ISL 1979

Prove that in the Euclidean plane every regular polygon having an even number of sides can be dissected into lozenges. (A lozenge is a quadrilateral whose four sides are all of equal length).

Prove that a pyramid $A_1A_2 \ldots A_{2k+1}S$ with equal lateral edges and equal space angles between adjacent lateral walls is regular.

The plane is divided into equal squares by parallel lines; i.e., a square net is given. Let $M$ be an arbitrary set of $n$ squares of this net. Prove that it is possible to choose no fewer than $n/4$ squares of $M$ in such a way that no two of them have a common point.

Let $S$ be a set of $n^2 + 1$ closed intervals ( $n$ a positive integer). Prove that at least one of the following assertions holds:(i) There exists a subset $S'$ of $n+1$ intervals from $S$ such that the intersection of the intervals in $S'$ is nonempty.(ii) There exists a subset $S''$ of $n + 1$ intervals from $S$ such that any two of the intervals in $S''$ are disjoint.

Let $Q$ be a square with side length $6$ . Find the smallest integer $n$ such that in $Q$ there exists a set $S$ of $n$ points with the property that any square with side $1$ completely contained in $Q$ contains in its interior at least one point from $S$ .

Show that for no integers $a \geq 1, n \geq 1$ is the sum $$1+\frac{1}{1+a}+\frac{1}{1+2a}+\cdots+\frac{1}{1+na}$$ an integer.

For $k = 1, 2, \ldots$ consider the $k$ -tuples $(a_1, a_2, \ldots, a_k)$ of positive integers such that $$a_1 + 2a_2 + \cdots + ka_k = 1979.$$ Show that there are as many such $k$ -tuples with odd $k$ as there are with even $k$ .

Let $E$ be the set of all bijective mappings from $\mathbb R$ to $\mathbb R$ satisfying $$f(t) + f^{-1}(t) = 2t, \forall t \in \mathbb R,$$ where $f^{-1}$ is the mapping inverse to $f$ . Find all elements of $E$ that are monotonic mappings.

Consider the set $E$ consisting of pairs of integers $(a, b)$ , with $a \geq 1$ and $b \geq 1$ , that satisfy in the decimal system the following properties:(i) $b$ is written with three digits, as $\overline{\alpha_2\alpha_1\alpha_0}$ , $\alpha_2 \neq 0$ ;(ii) $a$ is written as $\overline{\beta_p \ldots \beta_1\beta_0}$ for some $p$ ;(iii) $(a + b)^2$ is written as $\overline{\beta_p\ldots \beta_1 \beta_0 \alpha_2 \alpha_1 \alpha_0}.$ Find the elements of $E$ .

Let $a$ and $b$ be coprime integers, greater than or equal to $1$ . Prove that all integers $n$ greater than or equal to $(a - 1)(b - 1)$ can be written in the form: $$n = ua + vb, \text{with} (u, v) \in \mathbb N \times \mathbb N.$$

Let $M$ be a set of points in a plane with at least two elements. Prove that if $M$ has two axes of symmetry $g_1$ and $g_2$ intersecting at an angle $\alpha = q\pi$ , where $q$ is irrational, then $M$ must be infinite.

Let $n, k \ge 1$ be natural numbers. Find the number $A(n, k)$ of solutions in integers of the equation $$|x_1| + |x_2| +\cdots + |x_k| = n$$

Notice that in the fraction $\frac{16}{64}$ we can perform a simplification as $\cancel{\frac{16}{64}}=\frac 14$ obtaining a correct equality. Find all fractions whose numerators and denominators are two-digit positive integers for which such a simplification is correct.

Given a sequence $(a_n)$ , with $a_1 = 4$ and $a_{n+1} = a_n^2-2 (\forall n \in\mathbb{N})$ , prove that there is a triangle with side lengths $a_{n-1}, a_n, a_{n+1},$ and that its area is equal to an integer.

A regular tetrahedron $A_1B_1C_1D_1$ is inscribed in a regular tetrahedron $ABCD$ , where $A_1$ lies in the plane $BCD$ , $B_1$ in the plane $ACD$ , etc. Prove that $A_1B_1 \ge\frac{ AB}{3}$ .

Prove the following statement: If a polynomial $f(x)$ with real coefficients takes only nonnegative values, then there exists a positive integer $n$ and polynomials $g_1(x), g_2(x),\cdots, g_n(x)$ such that $$f(x) = g_1(x)^2 + g_2(x)^2 +\cdots+ g_n(x)^2$$

A polynomial $P(x)$ has degree at most $2k$ , where $k = 0, 1,2,\cdots$ . Given that for an integer $i$ , the inequality $-k \le i \le k$ implies $|P(i)| \le 1$ , prove that for all real numbers $x$ , with $-k \le x \le k$ , the following inequality holds: $$|P(x)| < (2k + 1)\dbinom{2k}{k}$$

Prove the following statement: There does not exist a pyramid with square base and congruent lateral faces for which the measures of all edges, total area, and volume are integers.

