Compete · Serbia
Opštinsko takmičenje (Municipal round) 2004
1Let \(E\) be a point of the diagonal \(AC\) of a rhombus \(ABCD\), with \(E \ne A\) and \(E \ne C\). Let \(N\) be the point of the line \(AB\) other than \(A\) for which \(EN = EA\), and let \(M\) be the …2Let \(a\), \(b\), \(c\) be positive integers such that all three of the numbers \[ p = b^{c} + a, \qquad q = a^{b} + c, \qquad r = c^{a} + b \] are prime. Prove that two of the numbers \(p\), \(q\), \(r\) …3Let \(q\) be an odd integer. Prove that the equation \[ x^{3} + 3x + q = 0 \] has no solutions in integers.4In how many ways can \(m\) distinct birds be placed into \(n\) distinct cages so that every cage contains at least one bird and at most two birds?5A mosquito sits on the lower left cell of a rectangular board of format \(2003 \times 2004\). It travels above the board in the following manner: taking off from the cell it occupies, it flies over \(99\) …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.