OSN 2021

On the whiteboard, the numbers are written sequentially: $1 \ 2 \ 3 \ 4 \ 5 \ 6 \ 7 \ 8 \ 9$ . Andi has to paste a $+$ (plus) sign or $-$ (minus) sign in between every two successive numbers, and compute the value. Determine the least odd positive integer that Andi can't get from this process.

Let $ABC$ be an acute triangle. Let $D$ and $E$ be the midpoint of segment $AB$ and $AC$ respectively. Suppose $L_1$ and $L_2$ are circumcircle of triangle $ABC$ and $ADE$ respectively. $CD$ intersects $L_1$ and $L_2$ at $M (M \not= C)$ and $N (N \not= D)$ . If $DM = DN$ , prove that $\triangle ABC$ is isosceles.

A natural number is called a prime power if that number can be expressed as $p^n$ for some prime $p$ and natural number $n$ . Determine the largest possible $n$ such that there exists a sequence of prime powers $a_1, a_2, \dots, a_n$ such that $a_i = a_{i - 1} + a_{i - 2}$ for all $3 \le i \le n$ .

Let $x,y$ and $z$ be positive reals such that $x + y + z = 3$ . Prove that $$2 \sqrt{x + \sqrt{y}} + 2 \sqrt{y + \sqrt{z}} + 2 \sqrt{z + \sqrt{x}} \le \sqrt{8 + x - y} + \sqrt{8 + y - z} + \sqrt{8 + z - x}$$

Let $P(x) = x^2 + rx + s$ be a polynomial with real coefficients. Suppose $P(x)$ has two distinct real roots, both of which are less than $-1$ and the difference between the two is less than $2$ . Prove that $P(P(x)) > 0$ for all real $x$ .

There are $n$ natural numbers written on the board. Every move, we could erase $a,b$ and change it to $\gcd(a,b)$ and $\text{lcm}(a,b) - \gcd(a,b)$ . Prove that in finite number of moves, all numbers in the board could be made to be equal.

Given $\triangle ABC$ with circumcircle $\ell$ . Point $M$ in $\triangle ABC$ such that $AM$ is the angle bisector of $\angle BAC$ . Circle with center $M$ and radius $MB$ intersects $\ell$ and $BC$ at $D$ and $E$ respectively, $(B \not= D, B \not= E)$ . Let $P$ be the midpoint of arc $BC$ in $\ell$ that didn't have $A$ . Prove that $AP$ angle bisector of $\angle DPE$ if and only if $\angle B = 90^{\circ}$ .

On a $100 \times 100$ chessboard, the plan is to place several $1 \times 3$ boards and $3 \times 1$ board, so that

• Each tile of the initial chessboard is covered by at most one small board.
• The boards cover the entire chessboard tile, except for one tile.
• The sides of the board are placed parallel to the chessboard.

Suppose that to carry out the instructions above, it takes $H$ number of $1 \times 3$ boards and $V$ number of $3 \times 1$ boards. Determine all possible pairs of $(H,V)$ .

Proposed by Muhammad Afifurrahman, Indonesia