Given points $A_1 = (x_1, y_1, z_1)$, $A_2 = (x_2, y_2, z_2)$, ..., $A_n = (x_n, y_n, z_n)$ let $P = (x, y, z)$ be the point which minimizes $\sum_{i=1}^n (|x - x_i| + |y - y_i| + |z - z_i|)$. Give an example (for each $n \gt 4$) of points $A_i$ for which the point $P$ lies outside the convex hull of the points $A_i$.
Given a point $p$ inside a convex polyhedron $P$. Show that there is a face $F$ of $P$ such that the foot of the perpendicular from $p$ to $F$ lies in the interior of $F$.
Two players play alternately. The first player is given a pair of positive integers $(x_1, y_1)$. Each player must replace the pair $(x_n, y_n)$ that he is given by a pair of non-negative integers $(x_{n+1}, y_{n+1})$ such that $x_{n+1} = \min(x_n, y_n)$ and $y_{n+1} = \max(x_n, y_n) - k \cdot x_{n+1}$ for some positive integer $k$. The first player to pass on a pair with $y_{n+1} = 0$ wins. Find for which values of $\frac{x_1}{y_1}$ the first player has a winning strategy.
$A$ and $B$ wish to divide a cake into two pieces. Each wants the largest piece he can get. The cake is a triangular prism with the triangular faces horizontal. $A$ chooses a point $P$ on the top face. $B$ then chooses a vertical plane through the point $P$ to divide the cake. $B$ chooses which piece to take. Which point $P$ should $A$ choose in order to secure as large a slice as possible?
$p(x_1, x_2, \dots, x_n)$ is a polynomial with integer coefficients. For each positive integer $r$, $k(r)$ is the number of $n$-tuples $(a_1, a_2, \dots, a_n)$ such that $0 \le a_i \le r - 1$ and $p(a_1, a_2, \dots, a_n)$ is prime to $r$. Show that if $u$ and $v$ are coprime then $k(u \cdot v) = k(u) \cdot k(v)$, and if $p$ is prime then $k(p^s) = p^{n(s-1)}k(p)$.
For every positive integer $n$ show that
$$
[\sqrt{n} + \sqrt{n+1}] = [\sqrt{4n+1}] = [\sqrt{4n+2}] = [\sqrt{4n+3}]
$$
where $[x]$ is the greatest integer less than or equal to $x$ (for example $[2.3]=2$, $[\pi]=3$, $[5]=5$).
Suppose $ABCD$ is a parallelogram and $E$ is a point between $B$ and $C$ on the line $BC$. If the triangles $DEC$, $BED$ and $BAD$ are isosceles what are the possible values for the angle $DAB$?
The number $1987$ can be written as a three digit number $x y z$ in some base $b$. If $x + y + z = 1 + 9 + 8 + 7$, determine all possible values of $x, y, z, b$.
On a large, flat field $n$ people are positioned so that for each person the distances to all the other people are different. Each person holds a water pistol and at a given signal fires and hits the person who is closest. When $n$ is odd show that there is at least one person left dry. Is this always true when $n$ is even?
If $a$ and $b$ are two different real numbers, solve the $$
\begin{aligned}
x+y & =1 \\
(a x+b y)^2 & \leq a^2 x+b^2 y
\end{aligned}
$$ system also solve the $$
\begin{aligned}
x+y & =1 \\
(a x+b y)^4 & \leq a^4 x+b^4 y
\end{aligned}
$$ system
A given triangle $T$ decomposes into triangles $T_{1}, T_{2}, \ldots, T_{n}$ so that:
a) No pair of triangles $T_{i}$ has common interior points.
b) The union of all triangles $T_{i}$ is $T$.
c) Each segment that is side of some triangle $T_{i}$, or is side of another triangle $T_{j}$, or it is side of the triangle $T$. Let $s$ be the total number of sides (each counted only once, even if it is common to two triangles), and $v$ the total number of vertices (each counted only once, even if it is common to several triangles). Prove that if $n$ is odd, there are several decompositions of this class, and they all have the same number $v$ and the same number $s$. Express $v$ and $s$ in function of $n$. also show that if $n$ is par, there is no such decomposition.
Let $p_{n}(k)$ be the number of permutations of the set $\{1,2, \ldots, n\}$, $n \geq 1$, which have exactly $k$ fixed points. Demonstrate that $$
\sum_{k=0}^{n} k p_{n}(k) = n!
$$ (Note: A $f$ permutation of a $S$ set is a bijective application of $S$ on itself. An element $i$ $S$ is called the permutation fixed point $f$ if $f(i) = i$.)
In a triangle $ABC$ we have points $D$ and $E$ respectively on $AB$ and $AC$. We know the measurement of the angles indicated below: $\widehat{ABE}=30^\circ$, $\widehat{EBC}=50^\circ$, $\widehat{ACD}=20^\circ$ and $\widehat{DCB}=60^\circ$. Find the value of the angle $\widehat{EDC}$.
Demonstrate that for all natural numbers $n\gt 1$ holds $$
1 \cdot \sqrt{\left(\begin{array}{l}
n \\
1
\end{array}\right)}+2 \cdot \sqrt{\left(\begin{array}{l}
n \\
2
\end{array}\right)}+\cdots+n \cdot \sqrt{\left(\begin{array}{l}
n \\
n
\end{array}\right)}\lt \sqrt{2^{n-1} n^{3}}
$$
Let $a, b, c$ be the lengths of the sides of a non-isosceles triangle. Three concentric circles of radii $a, b$ and $c$ are given.
a) How many equilateral triangles of different areas can be constructed, so that the lines that contain their sides are tangent to each circle?
b) Find the surfaces of these triangles.
In an acute triangle $ABC$ the inner bisector of the angle $A$ cuts to $BC$ at $L$ and cuts the circumscribed circumference of $ABC$ again on $N$. We draw perpendiculars from $L$ to $AB$ and $AC$, with feet $K$ and $M$, respectively. Prove that the quadrilateral $AKNM$ and triangle $ABC$ have the same area.
Let $x_{1}, x_{2}, \ldots, x_{n}$, real numbers that meet $x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=1$. Demonstrate that for each integer $k \geq 2$ there are integers not all null $a_{1}, a_{2}, \ldots, a_{n}$, such that $\left|a_{i}\right| \leq k-1$ for all $i$ and $$
\left|a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n}\right| \leq \frac{(k-1) \sqrt{n}}{k^{n}-1}
$$