Compete · Serbia
Republicko takmicenje (Republic round) 2000
1Let \(ABCD\) be a parallelogram whose interior angle at \(A\) is acute, and let \(E\) be a point of the plane such that \(EA \perp AB\) and \(EC \perp CB\). Prove that \[ \angle AED = \angle CEB . \]2An acute triangle \(ABC\) is given. Construct, with ruler and compass, a point \(P\) inside the triangle such that the rays \(AP\), \(BP\) and \(CP\) meet the circumcircle of \(ABC\) again in the three …3Let \(a_1, a_2, \ldots, a_n\) be positive real numbers with \(a_1 + a_2 + \cdots + a_n = 1\), and let \[ S = \sum_{i=1}^{n} \sum_{j=1}^{n} \frac{a_i a_j}{a_i + a_j} \] be the sum of all \(n^2\) expressions …4An island is inhabited by \(45\) chameleons: \(17\) yellow, \(15\) grey and \(13\) blue. They wander about and meet from time to time, never more than two at a time. When two chameleons of the same colour …5Let \(a\), \(b\), \(c\), \(d\), \(x\), \(y\) be positive real numbers such that \[ a + 2ay + y = b + 2bx + x \qquad \text{and} \qquad x + 2xd + d = y + 2yc + c . \] Prove that \[ a + 2ad + d = b + 2bc + c . \] …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.