Opštinsko takmičenje (Municipal round) 1997 · Lemma

1Number theoryFree answerFind all pairs \((n, m)\) of integers for which \[ 3n^2 + 2nm + 3 = m^2 + 10. \]2Number theoryProofLet \(a\) and \(b\) be arbitrary natural numbers, let \(M\) be their least common multiple and \(D\) their greatest common divisor. Prove that \[ a^n + b^n \le M^n + D^n \] holds for every natural number …3GeometryProofAn angle of \(7^\circ\) is given. Using only compass and straightedge, divide it into seven equal parts.4CombinatoricsFree answerThe caliph of Baghdad rewarded three wise men with ten purses: the first held \(0\) dinars, the second \(1\) dinar, the third \(2\) dinars, and so on up to the tenth, which held \(9\) dinars. The first …5Number theoryFree answerDetermine the smallest natural number the product of whose digits equals \(75600\).

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