Solve the system of equations $$\begin{aligned} &|a_1 - a_2| x_2 & +\, |a_1 - a_3| x_3 & + |a_1 - a_4| x_4 &= 1 \\ |a_2 - a_1| x_1 & & +\, |a_2 - a_3| x_3 & + |a_2 - a_3| x_3 &= 1 \\ |a_3 - a_1| x_1 & + |a_3 - a_2| x_2 & & &= 1 \\ |a_4 - a_1| x_1 & + |a_4 - a_2| x_2 & +\, |a_4 - a_3| x_3 & &= 1 \end{aligned}$$ where $a_1, a_2, a_3, a_4$ are four different real numbers.

In the interior of sides $BC, CA, AB$ of triangle $ABC$, any points $K, L, M$, respectively, are selected. Prove that the area of at least one of the triangles $AML, BKM, CLK$ is less than or equal to one quarter of the area of triangle $ABC$.

Let $ABCD$ be a parallelogram with side lengths $AB = a$, $AD = 1$, and with $\angle BAD = \alpha$. If $\triangle ABD$ is acute, prove that the four circles of radius 1 with centers $A, B, C, D$ cover the parallelogram if and only if $$a \leq \cos \alpha + \sqrt{3} \sin \alpha.$$

Prove that if one and only one edge of a tetrahedron is greater than 1, then its volume is $\leq 1/8$.

Let $k, m, n$ be natural numbers such that $m + k + 1$ is a prime greater than $n + 1$. Let $c_s = s(s + 1)$. Prove that the product $$(c_{m+1} - c_k)(c_{m+2} - c_k) \cdots (c_{m+n} - c_k)$$ is divisible by the product $c_1c_2\cdots c_n$.

Let $A_0B_0C_0$ and $A_1B_1C_1$ be any two acute-angled triangles. Consider all triangles $ABC$ that are similar to $\triangle A_1B_1C_1$ (so that vertices $A_1, B_1, C_1$ correspond to vertices $A, B, C$, respectively) and circumscribed about triangle $A_0B_0C_0$ (where $A_0$ lies on $BC$, $B_0$ on $CA$, and $AC_0$ on $AB$). Of all such possible triangles, determine the one with maximum area, and construct it.

Consider the sequence $\{c_n\}$, where $$\begin{aligned} c_1 &= a_1 + a_2 + \cdots + a_8 \\ c_2 &= a_1^2 + a_2^2 + \cdots + a_8^2 \\ &\cdots \\ c_n &= a_1^n + a_2^n + \cdots + a_8^n \\ &\cdots \end{aligned}$$ in which $a_1, a_2, \ldots, a_8$ are real numbers not all equal to zero. Suppose that an infinite number of terms of the sequence $\{c_n\}$ are equal to zero. Find all natural numbers $n$ for which $c_n = 0$.

In a sports contest, there were $m$ medals awarded on $n$ successive days ($n > 1$). On the first day, one medal and $1/7$ of the remaining $m - 1$ medals were awarded. On the second day, two medals and $1/7$ of the now remaining medals were awarded; and so on. On the $n$-th and last day, the remaining $n$ medals were awarded. How many days did the contest last, and how many medals were awarded altogether?

Prove that there is one and only one triangle whose side lengths are consecutive integers, and one of whose angles is twice as large as another.

Find all natural numbers $x$ such that the product of their digits (in decimal notation) is equal to $x^2 - 10x - 22$.

Consider the system of equations $$\begin{aligned} ax_1^2 + bx_1 + c &= x_2 \\ ax_2^2 + bx_2 + c &= x_3 \\ &\vdots \\ ax_{n-1}^2 + bx_{n-1} + c &= x_n \\ ax_n^2 + bx_n + c &= x_1, \end{aligned}$$ with unknowns $x_1, x_2, \ldots, x_n$, where $a, b, c$ are real and $a \neq 0$. Let $\Delta = (b-1)^2 - 4ac$. Prove that for this system

(a) if $\Delta < 0$, there is no solution,

(b) if $\Delta = 0$, there is exactly one solution,

(c) if $\Delta > 0$, there is more than one solution.

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which are the sides of a triangle.

Let $f$ be a real-valued function defined for all real numbers $x$ such that, for some positive constant $a$, the equation $$f(x + a) = \frac{1}{2} + \sqrt{f(x) - [f(x)]^2}$$ holds for all $x$.

(a) Prove that the function $f$ is periodic (i.e., there exists a positive number $b$ such that $f(x + b) = f(x)$ for all $x$).

