2010 USAJMO Problems
A permutation of the set of positive integers is a sequence such that each element of appears precisely one time as a term of the sequence. For example, is a permutation of . Let be the number of permutations of for which is a perfect square for all . Find with proof the smallest such that is a multiple of .
Let be an integer. Find, with proof, all sequences of positive integers with the following three properties:
- for all ;
- given any two indices and (not necessarily distinct) for which , there is an index such that .
Let be a convex pentagon inscribed in a semicircle of diameter . Denote by the feet of the perpendiculars from onto lines , respectively. Prove that the acute angle formed by lines and is half the size of , where is the midpoint of segment .
A triangle is called a parabolic triangle if its vertices lie on a parabola . Prove that for every nonnegative integer , there is an odd number and a parabolic triangle with vertices at three distinct points with integer coordinates with area .
Two permutations and of the numbers are said to intersect if for some value of in the range . Show that there exist permutations of the numbers such that any other such permutation is guaranteed to intersect at least one of these permutations.
Let be a triangle with . Points and lie on sides and , respectively, such that and . Segments and meet at . Determine whether or not it is possible for segments to all have integer lengths.
|2010 USAJMO (Problems • Resources)|
|1 • 2 • 3 • 4 • 5 • 6|
|All USAJMO Problems and Solutions|