Problem 1
1a. Peer and Solveig play a game with \(n\) coins. Peer flips one or more non-adjacent coins, and Solveig flips exactly two adjacent coins. Peer wins if all coins show \(K\). For which values of \(n \geq 2\) can Solveig prevent Peer from winning?
1b. In Duckville, the trophy is passed around among inhabitants. Gregers gets it back exactly every 2025 days. Hedvig has it today. For which values of \(n > 0\) is it impossible for Hedvig to receive the trophy again in \(n\) days?
Problem 2
2a. Eleven pupils each write a different positive integer on a sticky note. Prove that the teacher can always choose two or more notes so that the average of their numbers is not an integer.
2b. Find the integers \(a\) such that \(n! - a\) is a perfect square for infinitely many \(n\).
Problem 3
3a. In triangle \(ABC\), \(E\) and \(F\) are the feet of the altitudes from \(B\) and \(C\), respectively. \(P\) and \(Q\) are the projections of \(B\) and \(C\) onto line \(EF\). Show that \(PE = QF\).
3b. In an acute triangle \(ABC\), \(O\) is the circumcenter. Lines from \(O\) intersect the sides of the triangle. Show that if triangles \(ABC\) and \(DEF\) are similar, then triangle \(ABC\) must be equilateral.
Problem 4
4a. Find all polynomials \(P(x)\) that satisfy the functional equation: \[ P\left(\frac{1}{1+x}\right) = \frac{1}{1+P(x)} \] for all \(x \neq -1\).
4b. Find the largest real number \(C\) such that: \[ \frac{1}{x} + \frac{1}{2y} + \frac{1}{3z} \geq C \] for all \(x, y, z \neq 0\) satisfying: \[ \frac{x}{yz} + \frac{4y}{xz} + \frac{9z}{xy} = 24 \]
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