Okruzno takmicenje (Regional round) 2007 · Lemma

1Number theoryFree answerDetermine the remainder left by the number \[ 3^{1000} + 4^{1000} \] when it is divided by \(13\).2GeometryProofTwo circles touch each other internally at a point \(A\). Let \(AB\) be a diameter of the larger circle. Through the other endpoint \(B\) of this diameter a line is drawn which touches the smaller circle …3CombinatoricsFree answerAn entry of a permutation is called right-minimal if it is smaller than every entry standing to its right. For example, in the permutation \[ (2,\; 1,\; 4,\; 6,\; 3,\; 7,\; 8,\; 5) \] the right-minimal …4AlgebraProofLet \(a\) and \(b\) be real numbers with \(0 < b \leqslant a\). Prove that \[ \frac{1}{8} \cdot \frac{(a-b)^{2}}{a} \;\leqslant\; \frac{a+b}{2} - \sqrt{ab} \;\leqslant\; \frac{1}{8} \cdot \frac{(a-b)^{2}}{b} . \] …5Number theoryFree answerFor a natural number \(n\), determine the greatest common divisor of the two numbers \[ n^{2} + 1 \qquad \text{and} \qquad (n+1)^{2} + 1 , \] expressed in terms of \(n\).

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