math>n</math> pebbles is placed in a vertical column. This configuration is modified according to the following rules. A pebble can be moved if it is at the top of...)
• 17:34, 20 August 2008 (diff | hist) . . (0)‎ . . m 2001 IMO Shortlist Problems/C6 ‎ (corrected consequences of mindless copying and pasting :P) (current)
• 17:33, 20 August 2008 (diff | hist) . . (+1,061)‎ . . N 2001 IMO Shortlist Problems/C6 ‎ (New page: == Problem == For a positive integer <math>n</math> define a sequence of zeros and ones to be [i]balanced[/i] if it contains <math>n</math> zeros and <math>n</math> ones. Two balanced sequ...)
• 17:32, 20 August 2008 (diff | hist) . . (+448)‎ . . Nm 2001 IMO Shortlist Problems/C5 ‎ (New page: == Problem == Find all finite sequences <math>(x_0, x_1, \ldots,x_n)</math> such that for every <math>j</math>, <math>0 \leq j \leq n</math>, <math>x_j</math> equals the number of times <m...)

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