IrMO 2022

For n a positive integer, n! = 1 $\cdot$ 2 $\cdot$ 3 $\dots$ (n - 1) $\cdot$ n is the product of the positive integers from 1 to n.

Determine, with proof, all positive integers n for which n! + 3 is a power of 3.

Let ABCDbe a square and let $\Gamma$ denote the circle with diameter CD. A tangent line is drawn to the circle $\Gamma$ from B, meeting the circle $\Gamma$ at E and intersecting the segment AD at K.

Prove that |AD| = 4 |KD|.

Let $\mathbb{N}$ denote the strictly positive integers. A function $f$ : $\mathbb{N}$ $\to$ $\mathbb{N}$ has the following properties which hold for all $n \in$ $\mathbb{N}$ :

a) $f(n)$ < $f(n+1)$ ; b) $f(f(f(n)))$ = 4 $n$ Find $f(2022)$ .

Let $\triangle$ ABC be a triangle with circumcentre O. The perpendicular line from O to BC intersects line BC at M and line AC at P, and the perpendicular line from O to AC intersects line AC at N and line BC at Q. Let D be the intersection point of lines PQ and MN. construct the parallelogram PCQJ with PJ || CQ and QJ || CP. Prove the following:

a) The points A, B, O, P, Q, J are all on the same circle. b) line OD is perpendicular to line CJ.

Suppose a, b, c are real numbers such that a + b + c = 1. Prove that

$$a^3 + b^3 + c^3 + 3(1-a)(1-b)(1-c) = 1.$$

The four Vertices of a quadrilateral ABCD lie on the circle with diameter AB. The diagonals of ABCD intersect at E, and the lines AD and BC intersect at F. Line FE meets AB at K and line DK meets the circle again at L.

Prove that CL is perpendicular to AB.

The Equation AB X CD = EFGH, where each of the letters A, B, C, D, E, F, G, H represents a different digit and the values of A, C and E are all nonzero, has many solutions, e.g., 46 X 85 =3910. Find the smallest value of the four-digit number EFGH for which there is a solution.

Let k be a positive integer and let $x_0, x_1, x_2, \cdots$ be an infinite sequence defined by the relationship $$x_0 = 0$$ $$x_1 = 1$$ $$x_{n+1} = kx_n +x_{n-1}$$ For all n $\ge$ 1 (a) For the special case k = 1, prove that $x_{n-1}x_{n+1}$ is never a perfect square for n $\ge$ 2 (b) For the general case of integers k $\ge$ 1, prove that $x_{n-1}x_{n+1}$ is never a perfect square for n $\ge$ 2

Let $n \ge 5$ be an odd number and let $r$ be an integer such that $1\le r \le (n-1)/2$ . IN a sports tournament, $n$ players take part in a series of contests. In each contest, $2r+1$ players participate, and the scores obtained by the players are the numbers $$-r, -(r-1),\cdots, -1, 0, 1 \cdots, r-1, r$$ in some order. Each possible subset of $2r+1$ players takes part together in exactly one contest. let the final score of player $i$ be $S_i$ , for each $i=1, 2,\cdots,n$ . Define $N$ to be the smallest difference between the final scores of two players, i.e., $$N = \min_{i<j}|S_i - S_j|.$$ Determine, with proof, the maximum possible value of $N$ .