Problem 1
Let k and d be positive integers. Prove that there exists a positive integer N such that for every odd integer n > N, the digits in the base-2n representation of nk are all greater than d.
Problem 2
Let n and k be positive integers with k < n. Let P(x) be a polynomial of degree n with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers a₀, a₁, ..., ak such that the polynomial akxk + ... + a₁x + a₀ divides P(x), the product a₀a₁...ak is zero. Prove that P(x) has a nonreal root.
Problem 3
Alice the architect and Bob the builder play a game. Alice chooses two points P and Q and a subset 𝒮 of the plane. Bob places infinitely many cities so that no two are within distance 1 and no three are collinear. Two cities A and B are connected by a road if for every other city C, there exists R ∈ 𝒮 such that triangle PQR is directly similar to either ABC or BAC. Alice wins if the road network is connected and no roads cross. Who has a winning strategy?
Problem 4
Let H be the orthocenter of acute triangle ABC, let F be the foot of the altitude from C to AB, and let P be the reflection of H across BC. Suppose the circumcircle of triangle AFP intersects line BC at two distinct points X and Y. Prove that C is the midpoint of XY.
Problem 5
Determine, with proof, all positive integers k such that:
(1 / (n + 1)) × Σᵢ₌₀ⁿ (n choose i)^k
is an integer for every positive integer n.
Problem 6
Let m and n be positive integers with m ≥ n. There are m cupcakes of different flavors arranged around a circle and n people. Each person assigns nonnegative scores to the cupcakes. Suppose for each person P, the circle can be partitioned into n consecutive groups such that each group scores at least 1 for P. Prove that it’s possible to assign cupcakes so each person gets total score at least 1.