Let a quadratic polynomial $g(x) = ax^2 + bx + c$ be given and an integer $n \ge 1$ . Prove that there exists at most one polynomial $f(x)$ of $n$ th degree such that $f(g(x)) = g(f(x)).$

Let $a, b, c$ denote the lengths of the sides $BC,CA,AB$ , respectively, of a triangle $ABC$ . If $P$ is any point on the circumference of the circle inscribed in the triangle, show that $aPA^2+bPB^2+cPC^2$ is constant.

For any positive integer $n$ , we denote by $F(n)$ the number of ways in which $n$ can be expressed as the sum of three different positive integers, without regard to order. Thus, since $10 = 7+2+1 = 6+3+1 = 5+4+1 = 5+3+2$ , we have $F(10) = 4$ . Show that $F(n)$ is even if $n \equiv 2$ or $4 \pmod 6$ , but odd if $n$ is divisible by $6$ .

In the plane a circle $C$ of unit radius is given. For any line $l$ , a number $s(l)$ is defined in the following way: If $l$ and $C$ intersect in two points, $s(l)$ is their distance; otherwise, $s(l) = 0$ . Let $P$ be a point at distance $r$ from the center of $C$ . One defines $M(r)$ to be the maximum value of the sum $s(m) + s(n)$ , where $m$ and $n$ are variable mutually orthogonal lines through $P$ . Determine the values of $r$ for which $M(r) > 2$ .

Let there be given two sequences of integers $f_i(1), f_i(2), \cdots (i = 1, 2)$ satisfying: $(i) f_i(nm) = f_i(n)f_i(m)$ if $\gcd(n,m) = 1$ ; $(ii)$ for every prime $P$ and all $k = 2, 3, 4, \cdots$ , $f_i(P^k) = f_i(P)f_i(P^{k-1}) - P^2f(P^{k-2}).$ Moreover, for every prime $P$ : $(iii) f_1(P) = 2P,$ $(iv) f_2(P) < 2P.$ Prove that $|f_2(n)| < f_1(n)$ for all $n$ .

Let $ABC$ be an arbitrary triangle and let $S_1, S_2,\cdots, S_7$ be circles satisfying the following conditions: $S_1$ is tangent to $CA$ and $AB$ , $S_2$ is tangent to $S_1, AB$ , and $BC,$ $S_3$ is tangent to $S_2, BC$ , and $CA,$ .............................. $S_7$ is tangent to $S_6, CA$ and $AB.$ Prove that the circles $S_1$ and $S_7$ coincide.

Let a real number $\lambda > 1$ be given and a sequence $(n_k)$ of positive integers such that $\frac{n_{k+1}}{n_k}> \lambda$ for $k = 1, 2,\ldots$ Prove that there exists a positive integer $c$ such that no positive integer $n$ can be represented in more than $c$ ways in the form $n = n_k + n_j$ or $n = n_r - n_s$ .

An infinite increasing sequence of positive integers $n_j (j = 1, 2, \ldots )$ has the property that for a certain $c$ , $$\frac{1}{N}\sum_{n_j\le N} n_j \le c,$$ for every $N >0$ . Prove that there exist finitely many sequences $m^{(i)}_j (i = 1, 2,\ldots, k)$ such that $$\{n_1, n_2, \ldots \} =\bigcup_{i=1}^k\{m^{(i)}_1 ,m^{(i)}_2 ,\ldots\}$$ and $$m^{(i)}_{j+1} > 2m^{(i)}_j (1 \le i \le k, j = 1, 2,\ldots).$$

Let $a,b$ be coprime integers. Show that the equation $ax^2 + by^2 =z^3$ has an infinite set of solutions $(x,y,z)$ with $\{x,y,z\}\in\mathbb{Z}$ and each pair of $x,y$ mutually coprime.

Show that for every $n\in\mathbb{N}$ , $n\sqrt{2}-\lfloor n\sqrt{2}\rfloor>\frac{1}{2n \sqrt{2}}$ and that for every $\epsilon >0$ , there exists an $n\in\mathbb{N}$ such that $n\sqrt{2}-\lfloor n\sqrt{2}\rfloor < \frac{1}{2n \sqrt{2}}+\epsilon$ .

Let $M$ be a set and $A,B,C$ given subsets of $M$ . Find a necessary and sufficient condition for the existence of a set $X\subset M$ for which $(X\cup A)\backslash(X\cap B)=C$ . Describe all such sets.

Prove that there exists a $k_0\in\mathbb{N}$ such that for every $k\in\mathbb{N},k>k_0$ , there exists a finite number of lines in the plane not all parallel to one of them, that divide the plane exactly in $k$ regions. Find $k_0$ .

Determine the maximum value of $x^2 y^2 z^2 w$ for $\{x,y,z,w\}\in\mathbb{R}^{+} \cup\{0\}$ and $2x+xy+z+yzw=1$ .

