Izbirno tekmovanje (selection round) 2010 · Lemma

1AlgebraFree answerLet \(t\) be a real number and let \(a\) and \(b\) be positive real numbers satisfying \[ 2a^2 - 3abt + b^2 \;=\; 2a^2 + abt - b^2 \;=\; 0 . \] Determine the value of \(t\).2Number theoryFree answerFind all prime numbers \(p\), \(q\) and \(r\) for which \[ 15p + 7pq + qr = pqr . \]3GeometryProofLet \(ABC\) be an isosceles triangle with apex \(C\). Points \(D\) and \(E\) lie on the sides \(AC\) and \(BC\) respectively, and the bisector of the angle \(\angle DEB\) and the bisector of the angle …4Number theoryFree answerFind the smallest three-digit number with the following property: if its digits are written in the reverse order and the number so obtained is added to the original number, then every digit of the sum …5CombinatoricsFree answerVid cut a square \(ABCD\) of side length \(20\) units into \(400\) unit squares. Eva then picked four vertices of unit squares, all lying in the interior of \(ABCD\), that are the vertices of a rectangle …

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