ISL 1970

There is an equation $\sum_{i=1}^{n}{\frac{b_i}{x-a_i}}=c$ in $x$ , where all $b_i >0$ and $\{a_i\}$ is a strictly increasing sequence. Prove that it has $n-1$ roots such that $x_{n-1}\le a_n$ , and $a_i \le x_i$ for each $i\in\mathbb{N}, 1\le i\le n-1$ .

Let $ABCD$ be an arbitrary quadrilateral. Squares with centers $M_1, M_2, M_3, M_4$ are constructed on $AB,BC,CD,DA$ respectively, all outwards or all inwards. Prove that $M_1 M_3=M_2 M_4$ and $M_1 M_3\perp M_2 M_4$ .

Let $ABCD$ and $A'B'C'D'$ be two arbitrary squares in the plane that are oriented in the same direction. Prove that the quadrilateral formed by the midpoints of $AA',BB',CC',DD'$ is a square.

Each side of an arbitrary $\triangle ABC$ is divided into equal parts, and lines parallel to $AB,BC,CA$ are drawn through each of these points, thus cutting $\triangle ABC$ into small triangles. Points are assigned a number in the following manner: $(1)$ $A,B,C$ are assigned $1,2,3$ respectively $(2)$ Points on $AB$ are assigned $1$ or $2$ $(3)$ Points on $BC$ are assigned $2$ or $3$ $(4)$ Points on $CA$ are assigned $3$ or $1$ Prove that there must exist a small triangle whose vertices are marked by $1,2,3$ .

Let $1<n\in\mathbb{N}$ and $1\le a\in\mathbb{R}$ and there are $n$ number of $x_i, i\in\mathbb{N}, 1\le i\le n$ such that $x_1=1$ and $\frac{x_{i}}{x_{i-1}}=a+\alpha _ i$ for $2\le i\le n$ , where $\alpha _i\le \frac{1}{i(i+1)}$ . Prove that $\sqrt[n-1]{x_n}< a+\frac{1}{n-1}$ .

Let $E$ be a finite set, $P_E$ the family of its subsets, and $f$ a mapping from $P_E$ to the set of non-negative reals, such that for any two disjoint subsets $A,B$ of $E$ , $f(A\cup B)=f(A)+f(B)$ . Prove that there exists a subset $F$ of $E$ such that if with each $A \subset E$ , we associate a subset $A'$ consisting of elements of $A$ that are not in $F$ , then $f(A)=f(A')$ and $f(A)$ is zero if and only if $A$ is a subset of $F$ .

Let $\{n,p\}\in\mathbb{N}\cup \{0\}$ such that $2p\le n$ . Prove that $\frac{(n-p)!}{p!}\le \left(\frac{n+1}{2}\right)^{n-2p}$ . Determine all conditions under which equality holds.

A real function $f$ is defined for $0\le x\le 1$ , with its first derivative $f'$ defined for $0\le x\le 1$ and its second derivative $f''$ defined for $0<x<1$ . Prove that if $f(0)=f'(0)=f'(1)=f(1)-1 =0$ , then there exists a number $0<y<1$ such that $|f''(y)|\ge 4$ .

Consider a finite set of vectors in space $\{a_1, a_2, ... , a_n\}$ and the set $E$ of all vectors of the form $x=\sum_{i=1}^{n}{\lambda _i a_i}$ , where $\lambda _i \in \mathbb{R}^{+}\cup \{0\}$ . Let $F$ be the set consisting of all the vectors in $E$ and vectors parallel to a given plane $P$ . Prove that there exists a set of vectors $\{b_1, b_2, ... , b_p\}$ such that $F$ is the set of all vectors $y$ of the form $y=\sum_{i=1}^{p}{\mu _i b_i}$ , where $\mu _i \in \mathbb{R}^{+}\cup \{0\}$ .

Find a $n\in\mathbb{N}$ such that for all primes $p$ , $n$ is divisible by $p$ if and only if $n$ is divisible by $p-1$ .

