2009 USAMO Problems
Given circles and intersecting at points and , let be a line through the center of intersecting at points and and let be a line through the center of intersecting at points and . Prove that if and lie on a circle then the center of this circle lies on line .
Let be a positive integer. Determine the size of the largest subset of which does not contain three elements (not necessarily distinct) satisfying .
We define a chessboard polygon to be a polygon whose sides are situated along lines of the form or , where and are integers. These lines divide the interior into unit squares, which are shaded alternately grey and white so that adjacent squares have different colors. To tile a chessboard polygon by dominoes is to exactly cover the polygon by non-overlapping rectangles. Finally, a tasteful tiling is one which avoids the two configurations of dominoes shown on the left below. Two tilings of a rectangle are shown; the first one is tasteful, while the second is not, due to the vertical dominoes in the upper right corner.
a) Prove that if a chessboard polygon can be tiled by dominoes, then it can be done so tastefully.
b) Prove that such a tasteful tiling is unique.
For let , , ..., be positive real numbers such that
Prove that .
Trapezoid , with , is inscribed in circle and point lies inside triangle . Rays and meet again at points and , respectively. Let the line through parallel to intersect and at points and , respectively. Prove that quadrilateral is cyclic if and only if bisects .
Let be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that Suppose that is also an infinite, nonconstant sequence of rational numbers with the property that is an integer for all and . Prove that there exists a rational number such that and are integers for all and .
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