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- 15:31, 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...)
- 15:26, 30 December 2008 (diff | hist) . . (+41) . . m 1973 USAMO Problems/Problem 5 (→Problem)
- 15:24, 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...)
- 15:22, 30 December 2008 (diff | hist) . . (+13) . . m 1973 USAMO Problems/Problem 1 (→Solution)
- 15:21, 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==...)
- 15:18, 30 December 2008 (diff | hist) . . (-152) . . m 1973 USAMO Problems/Problem 2 (→Problem)
- 02:46, 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...)
- 02:42, 21 December 2008 (diff | hist) . . (-45) . . m 2002 USAMO Problems/Problem 3 (→Problem)
- 02:42, 21 December 2008 (diff | hist) . . (0) . . m 2002 USAMO Problems (→Problem 3)
- 02:40, 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 ...)
- 02:38, 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...)
- 02:37, 21 December 2008 (diff | hist) . . (+4) . . 2001 USAMO Problems/Problem 4
- 02:36, 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...)
- 02:35, 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...)
- 02:29, 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...)
- 02:19, 21 December 2008 (diff | hist) . . (+6) . . m 2002 USAMO Problems/Problem 6
- 02:18, 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...)
- 19:58, 18 December 2008 (diff | hist) . . (+801) . . 2003 USAMO Problems/Problem 5
- 19:54, 18 December 2008 (diff | hist) . . (+598) . . 2003 USAMO Problems/Problem 4
- 19:40, 18 December 2008 (diff | hist) . . (+4,499) . . 2003 USAMO Problems/Problem 6
- 18:43, 18 December 2008 (diff | hist) . . (+495) . . m Cauchy-Schwarz Inequality (→Introductory)
- 19:01, 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 ...)
- 18:59, 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...)
- 18:57, 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...)
- 18:56, 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)
- 18:55, 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...)
- 18:54, 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 ...)
- 18:51, 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 ...)
- 18:50, 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)
- 18:49, 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...)
- 18:47, 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)
- 18:46, 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)
- 18:44, 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...)
- 18:43, 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...)
- 18:39, 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...)
- 18:36, 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)
- 18:36, 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...)
- 18:34, 20 August 2008 (diff | hist) . . (0) . . m 2001 IMO Shortlist Problems/C6 (corrected consequences of mindless copying and pasting :P) (current)
- 18:33, 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...)
- 18:32, 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...)
- 18:26, 20 August 2008 (diff | hist) . . (+606) . . m 2001 IMO Shortlist Problems/A1
- 18:23, 20 August 2008 (diff | hist) . . (+509) . . Nm 2001 IMO Shortlist Problems/C4 (New page: == Problem == A set of three nonnegative integers <math>\{x,y,z\}</math> with <math>x < y < z</math> is called <i>historic</i> if <math>\{z - y,y - x\} = \{1776,2001\}</math>. Show that t...)
- 18:21, 20 August 2008 (diff | hist) . . (+579) . . Nm 2001 IMO Shortlist Problems/C3 (New page: == Problem == Define a <math>k</math>-<i>clique</i> to be a set of <math>k</math> people such that every pair of them are acquainted with each other. At a certain party, every pair of 3-cl...)
- 18:19, 20 August 2008 (diff | hist) . . (+650) . . Nm 2001 IMO Shortlist Problems/C2 (New page: == Problem == Let <math>n</math> be an odd integer greater than 1 and let <math>c_1, c_2, \ldots, c_n</math> be integers. For each permutation <math>a = (a_1, a_2, \ldots, a_n)</math> of <...)
- 18:17, 20 August 2008 (diff | hist) . . (+462) . . Nm 2001 IMO Shortlist Problems/A6 (New page: == Problem == Prove that for all positive real numbers <math>a,b,c</math>, <center><math>\frac {a}{\sqrt {a^2 + 8bc}} + \frac {b}{\sqrt {b^2 + 8ca}} + \frac {c}{\sqrt {c^2 + 8ab}} \geq 1.<...)
- 18:16, 20 August 2008 (diff | hist) . . (+549) . . Nm 2001 IMO Shortlist Problems/A5 (New page: == Problem == Find all positive integers <math>a_1, a_2, \ldots, a_n</math> such that <center><math>\frac {99}{100} = \frac {a_0}{a_1} + \frac {a_1}{a_2} + \cdots + \frac {a_{n - 1}}{a_n},...)
- 18:14, 20 August 2008 (diff | hist) . . (+445) . . Nm 2001 IMO Shortlist Problems/A4 (New page: == Problem == Find all functions <math>f: \mathbb{R} \rightarrow \mathbb{R}</math>, satisfying <center><math>f(xy)(f(x) - f(y)) = (x - y)f(x)f(y)</math></center> for all <math>x,y</math>. ...)
- 18:13, 20 August 2008 (diff | hist) . . (+61) . . Nm Category:Functional Equation (New page: This page lists all the functional equations in the AopsWiki.) (current)
- 18:06, 20 August 2008 (diff | hist) . . (+505) . . Nm 2001 IMO Shortlist Problems/A2 (New page: == Problem == Let <math>a_0, a_1, a_2, \ldots</math> be an arbitrary infinite sequence of positive numbers. Show that the inequality <math>1 + a_n > a_{n - 1} \sqrt [n]{2}</math> holds fo...)
- 18:01, 20 August 2008 (diff | hist) . . (+42) . . m 1999 IMO Problems/Problem 2
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