A set $G$ with elements $u,v,w...$ is a Group if the following conditions are fulfilled: $(\text{i})$ There is a binary operation $\circ$ defined on $G$ such that $\forall \{u,v\}\in G$ there is a $w\in G$ with $u\circ v = w$ . $(\text{ii})$ This operation is associative; i.e. $(u\circ v)\circ w = u\circ (v\circ w)$ $\forall\{u,v,w\}\in G$ . $(\text{iii})$ $\forall \{u,v\}\in G$ , there exists an element $x\in G$ such that $u\circ x = v$ , and an element $y\in G$ such that $y\circ u = v$ .

Let $K$ be a set of all real numbers greater than $1$ . On $K$ is defined an operation by $a\circ b = ab-\sqrt{(a^2-1)(b^2-1)}$ . Prove that $K$ is a Group.

Prove that the equation $4^x +6^x =9^x$ has no rational solutions.

Let there be given an acute angle $\angle AOB = 3\alpha$ , where $\overline{OA}= \overline{OB}$ . The point $A$ is the center of a circle with radius $\overline{OA}$ . A line $s$ parallel to $OA$ passes through $B$ . Inside the given angle a variable line $t$ is drawn through $O$ . It meets the circle in $O$ and $C$ and the given line $s$ in $D$ , where $\angle AOC = x$ . Starting from an arbitrarily chosen position $t_0$ of $t$ , the series $t_0, t_1, t_2, \ldots$ is determined by defining $\overline{BD_{i+1}}=\overline{OC_i}$ for each $i$ (in which $C_i$ and $D_i$ denote the positions of $C$ and $D$ , corresponding to $t_i$ ). Making use of the graphical representations of $BD$ and $OC$ as functions of $x$ , determine the behavior of $t_i$ for $i\to \infty$ .

The vertices of a given square are clockwise lettered $A,B,C,D$ . On the side $AB$ is situated a point $E$ such that $AE = AB/3$ . Starting from an arbitrarily chosen point $P_0$ on segment $AE$ and going clockwise around the perimeter of the square, a series of points $P_0, P_1, P_2, \ldots$ is marked on the perimeter such that $P_iP_{i+1} = AB/3$ for each $i$ . It will be clear that when $P_0$ is chosen in $A$ or in $E$ , then some $P_i$ will coincide with $P_0$ . Does this possibly also happen if $P_0$ is chosen otherwise?

Find for every value of $n$ a set of numbers $p$ for which the following statement is true: Any convex $n$ -gon can be divided into $p$ isosceles triangles.

Let $x, y, z$ be non-negative real numbers satisfying $$x^2 + y^2 + z^2 = 5 \text{ and } yz + zx + xy = 2.$$ Which values can the greatest of the numbers $x^2 -yz, y^2 - xz$ and $z^2 - xy$ have?

Solve the set of simultaneous equations \begin{align*} v^2+ w^2+ x^2+ y^2 &= 6 - 2u, u^2+ w^2+ x^2+ y^2 &= 6 - 2v, u^2+ v^2+ x^2+ y^2 &= 6- 2w, u^2+ v^2+ w^2+ y^2 &= 6 - 2x, u^2+ v^2+ w^2+ x^2 &= 6- 2y. \end{align*}

Find the greatest integer $A$ for which in any permutation of the numbers $1, 2, \ldots , 100$ there exist ten consecutive numbers whose sum is at least $A$ .

Let ABC be a triangle with angles $\alpha, \beta, \gamma$ commensurable with $\pi$ . Starting from a point $P$ interior to the triangle, a ball reflects on the sides of $ABC$ , respecting the law of reflection that the angle of incidence is equal to the angle of reflection. Prove that, supposing that the ball never reaches any of the vertices $A,B,C$ , the set of all directions in which the ball will move through time is finite. In other words, its path from the moment $0$ to infinity consists of segments parallel to a finite set of lines.

Let a cube of side $1$ be given. Prove that there exists a point $A$ on the surface $S$ of the cube such that every point of $S$ can be joined to $A$ by a path on $S$ of length not exceeding $2$ . Also prove that there is a point of $S$ that cannot be joined with $A$ by a path on $S$ of length less than $2$ .

