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- 18:33, 30 December 2008 (diff | hist) . . (+1) . . m 1975 USAMO Problems/Problem 1 (→Problem)
- 18:32, 30 December 2008 (diff | hist) . . (+487) . . N 1975 USAMO Problems/Problem 5 (New page: ==Problem== A deck of <math>n</math> playing cards, which contains three aces, is shuffled at random (it is assumed that all possible card distributions are equally likely). The cards are ...)
- 18:28, 30 December 2008 (diff | hist) . . (+2,181) . . N 1963 IMO Problems (New page: Problems of the 5th IMO 1963 Poland. == Day I == === Problem 1 === Find all real roots of the equation <center><math>\sqrt{x^2-p}+2\sqrt{x^2-1}=x</math>,</center>where <math>p</math...)
- 18:25, 30 December 2008 (diff | hist) . . (+732) . . N 1963 IMO Problems/Problem 6 (New page: ==Problem== Five students, <math>A,B,C,D,E</math>, took part in a contest. One prediction was that the contestants would finish in the order <math>ABCDE</math>. This prediction was very po...)
- 18:23, 30 December 2008 (diff | hist) . . (+190) . . N 1963 IMO Problems/Problem 5 (New page: ==Problem== Prove that <math>\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}</math>. ==Solution== {{solution}} ==See Also== {{IMO box|year=1963|num-b=4|num-a=6...)
- 18:21, 30 December 2008 (diff | hist) . . (+339) . . N 1963 IMO Problems/Problem 4 (New page: ==Problem== Find all solutions <math>x_1,x_2,x_3,x_4,x_5</math> of the system <center><math>\begin{eqnarray} x_5+x_2&=&yx_1\\ x_1+x_3&=&yx_2\\ x_2+x_4&=&yx_3\\ x_3+x_5&=&yx_4\\ x_4+x_1&=&y...)
- 18:19, 30 December 2008 (diff | hist) . . (+316) . . N 1963 IMO Problems/Problem 3 (New page: ==Problem== In an <math>n</math>-gon all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation <center><math>a_1\ge a_2\ge \cdots \ge a_n</math>.</cente...)
- 18:17, 30 December 2008 (diff | hist) . . (+336) . . N 1963 IMO Problems/Problem 2 (New page: ==Problem== Point <math>A</math> and segment <math>BC</math> are given. Determine the locus of points in space which are the vertices of right angles with one side passing through <math>A<...)
- 18:15, 30 December 2008 (diff | hist) . . (+243) . . N 1963 IMO Problems/Problem 1 (New page: ==Problem== Find all real roots of the equation <center><math>\sqrt{x^2-p}+2\sqrt{x^2-1}=x</math>,</center>where <math>p</math> is a real parameter. ==Solution== {{solution}} ==See Also...)
- 18:10, 30 December 2008 (diff | hist) . . (+422) . . N 1962 IMO Problems/Problem 7 (New page: ==Problem== The tetrahedron <math>SABC</math> has the following property: there exist five spheres, each tangent to the edges <math>SA, SB, SC, BC, CA, AB</math>, or to their extensions. ...)
- 18:09, 30 December 2008 (diff | hist) . . (+374) . . N 1962 IMO Problems/Problem 6 (New page: ==Problem== Consider an isosceles triangle. Let <math>r</math> be the radius of its circumscribed circle and <math>\rho</math> the radius of its inscribed circle. Prove that the distance <...)
- 18:09, 30 December 2008 (diff | hist) . . (+346) . . N 1962 IMO Problems/Problem 5 (New page: ==Problem== On the circle <math>K</math> there are given three distinct points <math>A,B,C</math>. Construct (using only straightedge and compass) a fourth point <math>D</math> on <math>K<...)
- 18:08, 30 December 2008 (diff | hist) . . (+155) . . N 1962 IMO Problems/Problem 4 (New page: ==Problem== Solve the equation <math>cos^2{x}+cos^2{2x}+cos^2{3x}=1</math>. ==Solution== {{solution}} ==See Also== {{IMO box|year=1962|num-b=3|num-a=5}})
- 18:08, 30 December 2008 (diff | hist) . . (+841) . . N 1962 IMO Problems/Problem 3 (New page: ==Problem== Consider the cube <math>ABCDA'B'C'D'</math>(<math>ABCD</math> and <math>A'B'C'D'</math> are the upper and lower bases, respectively, and edges <math>AA'</math>, <math>BB'</math...)
- 18:07, 30 December 2008 (diff | hist) . . (+238) . . N 1962 IMO Problems/Problem 2 (New page: ==Problem== Determine all real numbers <math>x</math> which satisfy the inequality: <center> <math>\sqrt{\sqrt{3-x}-\sqrt{x+1}}>\dfrac{1}{2}</math> </center> ==Solution== {{solution}} =...)
