IrMO 2021

Let $N = 15! = 15\cdot 14\cdot 13 ... 3\cdot 2\cdot 1$ . Prove that $N$ can be written as a product of nine different integers all between $16$ and $30$ inclusive.

An isosceles triangle $ABC$ is inscribed in a circle with $\angle ACB = 90^o$ and $EF$ is a chord of the circle such that neither E nor $F$ coincide with $C$ . Lines $CE$ and $CF$ meet $AB$ at $D$ and $G$ respectively. Prove that $|CE|\cdot |DG| = |EF| \cdot |CG|$ .

For each integer $n \ge 100$ we define $T(n)$ to be the number obtained from $n$ by moving the two leading digits to the end. For example, $T(12345) = 34512$ and $T(100) = 10$ . Find all integers $n \ge 100$ for which $n + T(n) = 10n$ .

You have a $3 \times 2021$ chessboard from which one corner square has been removed. You also have a set of $3031$ identical dominoes, each of which can cover two adjacent chessboard squares. Let $m$ be the number of ways in which the chessboard can be covered with the dominoes, without gaps or overlaps. What is the remainder when $m$ is divided by $19$ ?

The function $g : [0, \infty) \to [0, \infty)$ satisfies the functional equation: $g(g(x)) = \frac{3x}{x + 3}$ , for all $x \ge 0$ . You are also told that for $2 \le x \le 3$ : $g(x) = \frac{x + 1}{2}$ . (a) Find $g(2021)$ . (b) Find $g(1/2021)$ .

A sequence whose first term is positive has the property that any given term is the area of an equilateral triangle whose perimeter is the preceding term. If the first three terms form an arithmetic progression, determine all possible values of the first term.

Each square of an $n \times n$ grid is coloured either blue or red, where $n$ is a positive integer. There are $k$ blue cells in the grid. Pat adds the sum of the squares of the numbers of blue cells in each row to the sum of the squares of the numbers of blue cells in each column to form $S_B$ . He then performs the same calculation on the red cells to compute $S_R$ . If $S_B- S_R = 50$ , determine (with proof) all possible values of $k$ .

A point $C$ lies on a line segment $AB$ between $A$ and $B$ and circles are drawn having $AC$ and $CB$ as diameters. A common tangent to both circles touches the circle with $AC$ as diameter at $P \ne C$ and the circle with $CB$ as diameter at $Q \ne C$ . Prove that $AP, BQ$ and the common tangent to both circles at $C$ all meet at a single point which lies on the circumference of the circle with $AB$ as diameter.

Suppose the real numbers $a, A, b, B$ satisfy the inequalities: $$|A - 3a| \le 1 - a\,\, , \,\, |B -3b| \le 1 - b$$ , and $a, b$ are positive. Prove that $$\left|\frac{AB}{3}- 3ab\right | - 3ab \le 1 - ab.$$

Let $P_{1}, P_{2}, \ldots, P_{2021}$ be 2021 points in the quarter plane $\{(x, y): x \geq 0, y \geq 0\}$ . The centroid of these 2021 points lies at the point $(1,1)$ .

Show that there are two distinct points $P_{i}, P_{j}$ such that the distance from $P_{i}$ to $P_{j}$ is no more than $\sqrt{2} / 20$ .