Compete · Serbia
Okruzno takmicenje (Regional round) 2009
1Let \(M\) and \(N\) be two distinct points, neither of which lies on a given line \(p\). Construct a triangle \(ABC\) whose side \(AB\) lies on \(p\) and for which \(M\) and \(N\) are the feet of the altitudes …2Let \(p\), \(q\), \(r\) be real numbers such that \[ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 0 \qquad \text{and} \qquad p + q + r = 1 . \] Prove that for all real numbers \(a\), \(b\), \(c\), \[ a^{2} + b^{2} + c^{2} = (pa + qb + rc)^{2} + (qa + rb + pc)^{2} + (ra + pb + qc)^{2} . \] …3For every point of the first quadrant, determine the line through that point which, together with the positive parts of the coordinate axes, bounds a triangle of the smallest possible area.4Determine all natural numbers \(n\) for which the following assertion is true: a natural number \(x\) is divisible by \(n\) if and only if the sum of the digits of \(x\) is divisible by \(n\).5Two teams, each consisting of \(6\) footballers, have at their disposal \(4\) pairs of shorts and \(4\) jerseys in each of the colours red, blue and white. In how many ways can the footballers dress for …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.