MathDB

B3

Part of 2011 Princeton University Math Competition

Problems(3)

2011 PUMaC Algebra B3

Source:

9/24/2019
Let f(x)=x3−7x2+16x−10f(x) = x^3-7x^2+16x-10. As xx ranges over all integers, find the sum of distinct prime values taken on by f(x)f(x).
algebra
2011 PUMaC Geometry B3

Source:

9/24/2019
Let ABCDABCD be a trapezoid with ADAD parallel to BCBC, AD=2AD = 2, and BC=1BC = 1. Let MM be the midpoint of ADAD, and let PP be the intersection of BDBD with CMCM. Extend APAP to meet segment CDCD at point QQ. If the ratio CQ/QD=a/bCQ/QD = a/b, where aa and bb are positive integers and gcd(a,b)=1\text{gcd}(a, b) = 1, find a+ba + b.
geometry
2011 PUMaC Individual Finals B3

Source:

9/24/2019
In a kk-player tournament for k>1k > 1, every player plays every other player exactly once. Find with proof the smallest value of kk such that it is possible that for any two players, there was a third player who beat both of them.
combinatorics