Subcontests
(5)Partitioning Region into Paths
Let n be a positive integer. Denote by Sn the set of points (x,y) with integer coordinates such that ∣x∣+y+21<n. A path is a sequence of distinct points (x1,y1),(x2,y2),…,(xℓ,yℓ) in Sn such that, for i=2,…,ℓ, the distance between (xi,yi) and (xi−1,yi−1) is 1 (in other words, the points (xi,yi) and (xi−1,yi−1) are neighbors in the lattice of points with integer coordinates). Prove that the points in Sn cannot be partitioned into fewer than n paths (a partition of Sn into m paths is a set P of m nonempty paths such that each point in Sn appears in exactly one of the m paths in P). Product of Consecutive Integers
Prove that for each positive integer n, there are pairwise relatively prime integers k0,k1,…,kn, all strictly greater than 1, such that k_0k_1\ldots k_n\minus{}1 is the product of two consecutive integers. Cyclic Quad
Let ABC be an acute, scalene triangle, and let M, N, and P be the midpoints of BC, CA, and AB, respectively. Let the perpendicular bisectors of AB and AC intersect ray AM in points D and E respectively, and let lines BD and CE intersect in point F, inside of triangle ABC. Prove that points A, N, F, and P all lie on one circle.