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USAJMO Problem 1
Given a triangle , let and be points on segments and , respectively, such that . Let and be distinct points on segment such that lies between and , , and . Prove that , , , are concyclic (in other words, these four points lie on a circle).
Find all integers such that among any positive real numbers , , , with there exist three that are the side lengths of an acute triangle.
Let , , be positive real numbers. Prove that
We proceed to prove that
(then the inequality in question is just the cyclic sum of both sides, since )
Indeed, by AP-GP, we have
Summing up, we have
which is equivalent to:
Dividing from both sides, the desired inequality is proved.
Let be an irrational number with , and draw a circle in the plane whose circumference has length 1. Given any integer , define a sequence of points , , , as follows. First select any point on the circle, and for define as the point on the circle for which the length of arc is , when travelling counterclockwise around the circle from to . Supose that and are the nearest adjacent points on either side of . Prove that .
Use mathematical induction. For it is true because one point can't be closest to in both ways, and that . Suppose that for some , the nearest adjacent points and on either side of satisfy . Then consider the nearest adjacent points and on either side of . It is by the assumption of the nearness we can see that either still holds, or jumps into the interior of the arc , so that or equals two . Let's consider the following two cases.
(i) Suppose .
Since the length of the arc is (where equals to subtracted by the greatest integer not exceeding ) and length of the arc is , we now consider a point which is defined by traveling clockwise on the circle such that the length of arc is . We claim that is in the interior of the arc . Algebraically, it is equivalent to either or .
Suppose the latter fails, i.e. . Then suppose and , where , are integers and ( is not zero because is irrational). We now have and
Therefore is either closer to than on the side, or closer to than on the side. In other words, is the closest adjacent point of on the side, or the closest adjacent point of on the side. Hence or is , therefore .
(ii) Suppose Then either when and , or when one of or is .
In either case, is true.
For distinct positive integers , , define to be the number of integers with such that the remainder when divided by 2012 is greater than that of divided by 2012. Let be the minimum value of , where and range over all pairs of distinct positive integers less than 2012. Determine .
Let be a point in the plane of triangle , and a line passing through . Let , , be the points where the reflections of lines , , with respect to intersect lines , , , respectively. Prove that , , are collinear.
给定三角形 , 点, 分别落在边 和 上，使得 . 令 , 为 上相异的两点，使得 落在 和 之间，而且有 , . 证明 , , , 四点共圆 （即，存在一个圆使得这四点同时落在其上。)
找出所有满足如下条件的正整数 ： 对于任意的个正实数 , , , ，只要 就存在其中的三个数，它们能构成锐角三角形的三边长。
找出所有满足如下条件的函数 (其中 为正整数集) 使得 对任意正整数成立，而且对任意相异正整数， 能被 整除。
Two people play a game with a bar of chocolate made of 60 pieces, in a 6 × 10 rectangle. The first person breaks off a part of the chocolate bar along the grooves dividing the pieces, and discards (or eats) the part he broke off. Then the second person breaks off a part of the remaining part and discards her part. The game continues until one piece is left. The winner is the one who leaves the other with the single piece (i.e. is the last to move). Which person has a perfect winning strategy?
Let be positive real numbers such that . Prove that with equality if and only if .