Prove that the equation $$x^3 - 3 \tan\frac{\pi}{12} x^2 - 3x + \tan\frac{\pi}{12}= 0$$ has one root $x_1 = \tan \frac{\pi}{36}$ , and find the other roots.

If $a, b, c$ are side lengths of a triangle, prove that $$(a + b)(b + c)(c + a) \geq 8(a + b - c)(b + c - a)(c + a - b).$$

Let $M$ be an interior point of tetrahedron $V ABC$ . Denote by $A_1,B_1, C_1$ the points of intersection of lines $MA,MB,MC$ with the planes $VBC,V CA,V AB$ , and by $A_2,B_2, C_2$ the points of intersection of lines $V A_1, VB_1, V C_1$ with the sides $BC,CA,AB$ .(a) Prove that the volume of the tetrahedron $V A_2B_2C_2$ does not exceed one-fourth of the volume of $V ABC$ .(b) Calculate the volume of the tetrahedron $V_1A_1B_1C_1$ as a function of the volume of $V ABC$ , where $V_1$ is the point of intersection of the line $VM$ with the plane $ABC$ , and $M$ is the barycenter of $V ABC$ .

Given a triangle $ABC$ and a plane $\pi$ having no common points with the triangle, find a point $M$ such that the triangle determined by the points of intersection of the lines $MA,MB,MC$ with $\pi$ is congruent to the triangle $ABC$ .

Given a polynomial $$P(x) = ab(a - c)x^3 + (a^3 - a^2c + 2ab^2 - b^2c + abc)x^2 +(2a^2b + b^2c + a^2c + b^3 - abc)x + ab(b + c),$$ where $a, b, c \neq 0$ , prove that $P(x)$ is divisible by $$Q(x) = abx^2 + (a^2 + b^2)x + ab$$ and conclude that $P(x_0)$ is divisible by $(a + b)^3$ for $x_0 = (a + b + 1)^n, n \in \mathbb N$ .

Let a polynomial $p(x)$ with integer coefficients take the value $5$ for five different integer values of $x.$ Prove that $p(x)$ does not take the value $8$ for any integer $x.$

For $n \in \mathbb N$ , let $f(n)$ be the number of positive integers $k \leq n$ that do not contain the digit $9$ . Does there exist a positive real number $p$ such that $\frac{f(n)}{n} \geq p$ for all positive integers $n$ ?

A square $ABCD$ is divided into $(n - 1)^2$ congruent squares, with sides parallel to the sides of the given square. Consider the grid of all $n^2$ corners obtained in this manner. Determine all integers $n$ for which it is possible to construct a non-degenerate parabola with its axis parallel to one side of the square and that passes through exactly $n$ points of the grid.

Let $P,Q,R$ be polynomials and let $S(x) = P(x^3) + xQ(x^3) + x^2R(x^3)$ be a polynomial of degree $n$ whose roots $x_1,\ldots, x_n$ are distinct. Construct with the aid of the polynomials $P,Q,R$ a polynomial $T$ of degree $n$ that has the roots $x_1^3 , x_2^3 , \ldots, x_n^3.$

A turtle runs away from an UFO with a speed of $0.2 \ m/s$ . The UFO flies $5$ meters above the ground, with a speed of $20 \ m/s$ . The UFO's path is a broken line, where after flying in a straight path of length $\ell$ (in meters) it may turn through for any acute angle $\alpha$ such that $\tan \alpha < \frac{\ell}{1000}$ . When the UFO's center approaches within $13$ meters of the turtle, it catches the turtle. Prove that for any initial position the UFO can catch the turtle.

Let the numbers $1, 2, \ldots , n^2$ be written in the cells of an $n \times n$ square board so that the entries in each column are arranged increasingly. What are the smallest and greatest possible sums of the numbers in the $k^{th}$ row? ( $k$ a positive integer, $1 \leq k \leq n$ .)