Welcome to the
Princeton Math Circle!
Princeton Math Circle is a weekly program for the high school and the middle school students of Princeton and its surrounding communities. It meets on Saturdays from 2:00 pm to 5:00 pm for middle-school students and from 3:00 pm to 6:00 pm for high school students at Fine Hall on the Princeton University campus.
The program is sponsored by the members of the Princeton University. Its mission is to:
This program is NOT to supplement what is learned at school, or to improve test scores, or grades, but rather means to foster interest in mathematics among local school kids to help them become tomorrow's scientists, economists, engineers and mathematicians.
Let a1, a2, a3, ..., an be distinct positive integers and let M be a set of n - 1 positive integers not containing s = a1 + a2 + a3 + ....+ an. A grasshopper is to jump along the real axis, starting at the point 0 and making n jumps to the right with lengths a1, a2, a3, ...., an in some order.
Prove that the order can be chosen in such a way that the grasshopper never lands on any point in M. [IMO, 2009]
Let ABCD be a convex quadrilateral with BA different from BC. Denote the incircles of triangles ABC and ADC by k1 and k2 respectively. Suppose that there exists a circle k tangent to ray BA beyond A and to the ray BC beyond C, which is also tangent to the lines AD and CD.
Prove that the common external tangents to k1 and k2 intersects on k. [IMO, 2008]
Let P(x) be a polynomial of degree n > 1 with integer coefficients and let k be a positive integer. Consider the polynomial Q(x) = P(P(...P(P(x))...)), where P occurs k times.
Prove that there are at most n integers t such that Q(t) = t. [IMO, 2006]
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