Compete · Serbia
Opštinsko takmičenje (Municipal round) 2016
1Let \(A\), \(B\) and \(C\) be finite sets whose sizes satisfy \[ |A \triangle C| + |B \triangle C| = |A \triangle B|. \] Prove that \(C\) is then trapped between the intersection and the union of the other …2A row contains \(2016\) chairs. Each chair is to be painted either red or blue. In how many ways can this be done so that the number of neighbouring pairs of chairs painted in the same colour is even?3Let \(ABC\) be a triangle. On side \(AB\) choose points \(C_1\) and \(C_2\) with \[ AC_1 = \tfrac{2015}{3015}\,AB, \qquad AC_2 = \tfrac{2015}{3014}\,AB; \] on side \(BC\) choose points \(A_1\) and \(A_2\) …4Consider strictly increasing sequences \(a_1, a_2, a_3, \dots\) of prime numbers in which any two consecutive terms differ by \(2\) or by \(4\); that is, \[ a_{i+1} - a_i \in \{2, 4\} \quad \text{for every } i. \] …5Let \(ABCDE\) be a convex pentagon whose five sides all have the same length. Suppose that two of its diagonals meet at an angle of \(60^\circ\). Prove that the pentagon has two parallel sides.
Work through the paper in order - each problem opens in the workspace with this paper as its trail.