2007 USAMO Problems
Let be a positive integer. Define a sequence by setting and, for each , letting be the unique integer in the range for which is divisible by . For instance, when the obtained sequence is . Prove that for any the sequence eventually becomes constant.
A square grid on the Euclidean plane consists of all points , where and are integers. Is it possible to cover all grid points by an infinite family of discs with non-overlapping interiors if each disc in the family has radius at least 5?
Let be a set containing elements, for some positive integer . Suppose that the -element subsets of are partitioned into two classes. Prove that there are at least pairwise disjoint sets in the same class.
An animal with cells is a connected figure consisting of equal-sized cells. The figure below shows an 8-cell animal.
A dinosaur is an animal with at least 2007 cells. It is said to be primitive if its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.
Animals are also called polyominoes. They can be defined inductively. Two cells are adjacent if they share a complete edge. A single cell is an animal, and given an animal with cells, one with cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.
Prove that for every nonnegative integer , the number is the product of at least (not necessarily distinct) primes.
Let be an acute triangle with , , and being its incircle, circumcircle, and circumradius, respectively. Circle is tangent internally to at and tangent externally to . Circle is tangent internally to at and tangent internally to . Let and denote the centers of and , respectively. Define points , , , analogously. Prove that with equality if and only if triangle is equilateral.
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