PL-OM 1983

On the plane are given a convex $n$ -gon $P_1P_2....P_n$ and a point $Q$ inside it, not lying on any of its diagonals. Prove that if $n$ is even, then the number of triangles $P_iP_jP_k$ containing the point $Q$ is even.

Let be given an irrational number $a$ in the interval $(0,1)$ and a positive integer $N$ . Prove that there exist positive integers $p,q,r,s$ such that $\frac{p}{q} < a <\frac{r}{s}, \frac{r}{s} -\frac{p}{q}<\frac{1}{N}$ , and $rq- ps = 1$ .

Consider the following one-player game on an infinite chessboard. If two horizontally or vertically adjacent squares are occupied by a pawn each, and a square on the same line that is adjacent to one of them is empty, then it is allowed to remove the two pawns and place a pawn on the third (empty) square. Prove that if in the initial position all the pawns were forming a rectangle with the number of squares divisible by $3$ , then it is not possible to end the game with only one pawn left on the board.

Prove that if natural numbers $a,b,c,d$ satisfy the equality $ab = cd$ , then $\frac{gcd(a,c)gcd(a,d)}{gcd(a,b,c,d)}= a$

On the plane are given unit vectors $\overrightarrow{a_1},\overrightarrow{a_2},\overrightarrow{a_3}$ . Show that one can choose numbers $c_1,c_2,c_3 \in \{-1,1\}$ such that the length of the vector $c_1\overrightarrow{a_1}+c_2\overrightarrow{a_2}+c_3\overrightarrow{a_3}$ is at least $2$ .

Prove that if all dihedral angles of a tetrahedron are acute, then all its faces are acute-angled triangles.