Compete · Serbia
Okruzno takmicenje (Regional round) 2002
1Let \(n\) be a natural number. Prove that \[ (n+1)^{3n} - n^{2n}(n+3)^n \] is divisible by \(3n+1\).2Prove that the number \[ \sqrt{1 + \sqrt{2 + \cdots + \sqrt{n}}} \] is irrational for every natural number \(n \geq 2\).3In a pentagon \(ABCDE\) all five sides are congruent to one another, and \[ \angle BAE = 2 \angle CAD . \] Determine \(\angle BAE\).4Over each side of a convex quadrilateral, as a diameter, a circle is constructed. Prove that these four circles cover the quadrilateral.5Ten teams took part in a volleyball tournament, and every team played exactly one match against each of the others. When the tournament ended, the first team had \(x_1\) wins and \(y_1\) losses, the second …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.