Compete · Slovenia
Drzavno tekmovanje (national round) 2021
1Each of the five figures drawn below is bounded entirely by semicircular arcs, and the largest arc is the same in all five figures. Among the figures with the smallest perimeter, which one has the largest …2The digits of a five-digit number add up to \(44\). What is the product of the digits of that number? A \(2^{3} \cdot 3^{8}\) B \(2^{3} \cdot 9^{3}\) C \(8 \cdot 4^{9}\) D \(8 \cdot 3^{4}\) E None of the …3Peter covered a wound with two rectangular plasters, laid across each other as in the picture. The region covered by both plasters at once has area \(40 \ \mathrm{cm}^{2}\) and perimeter \(30 \ \mathrm{cm}\). …4Prove that there are no natural numbers \(a\) and \(b\) satisfying \[ \sqrt{a} + \sqrt{b} = \sqrt{2021} . \]5Find all real numbers \(x\), \(y\), \(z\) that satisfy the system \[ \frac{3xy}{x-y} = 2, \qquad \frac{2yz}{y+2z} = 3, \qquad \frac{xz}{z-4x} = 3 . \]6Natasa glued a square and an equilateral triangle of the same side length into a pentagon. Out of seven copies of that pentagon she built the figure shown on the right, which sits inside a large square. …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.