Izbirno tekmovanje (selection round) 2007 · Lemma

1GeometryFree answerTwo concentric circles of radii \(7\) cm and \(11\) cm are drawn in the plane. The smaller circle cuts a chord of the larger circle into three pieces of equal length. How long is that chord?2Number theoryFree answerA competition paper had \(20\) problems. Each problem answered correctly was worth \(8\) points, each problem answered incorrectly cost \(5\) points, and a problem left blank scored \(0\). Tine handed …3GeometryFree answerIn the right triangle \(ABC\) the right angle is at \(C\), the hypotenuse \(AB\) measures \(1\) dm, and \(\angle BAC = 30^\circ\). A point \(D\) inside the triangle satisfies \[ \angle BDC = 90^\circ \qquad\text{and}\qquad \angle ACD = \angle DBA . \] …4AlgebraProofLet \(a\) and \(b\) be arbitrary nonnegative real numbers. Prove that \[ (ab+1)(a+b) \ge 4ab , \] and determine all pairs \((a,b)\) for which equality holds.5Number theoryFree answerThe edge of a wooden cube is a natural number \(a > 2\). The cube is painted all over and then cut into unit cubes. It turns out that the number of unit cubes with exactly two painted faces divides the …

Work through the paper in order - each problem opens in the workspace with this paper as its trail.