2007 USA TST Problems
Problem 1
Circles and intersect at and . and are chords of and , respectively, such that is on segment and on ray . Lines and intersect at . Let the line through parallel to intersect again at , and let the line through parallel to intersect again at . Prove are collinear.
Problem 2
Let , be two nonincreasing sequences of reals such that , , , and For any real number , the number of pairs such that is equal to the number of pairs such that . Prove that for .
Problem 3
For some , is irrational. If, for some positive integer , and are both rational, then show .
Problem 4
Are there two positive integers such that, for each positive integer , is not divisible by ?
Problem 5
Let the tangents at and to the circumcircle of meet at . Let the perpendicular to at meet ray at . Let lie on such that and so that lies between and . Prove that .
Problem 6
For any polynomial , let be the remainder mod from 0 to 1023, inclusive, of for . Call the set the remainder sequence of . Call a remaidner sequence complete if it is a permutation of . Show that the number of complete remainder sequences is at most .