IOQM 2026 class 1 - Parity and power of integers
Q1: Find all nonnegative integers \( n \) such that there are integers \( a \) and \( b \) with the property \[ n^2 = a + b \quad \text{and} \quad n^3 = a^2 + b^2. \] Solution: Using QM-AM inequality, \[ \frac{a + b}{2} \le \sqrt{\frac{a^2 + b^2}{2}}. \] Substitute \( a + b = n^2 \) and \( a^2 + b^2 = n^3 \): \[ \frac{n^2}{2} \le \sqrt{\frac{n^3}{2}}. \] Squaring both sides: \[ \frac{n^4}{4} \le \frac{n^3}{2}. \] Simplify: \[ \frac{n}{2} \le 1. \] Thus, \[ n \le 2. \] So \( n = 0, 1, 2 \). Q2: In each box of a \(15 \times 15\) square table one of the numbers \(1,2,3,\dots,15\) is written. Boxes which are symmetric with respect to one of the main diagonals contain equal numbers, and no row or column contains two copies of the same number. Show that no two of the numbers along the main diagonal are the same. Solution: Sample 3x3 matrix: [1,2,3] [3,1,2] [2,3,1] Consider a smaller \(3 \times 3\) matrix to understand the structure. Let the matrix be symmetric about a...
