Compete · Serbia
Opštinsko takmičenje (Municipal round) 2014
1Real numbers \(a\), \(b\), \(c\) satisfy the three inequalities \[ |b - c| \ge |a|, \qquad |c - a| \ge |b|, \qquad |a - b| \ge |c|. \] Prove that one of the numbers \(a\), \(b\), \(c\) is equal to the …2A circle is inscribed in triangle \(ABC\), whose sides have lengths \(BC = a\), \(CA = b\) and \(AB = c\). A line tangent to this circle meets the side \(AC\) at the point \(P\) and the side \(BC\) at …3Find all solutions of the equation \[ 6\left(6a^{2} + 3b^{2} + c^{2}\right) = 5d^{2} \] in integers \(a\), \(b\), \(c\), \(d\).4Let \(ABCD\) be a quadrilateral such that \[ \angle BCA + \angle CAD = 180^{\circ} \qquad\text{and}\qquad AB = AD + BC. \] Prove that \(\angle BAC + \angle ACD = \angle CDA\).5Ten teams took part in a volleyball tournament, and each team played exactly one match against each of the other nine. When the tournament ended, the first team had \(x_{1}\) wins and \(y_{1}\) losses, …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.