PL-OM 1991

Prove or disprove that there exist two tetrahedra $T_1$ and $T_2$ such that: (i) the volume of $T_1$ is greater than that of $T_2$ ; (ii) the area of any face of $T_1$ does not exceed the area of any face of $T_2$ .

Let $X$ be the set of all lattice points in the plane (points $(x, y)$ with $x, y \in \mathbb{Z}$ ). A path of length $n$ is a chain $(P_0, P_1, ... , P_n)$ of points in $X$ such that $P_{i-1}P_i = 1$ for $i = 1, ... , n$ . Let $F(n)$ be the number of distinct paths beginning in $P_0=(0,0)$ and ending in any point $P_n$ on line $y = 0$ . Prove that $F(n) = \binom{2n}{n}$

Define $$N=\sum\limits_{k=1}^{60}e_k k^{k^k}$$ where $e_k \in \{-1, 1\}$ for each $k$ . Prove that $N$ cannot be the fifth power of an integer.

On the Cartesian plane consider the set $V$ of all vectors with integer coordinates. Determine all functions $f : V \rightarrow \mathbb{R}$ satisfying the conditions: (i) $f(v) = 1$ for each of the four vectors $v \in V$ of unit length. (ii) $f(v+w) = f(v)+f(w)$ for every two perpendicular vectors $v, w \in V$ (Zero vector is considered to be perpendicular to every vector).

Two noncongruent circles $k_1$ and $k_2$ are exterior to each other. Their common tangents intersect the line through their centers at points $A$ and $B$ . Let $P$ be any point of $k_1$ . Prove that there is a diameter of $k_2$ with one endpoint on line $PA$ and the other on $PB$ .

If $x, y, z$ are real numbers satisfying $x^2 +y^2 +z^2 = 2$ , prove the inequality $$x + y + z \leq 2 + xyz$$ When does equality occur?