Okruzno takmicenje (Regional round) 2017 · Lemma

1Logic and setsProofLet \(A\), \(B\), \(C\) and \(D\) be finite sets such that \(D \subseteq A \cup B\), \(D \subseteq C\) and \[ |A \triangle B| + |B \setminus C| + |C \setminus D| + |B \cap D| = |A| . \] (a) Prove that …2GeometryProofLet \(k\) be a circle with centre \(O\), and let \(T\) be a point outside it. The two tangents drawn from \(T\) touch \(k\) at the points \(A\) and \(B\). Let \(k'\) be the circle with centre \(T\) that …3CombinatoricsFree answerSixteen teams take part in a basketball tournament played as a double round robin: every two teams meet exactly twice. The eight best-placed teams qualify for the next tournament. Teams are ranked by the …4Number theoryFree answerDetermine all natural numbers \(n\) with all of the following properties: \(n\) is divisible by \(2\) but not by \(4\); the sum of the digits of \(n\) equals \(6\); the number of divisors of \(n\) equals …5GeometryProofThe side lengths of a certain triangle are mutually distinct natural numbers, and its area is a natural number as well. Must that triangle be right-angled?

Work through the paper in order - each problem opens in the workspace with this paper as its trail.