Okruzno takmicenje (Regional round) 1997 · Lemma

1AlgebraProofLet \(a, b, c, d\) be numbers satisfying \[ a^2 + b^2 + (a+b)^2 = c^2 + d^2 + (c+d)^2 . \] Prove that then \[ a^4 + b^4 + (a+b)^4 = c^4 + d^4 + (c+d)^4 . \]2Number theoryFree answerIt is known that \[ 35! = 10333147966386144929\,ab\,6651337523200000000 , \] where the letters \(a\) and \(b\) stand for two unknown decimal digits. Determine these two digits.3GeometryFree answerIn triangle \(ABC\) the bisector of the angle \(CAB\) meets the side \(BC\) at the point \(N\), and the bisector of the angle \(CBA\) meets the side \(AC\) at the point \(P\), where \[ PN = a . \] Let …4CombinatoricsFree answerIn how many ways can a king, a queen, two rooks, two bishops and two knights be placed on the eight squares of the first rank of a chessboard so that the two rooks stand on opposite sides of the king, …5GeometryProofA quadrilateral \(ABCD\) satisfies \[ AD = BC \qquad \text{and} \qquad \angle DAB > \angle ABC . \] Prove that then \(\angle BCD > \angle CDA\).

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