Chapter: Algebra
Problem 1: Let \(a, b, c\) be positive integers such that \(a^2 + b^2 + c^2\) is divisible by \(ab + bc + ca\). Prove that \(a = b = c\).
Points: 7
Chapter: Number Theory
Problem 2: Prove that there are infinitely many primes of the form \(4k + 3\).
Points: 7
Chapter: Combinatorics
Problem 3: In a party of \(2n\) people, each person shakes hands with exactly \(n\) others. Show that there are an even number of people who shake hands with each other.
Points: 7
Chapter: Geometry
Problem 4: Let \(ABC\) be an acute-angled triangle with \(AB \neq AC\). The circle with center at the midpoint of \(BC\) and radius \(BC\) intersects the line \(AB\) at \(P\) and the line \(AC\) at \(Q\). Prove that the line \(PQ\) is perpendicular to \(BC\).
Points: 7
Chapter: Probability
Problem 5: Consider a sequence of \(2n\) coin flips. Prove that the number of heads and tails are equal in at least one of the first \(2n\) prefixes.
Points: 7
Chapter: Calculus
Problem 6: Evaluate the limit \( \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{k}{n^2 + k^2} \right) \).
Points: 7