MathDB

2

Part of 1991 Polish MO Finals

Problems(2)

Lattice points and path

Source: Problem 2, Polish NO 1991

10/1/2005
Let XX be the set of all lattice points in the plane (points (x,y)(x, y) with x,y∈Zx, y \in \mathbb{Z}). A path of length nn is a chain (P0,P1,...,Pn)(P_0, P_1, ... , P_n) of points in XX such that Pi−1Pi=1P_{i-1}P_i = 1 for i=1,...,ni = 1, ... , n. Let F(n)F(n) be the number of distinct paths beginning in P0=(0,0)P_0=(0,0) and ending in any point PnP_n on line y=0y = 0. Prove that F(n)=(2nn)F(n) = \binom{2n}{n}
combinatorics unsolvedcombinatorics
Two noncongruent circles

Source: Problem 5, Polish NO 1991

10/1/2005
Two noncongruent circles k1k_1 and k2k_2 are exterior to each other. Their common tangents intersect the line through their centers at points AA and BB. Let PP be any point of k1k_1. Prove that there is a diameter of k2k_2 with one endpoint on line PAPA and the other on PBPB.
geometry solvedgeometry