Opštinsko takmičenje (Municipal round) 2017 · Lemma

1CombinatoricsProofA park has the shape of a square with side \(1\) km. Inside it grow \(4567\) trees, each of diameter at most \(50\) cm, and each tree lies entirely within the park. Prove that the park contains a \(10\) …2GeometryFree answerIn a quadrilateral \(ABCD\), \[ \angle ABC = 104^\circ, \qquad \angle ADC = 128^\circ, \qquad AB = BC = 2. \] Compute the length of the diagonal \(BD\).3Number theoryProofSolve the equation \[ 12^{x} + 10^{y} = 7102^{z} \] in the set of natural numbers.4GeometryFree answerLet \(A\), \(B\), \(C\), \(D\) be four points in the plane, no three of them collinear. Every choice of three of these points forms a triangle, so the four points determine \(12\) angles in all. Write …5AlgebraFree answerDetermine how many distinct solutions the equation \[ \Bigl| \bigl| \cdots \bigl| \bigl| |x| - 1 \bigr| - 2 \bigr| - \cdots - 2016 \bigr| - 2017 \Bigr| = 2017 \] has, where the constants \(1, 2, \dots, 2017\) …

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