2012 Indonesia MO Problems
Show that for any positive integers and , the number is an even non-negative integer.
Let be an integer, and let be positive real numbers such that . Prove that
Given an acute triangle with that has circumcenter . Line and meet the bisector of at and , respectively. Moreover, line and meet at . Show that is perpendicular to .
Given distinct points on the Cartesian plane. For any permutation of define the shadow of a point as follows: Point is rotated by around resulting , point is rotated by around resulting , ..., point is rotated by around resulting . Then, is called the shadow of with respect to the permutation . Let be the number of different shadows of up to all permutations of . Determine the maximum value of .
Given positive integers and . Let and be two collections of numbers of and , arranged in rows and columns. An example of such collections for and is Let those two collections satisfy the following properties: (i) On each row of , from left to right, the numbers are non-increasing, (ii) On each column of , from top to bottom, the numbers are non-increasing, (iii) The sum of numbers on the row in equals to the same row in , (iv) The sum of numbers on the column in equals to the same column in . Show that the number on row and column of equals to the number on row and column of for and .
Let be the set of all positive real numbers. Show that there is no function satisfying for all positive real numbers and .
Let be a positive integer. Show that the equation have solution of pairs of positive integers if and only if is divisible by some perfect square greater than .
Given a triangle , let the bisector of meets the side and circumcircle of triangle at and , respectively. Let and be the midpoints of and , respectively. Circumcircle of triangle meets at . Circle passing through that is tangent to at meets line and side respectively at and . Show that the four points lie on the same line.
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