- 17:57, 30 December 2008 (diff | hist) . . (+409) . . N 1962 IMO Problems/Problem 1 (New page: ==Problem== Find the smallest natural number <math>n</math> which has the following properties: (a) Its decimal representation has 6 as the last digit. (b) If the last digit 6 is erased ...)
- 17:56, 30 December 2008 (diff | hist) . . (-2) . . m 1962 IMO Problems (→Problem 1)
- 17:55, 30 December 2008 (diff | hist) . . (+1,325) . . 1962 IMO Problems
- 17:46, 30 December 2008 (diff | hist) . . (+587) . . N 1960 IMO Problems/Problem 7 (New page: ==Problem== An isosceles trapezoid with bases <math>a</math> and <math>c</math> and altitude <math>h</math> is given. a) On the axis of symmetry of this trapezoid, find all points <math>P...)
- 17:45, 30 December 2008 (diff | hist) . . (-13) . . m 1960 IMO Problems/Problem 6 (→See Also)
- 17:18, 30 December 2008 (diff | hist) . . (+664) . . N 1975 USAMO Problems/Problem 4 (New page: ==Problem== Two given circles intersect in two points <math>P</math> and <math>Q</math>. Show how to construct a segment <math>AB</math> passing through <math>P</math> and terminating on t...)
- 17:12, 30 December 2008 (diff | hist) . . (+302) . . N 1975 USAMO Problems/Problem 3 (New page: ==Problem== If <math>P(x)</math> denotes a polynomial of degree <math>n</math> such that <math>P(k)=k/(k+1)</math> for <math>k=0,1,2,\ldots,n</math>, determine <math>P(n+1)</math>. ==Solu...)
- 17:11, 30 December 2008 (diff | hist) . . (+390) . . N 1975 USAMO Problems/Problem 2 (New page: ==Problem== Let <math>A,B,C,D</math> denote four points in space and <math>AB</math> the distance between <math>A</math> and <math>B</math>, and so on. Show that <center><math>AC^2+BD^2+AD...)
- 17:09, 30 December 2008 (diff | hist) . . (+538) . . N 1975 USAMO Problems/Problem 1 (New page: ==Problem== (a) Prove that <center><math>[5x]+[5y]\ge [3x+y]+[3y+x]</math>,</center>where <math>x,y\ge 0</math> and <math>[u]</math> denotes the greatest integer <math>\le u</math> (e.g., ...)
- 17:04, 30 December 2008 (diff | hist) . . (+427) . . N 1975 USAMO (New page: The '''Fourth USAMO''' was held May 6, 1975. There were five problems. * 1975 USAMO Problems * Problem 1 * [[1975 USAMO Problems/Problem 2 | P...)
- 17:02, 30 December 2008 (diff | hist) . . (+2,660) . . N 1974 USAMO Problems (New page: Problems from the '''1974 USAMO'''. ==Problem 1== Let <math>a</math>, <math>b</math>, and <math>c</math> denote three distinct integer...)
- 17:00, 30 December 2008 (diff | hist) . . (-12) . . m 1974 USAMO Problems/Problem 4 (→See also)
- 17:00, 30 December 2008 (diff | hist) . . (+774) . . N 1974 USAMO Problems/Problem 4 (New page: ==Problem== A father, mother and son hold a family tournament, playing a two person board game with no ties. The tournament rules are: (i) The weakest player chooses the first two contest...)
- 16:56, 30 December 2008 (diff | hist) . . (+336) . . N 1974 USAMO Problems/Problem 3 (New page: ==Problem== Two boundary points of a ball of radius 1 are joined by a curve contained in the ball and having length less than 2. Prove that the curve is contained entirely within some hemi...)
- 16:54, 30 December 2008 (diff | hist) . . (+294) . . N 1974 USAMO Problems/Problem 2 (New page: ==Problem== Prove that if <math>a</math>, <math>b</math>, and <math>c</math> are positive real numbers, then <center><math>a^ab^bc^c\ge (abc)^{(a+b+c)/3}</math></center> ==Solution== {{so...)
- 15:58, 30 December 2008 (diff | hist) . . (-3) . . m 1972 USAMO Problems (→See also)
- 15:57, 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 ...)
- 15:52, 30 December 2008 (diff | hist) . . (+4) . . m 1972 USAMO Problems/Problem 5 (→See also)
- 15:52, 30 December 2008 (diff | hist) . . (+21) . . m 1972 USAMO Problems/Problem 5 (→See also)
- 15:52, 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>, ...)
- 15:42, 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...)
- 15:39, 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...)
- 15:35, 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== ...)
- 15:33, 30 December 2008 (diff | hist) . . (0) . . m 1972 USAMO Problems/Problem 3 (→See also)
- 15:33, 30 December 2008 (diff | hist) . . (0) . . m 1972 USAMO Problems/Problem 3 (→See also)
- 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 ...)
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