There are two non-decreasing sequences $\{a_i\}$ and $\{b_i\}$ of $n$ real numbers each, such that $a_i\le a_{i+1}$ for each $1\le i\le n-1$ , and $b_i\le b_{i+1}$ for each $1\le i\le n-1$ , and $\sum_{k=1}^{m}{a_k}\ge \sum_{k=1}^{m}{b_k}$ where $m\le n$ with equality for $m=n$ . For a convex function $f$ defined on the real numbers, prove that $\sum_{k=1}^{n}{f(a_k)}\le \sum_{k=1}^{n}{f(b_k)}$ .

$T$ is a given triangle with vertices $P_1,P_2,P_3$ . Consider an arbitrary subdivision of $T$ into finitely many subtriangles such that no vertex of a subtriangle lies strictly between two vertices of another subtriangle. To each vertex $V$ of the subtriangles there is assigned a number $n(V)$ according to the following rules: $(\text{i})$ If $V$ = $P_i$ , then $n(V) = i$ . $(\text{ii})$ If $V$ lies on the side $P_i P_j$ of $T$ , then $n(V) = i$ or $j$ . $(\text{iii})$ If $V$ lies inside the triangle $T$ , then $n(V)$ is any of the numbers $1,2,3$ . Prove that there exists at least one subtriangle whose vertices are numbered $1, 2, 3$ .

Let the sequence $\{a_i\}$ of $n$ positive reals denote the lengths of the sides of an arbitrary $n$ -gon. Let $s=\sum_{i=1}^{n}{a_i}$ . Prove that $2\ge \sum_{i=1}^{n}{\frac{a_i}{s-a_i}}\ge \frac{n}{n-1}$ .

From point $P$ on arc $BC$ of the circumcircle about triangle $ABC$ , $PX$ is constructed perpendicular to $BC$ , $PY$ is perpendicular to $AC$ , and $PZ$ perpendicular to $AB$ (all extended if necessary). Prove that $\frac{BC}{PX}=\frac{AC}{PY}+\frac{AB}{PZ}$ .

Given a function $f$ such that $f(x)\le x\forall x\in\mathbb{R}$ and $f(x+y)\le f(x)+f(y)\forall \{x,y\}\in\mathbb{R}$ , prove that $f(x)=x\forall x\in\mathbb{R}$ .

There are $1979$ equilateral triangles: $T_1,T_2, . . . ,T_{1979}$ . A side of triangle $T_k$ is equal to $\frac{1}{k}$ , $k = 1,2, . . . ,1979$ . At what values of a number $a$ can one place all these triangles into the equilateral triangle with side length $a$ so that they don't intersect (points of contact are allowed)?

Let $f (x)$ be a polynomial with integer coefficients. Prove that if $f (x)= 1979$ for four different integer values of $x$ , then $f (x)$ cannot be equal to $2\times 1979$ for any integral value of $x$ .

In a plane a finite number of equal circles are given. These circles are mutually nonintersecting (they may be externally tangent). Prove that one can use at most four colors for coloring these circles so that two circles tangent to each other are of different colors. What is the smallest number of circles that requires four colors?

Given an equilateral triangle $ABC$ of side $a$ in a plane, let $M$ be a point on the circumcircle of the triangle. Prove that the sum $s = MA^4 +MB^4 +MC^4$ is independent of the position of the point $M$ on the circle, and determine that constant value as a function of $a$ .

Given an equilateral triangle $ABC$ , let $M$ be an arbitrary point in space. $(\text{a})$ Prove that one can construct a triangle from the segments $MA, MB, MC$ . $(\text{b})$ Suppose that $P$ and $Q$ are two points symmetric with respect to the center $O$ of $ABC$ . Prove that the two triangles constructed from the segments $PA,PB,PC$ and $QA,QB,QC$ are of equal area.

Suppose that a triangle whose sides are of integer lengths is inscribed in a circle of diameter $6.25$ . Find the sides of the triangle.

By $h(n)$ , where $n$ is an integer greater than $1$ , let us denote the greatest prime divisor of the number $n$ . Are there infinitely many numbers $n$ for which $h(n) < h(n+1)< h(n+2)$ holds?

Denote the number of different prime divisors of the number $n$ by $\omega (n)$ , where $n$ is an integer greater than $1$ . Prove that there exist infinitely many numbers $n$ for which $\omega (n)< \omega (n+1)<\omega (n+2)$ holds.

Let $S$ be a unit circle and $K$ a subset of $S$ consisting of several closed arcs. Let $K$ satisfy the following properties: $(\text{i})$ $K$ contains three points $A,B,C$ , that are the vertices of an acute-angled triangle $(\text{ii})$ For every point $A$ that belongs to $K$ its diametrically opposite point $A'$ and all points $B$ on an arc of length $\frac{1}{9}$ with center $A'$ do not belong to $K$ . Prove that there are three points $E,F,G$ on $S$ that are vertices of an equilateral triangle and that do not belong to $K$ .

Let $\Pi$ be the set of rectangular parallelepipeds that have at least one edge of integer length. If a rectangular parallelepiped $P_0$ can be decomposed into parallelepipeds $P_1,P_2, . . . ,P_N\in \Pi$ , prove that $P_0\in \Pi$ .