2007 Indonesia MO Problems
Let be a triangle with . Let point on side such that is the altitude, point on side such that , and point is the intersection of and . Prove that .
For every positive integer , denote the number of positive divisors of and denote the sum of all positive divisors of . For example, and . Let be a positive integer greater than .
(a) Prove that there are infinitely many positive integers which satisfy .
(b) Prove that there are finitely many positive integers which satisfy .
Let be positive real numbers which satisfy . Prove that these three inequalities hold: , , .
A 10-digit arrangement is called beautiful if (i) when read left to right, form an increasing sequence, and form a decreasing sequence, and (ii) is not the leftmost digit. For example, is a beautiful arrangement. Determine the number of beautiful arrangements.
Let , be two positive integers and a 'chessboard' with rows and columns. Let denote the maximum value of rooks placed on such that no two of them attack each other.
(a) Determine .
(b) How many ways to place rooks on such that no two of them attack each other?
[Note: In chess, a rook moves and attacks in a straight line, horizontally or vertically.]
Find all triples of real numbers which satisfy the simultaneous equations
Points are on circle , such that is the diameter of , but is not the diameter. Given also that and are on different sides of . The tangents of at and intersect at . Points and are the intersections of line with line and line with line , respectively.
(a) Prove that , , and are collinear.
(b) Prove that is perpendicular to line .
Let and be two positive integers. If there are infinitely many integers such that is a perfect square, prove that .
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