(b) For $a = 1$, give an example of a non-constant function with the required properties.

For every natural number $n$, evaluate the sum $$\sum_{k=0}^{\infty} \left[ \frac{n + 2^k}{2^{k+1}} \right] = \left[ \frac{n + 1}{2} \right] + \left[ \frac{n + 2}{4} \right] + \cdots + \left[ \frac{n + 2^k}{2^{k+1}} \right] + \cdots$$

(The symbol $[x]$ denotes the greatest integer not exceeding $x$.)

Prove that there are infinitely many natural numbers $a$ with the following property: the number $z = n^4 + a$ is not prime for any natural number $n$.

Let $a_1, a_2, \cdots, a_n$ be real constants, $x$ a real variable, and

$$f(x) = \cos(a_1 + x) + \frac{1}{2}\cos(a_2 + x) + \frac{1}{4}\cos(a_3 + x) + \cdots + \frac{1}{2^{n-1}}\cos(a_n + x).$$

Given that $f(x_1) = f(x_2) = 0$, prove that $x_2 - x_1 = m\pi$ for some integer $m$.

For each value of $k = 1, 2, 3, 4, 5$, find necessary and sufficient conditions on the number $a > 0$ so that there exists a tetrahedron with $k$ edges of length $a$, and the remaining $6 - k$ edges of length 1.

A semicircular arc $\gamma$ is drawn on $AB$ as diameter. $C$ is a point on $\gamma$ other than $A$ and $B$, and $D$ is the foot of the perpendicular from $C$ to $AB$. We consider three circles, $\gamma_1, \gamma_2, \gamma_3$, all tangent to the line $AB$. Of these, $\gamma_1$ is inscribed in $\triangle ABC$, while $\gamma_2$ and $\gamma_3$ are both tangent to $CD$ and to $\gamma$, one on each side of $CD$. Prove that $\gamma_1, \gamma_2$ and $\gamma_3$ have a second tangent in common.

Given $n > 4$ points in the plane such that no three are collinear. Prove that there are at least $\binom{n-3}{2}$ convex quadrilaterals whose vertices are four of the given points.

Prove that for all real numbers $x_1, x_2, y_1, y_2, z_1, z_2$, with $x_1 > 0$, $x_2 > 0$, $x_1y_1 - z_1^2 > 0$, $x_2y_2 - z_2^2 > 0$, the inequality

$$\frac{8}{(x_1 + x_2)(y_1 + y_2) - (z_1 + z_2)^2} \leq \frac{1}{x_1y_1 - z_1^2} + \frac{1}{x_2y_2 - z_2^2}$$

is satisfied. Give necessary and sufficient conditions for equality.

Let $M$ be a point on the side $AB$ of $\triangle ABC$. Let $r_1, r_2$ and $r$ be the radii of the inscribed circles of triangles $AMC, BMC$ and $ABC$. Let $q_1, q_2$ and $q$ be the radii of the escribed circles of the same triangles that lie in the angle $ACB$. Prove that $$\frac{r_1}{q_1} \cdot \frac{r_2}{q_2} = \frac{r}{q}.$$

Let $a, b$ and $n$ be integers greater than 1, and let $a$ and $b$ be the bases of two number systems. $A_{n-1}$ and $A_n$ are numbers in the system with base $a$, and $B_{n-1}$ and $B_n$ are numbers in the system with base $b$; these are related as follows: $$A_n = x_n x_{n-1} \cdots x_0, \quad A_{n-1} = x_{n-1} x_{n-2} \cdots x_0,$$ $$B_n = x_n x_{n-1} \cdots x_0, \quad B_{n-1} = x_{n-1} x_{n-2} \cdots x_0,$$ $$x_n \neq 0, \quad x_{n-1} \neq 0.$$

Prove: $$\frac{A_{n-1}}{A_n} < \frac{B_{n-1}}{B_n} \text{ if and only if } a > b.$$

The real numbers $a_0, a_1, \ldots, a_n, \ldots$ satisfy the condition: $$1 = a_0 \leq a_1 \leq a_2 \leq \cdots \leq a_n \leq \cdots.$$

The numbers $b_1, b_2, \ldots, b_n, \ldots$ are defined by $$b_n = \sum_{k=1}^{n} \left(1 - \frac{a_{k-1}}{a_k}\right) \frac{1}{\sqrt{a_k}}.$$

(a) Prove that $0 \leq b_n < 2$ for all $n$.

