Compete · Serbia
Okruzno takmicenje (Regional round) 2020
1A relation \(\diamond\) is defined on the set \(\mathbb{R}\) of real numbers by \[ a \diamond b \quad \text{if and only if} \quad |a - 1| + |b - 2| \leqslant 1 . \] Suppose the real numbers \(x\) and \(y\) …2Every positive integer is painted in one of two colours, one of which is called red. The painting is periodic with period \(d\): the numbers \(x\) and \(x + d\) always receive the same colour. Suppose …3Find all points \(X\) inside the square \(ABCD\) for which \[ AX + CX = BX + DX . \]4A collection of \(2020\) consecutive positive integers is split into two subsets of \(1010\) numbers each. Can the least common multiple of all the numbers in the first subset be equal to the least common …5Determine the smallest possible value of the expression \[ F = \max\{x,\, 1 - y\} + \max\{y,\, 2 - z\} + \max\{z,\, 3 - x\}, \] where \(x\), \(y\), \(z\) are real numbers, and find all triples \((x, y, z)\) …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.