(b) Given $c$ with $0 \leq c < 2$, prove that there exist numbers $a_0, a_1, \ldots$ with the above properties such that $b_n > c$ for large enough $n$.

Find the set of all positive integers $n$ with the property that the set $\{n, n + 1, n + 2, n + 3, n + 4, n + 5\}$ can be partitioned into two sets such that the product of the numbers in one set equals the product of the numbers in the other set.

In the tetrahedron $ABCD$, angle $BDC$ is a right angle. Suppose that the foot $H$ of the perpendicular from $D$ to the plane $ABC$ is the intersection of the altitudes of $\triangle ABC$. Prove that $$(AB + BC + CA)^2 \leq 6(AD^2 + BD^2 + CD^2).$$

For what tetrahedra does equality hold?

In a plane there are 100 points, no three of which are collinear. Consider all possible triangles having these points as vertices. Prove that no more than 70% of these triangles are acute-angled.

Prove that the following assertion is true for $n = 3$ and $n = 5$, and that it is false for every other natural number $n > 2$: If $a_1, a_2, \ldots, a_n$ are arbitrary real numbers, then $$(a_1 - a_2)(a_1 - a_3) \cdots (a_1 - a_n) + (a_2 - a_1)(a_2 - a_3) \cdots (a_2 - a_n)$$ $$+ \cdots + (a_n - a_1)(a_n - a_2) \cdots (a_n - a_{n-1}) \geq 0$$

Consider a convex polyhedron $P_1$ with nine vertices $A_1A_2, \ldots, A_9$; let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves vertex $A_1$ to $A_i$ $(i = 2, 3, \ldots, 9)$. Prove that at least two of the polyhedra $P_1, P_2, \ldots, P_9$ have an interior point in common.

Prove that the set of integers of the form $2^k - 3$ $(k = 2, 3, \ldots)$ contains an infinite subset in which every two members are relatively prime.

All the faces of tetrahedron $ABCD$ are acute-angled triangles. We consider all closed polygonal paths of the form $XYZTX$ defined as follows: $X$ is a point on edge $AB$ distinct from $A$ and $B$; similarly, $Y, Z, T$ are interior points of edges $BC$, $CD$, $DA$, respectively. Prove:

(a) If $\angle DAB + \angle BCD \neq \angle CDA + \angle ABC$, then among the polygonal paths, there is none of minimal length.

(b) If $\angle DAB + \angle BCD = \angle CDA + \angle ABC$, then there are infinitely many shortest polygonal paths, their common length being $2AC\sin(\alpha/2)$, where $\alpha = \angle BAC + \angle CAD + \angle DAB$.

Prove that for every natural number $m$, there exists a finite set $S$ of points in a plane with the following property: For every point $A$ in $S$, there are exactly $m$ points in $S$ which are at unit distance from $A$.

Let $A = (a_{ij})$ $(i, j = 1, 2, \ldots, n)$ be a square matrix whose elements are non-negative integers. Suppose that whenever an element $a_{ij} = 0$, the sum of the elements in the $i$th row and the $j$th column is $\geq n$. Prove that the sum of all the elements of the matrix is $\geq n^2/2$.

Prove that from a set of ten distinct two-digit numbers (in the decimal system), it is possible to select two disjoint subsets whose members have the same sum.

Prove that if $n \geq 4$, every quadrilateral that can be inscribed in a circle can be dissected into $n$ quadrilaterals each of which is inscribable in a circle.

Let $m$ and $n$ be arbitrary non-negative integers. Prove that $$\frac{(2m)!(2n)!}{m!n!(m+n)!}$$ is an integer. $(0! = 1.)$

Find all solutions $(x_1, x_2, x_3, x_4, x_5)$ of the system of inequalities $$(x_1^2 - x_3x_5)(x_2^2 - x_3x_5) \leq 0$$ $$(x_2^2 - x_4x_1)(x_3^2 - x_4x_1) \leq 0$$ $$(x_3^2 - x_5x_2)(x_4^2 - x_5x_2) \leq 0$$ $$(x_4^2 - x_1x_3)(x_5^2 - x_1x_3) \leq 0$$ $$(x_5^2 - x_2x_4)(x_1^2 - x_2x_4) \leq 0$$ where $x_1, x_2, x_3, x_4, x_5$ are positive real numbers.

Let $f$ and $g$ be real-valued functions defined for all real values of $x$ and $y$, and satisfying the equation $$f(x + y) + f(x - y) = 2f(x)g(y)$$ for all $x, y$. Prove that if $f(x)$ is not identically zero, and if $|f(x)| \leq 1$ for all $x$, then $|g(y)| \leq 1$ for all $y$.

Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.

Point $O$ lies on line $g$; $\overrightarrow{OP_1}, \overrightarrow{OP_2}, \ldots, \overrightarrow{OP_n}$ are unit vectors such that points $P_1, P_2, \ldots, P_n$ all lie in a plane containing $g$ and on one side of $g$. Prove that if $n$ is odd, $$\left|\overrightarrow{OP_1} + \overrightarrow{OP_2} + \cdots + \overrightarrow{OP_n}\right| \geq 1$$

Here $\left|\overrightarrow{OM}\right|$ denotes the length of vector $\overrightarrow{OM}$.

Determine whether or not there exists a finite set $M$ of points in space not lying in the same plane such that, for any two points $A$ and $B$ of $M$, one can select two other points $C$ and $D$ of $M$ so that lines $AB$ and $CD$ are parallel and not coincident.

Let $a$ and $b$ be real numbers for which the equation $$x^4 + ax^3 + bx^2 + ax + 1 = 0$$ has at least one real solution. For all such pairs $(a,b)$, find the minimum value of $a^2 + b^2$.

A soldier needs to check on the presence of mines in a region having the shape of an equilateral triangle. The radius of action of his detector is equal to half the altitude of the triangle. The soldier leaves from one vertex of the triangle. What path should he follow in order to travel the least possible distance and still accomplish his mission?

$G$ is a set of non-constant functions of the real variable $x$ of the form $$f(x) = ax + b, \text{ } a \text{ and } b \text{ are real numbers,}$$ and $G$ has the following properties:

(a) If $f$ and $g$ are in $G$, then $g \circ f$ is in $G$; here $(g \circ f)(x) = g[f(x)]$.

(b) If $f$ is in $G$, then its inverse $f^{-1}$ is in $G$; here the inverse of $f(x) = ax + b$ is $f^{-1}(x) = (x - b)/a$.

(c) For every $f$ in $G$, there exists a real number $x_f$ such that $f(x_f) = x_f$.

Prove that there exists a real number $k$ such that $f(k) = k$ for all $f$ in $G$.

Let $a_1, a_2, \ldots, a_n$ be $n$ positive numbers, and let $q$ be a given real number such that $0 < q < 1$. Find $n$ numbers $b_1, b_2, \ldots, b_n$ for which

(a) $a_k < b_k$ for $k = 1, 2, \ldots, n$,

(b) $q < \frac{b_{k+1}}{b_k} < \frac{1}{q}$ for $k = 1, 2, \ldots, n - 1$,

(c) $b_1 + b_2 + \cdots + b_n < \frac{1+q}{1-q}(a_1 + a_2 + \cdots + a_n)$.

Three players $A$, $B$ and $C$ play the following game: On each of three cards an integer is written. These three numbers $p$, $q$, $r$ satisfy $0 < p < q < r$. The three cards are shuffled and one is dealt to each player. Each then receives the number of counters indicated by the card he holds. Then the cards are shuffled again; the counters remain with the players.

This process (shuffling, dealing, giving out counters) takes place for at least two rounds. After the last round, $A$ has 20 counters in all, $B$ has 10 and $C$ has 9. At the last round $B$ received $r$ counters. Who received $q$ counters on the first round?

In the triangle $ABC$, prove that there is a point $D$ on side $AB$ such that $CD$ is the geometric mean of $AD$ and $DB$ if and only if $$\sin A \sin B \leq \sin^2 \frac{C}{2}.$$

Prove that the number $\sum_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k}$ is not divisible by 5 for any integer $n \geq 0$.

Consider decompositions of an $8 \times 8$ chessboard into $p$ non-overlapping rectangles subject to the following conditions:

(i) Each rectangle has as many white squares as black squares.

(ii) If $a_i$ is the number of white squares in the $i$-th rectangle, then $a_1 < a_2 < \cdots < a_p$. Find the maximum value of $p$ for which such a decomposition is possible. For this value of $p$, determine all possible sequences $a_1, a_2, \ldots, a_p$.

Determine all possible values of $$S = \frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d}$$ where $a$, $b$, $c$, $d$ are arbitrary positive numbers.

Let $P$ be a non-constant polynomial with integer coefficients. If $n(P)$ is the number of distinct integers $k$ such that $(P(k))^2 = 1$, prove that $n(P) - \deg(P) \leq 2$, where $\deg(P)$ denotes the degree of the polynomial $P$.