International Competitions

Let $\mathbb{R}^+$ be the set of positive real numbers. Let $f\colon \mathbb{R}^{+}\to \mathbb{R}^{+}$ be a function such that for all $x,y\in \mathbb{R}^{+}$ it holds that

$$y f^{2025}(x) \geq x f(y).$$

Show that there exists a positive integer $n_0$ such that for all positive integers $n \geq n_0$ and for all $x \in \mathbb{R}^+$ it holds that

$$f^n(x) \geq x.$$

Remark. Here $f^n$ denotes the function $f$ applied $n$ times, this means $f^n(x) = \underbrace{f(f(\ldots f(x)\ldots))}_{n \text{ times}}$.

On an infinite square grid, on which some unit squares are coloured red, a ruby rook is a piece which, in one move, can travel any number of squares in one direction parallel to one of the grid lines (either vertically or horizontally), while remaining on red squares at all times throughout the move.

Starting with an uncoloured infinite square grid, Alice performs the following procedure: First, she colours at most 2025 of the unit squares red. Afterwards, she places some ruby rooks on distinct red unit squares, such that the following two rules are satisfied:

• No ruby rook can reach another ruby rook in one move.
• Every ruby rook can reach every other ruby rook in two moves.

Find the maximum possible number of ruby rooks that Alice can place during this procedure.

Prove: The sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space.

Prove that for every natural number $n$, and for every real number $x \neq k\pi / 2^t$ ($t = 0,1,\dots,n; k$ any integer) $$\frac{1}{\sin 2x} + \frac{1}{\sin 4x} + \cdots + \frac{1}{\sin 2^n x} = \cot x - \cot 2^n x.$$

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 $ABC$.

Determine all values $x$ in the interval $0 \leq x \leq 2\pi$ which satisfy the inequality $$2 \cos x \leq \left| \sqrt{1 + \sin 2x} - \sqrt{1 - \sin 2x} \right| \leq \sqrt{2}.$$

Consider the system of equations $$a_{11}x_1 + a_{12}x_2 + a_{13}x_3 = 0$$ $$a_{21}x_1 + a_{22}x_2 + a_{23}x_3 = 0$$ $$a_{31}x_1 + a_{32}x_2 + a_{33}x_3 = 0$$

with unknowns $x_1, x_2, x_3$. The coefficients satisfy the conditions:

(a) $a_{11}, a_{22}, a_{33}$ are positive numbers;

(b) the remaining coefficients are negative numbers;

(c) in each equation, the sum of the coefficients is positive.

Prove that the given system has only the solution $x_1 = x_2 = x_3 = 0$.

Given the tetrahedron $ABCD$ whose edges $AB$ and $CD$ have lengths $a$ and $b$ respectively. The distance between the skew lines $AB$ and $CD$ is $d$, and the angle between them is $\omega$. Tetrahedron $ABCD$ is divided into two solids by plane $\varepsilon$, parallel to lines $AB$ and $CD$. The ratio of the distances of $\varepsilon$ from $AB$ and $CD$ is equal to $k$. Compute the ratio of the volumes of the two solids obtained.

Find all sets of four real numbers $x_1, x_2, x_3, x_4$ such that the sum of any one and the product of the other three is equal to 2.

Consider $\triangle OAB$ with acute angle $AOB$. Through a point $M \neq O$ perpendiculars are drawn to $OA$ and $OB$, the feet of which are $P$ and $Q$ respectively. The point of intersection of the altitudes of $\triangle OPQ$ is $H$. What is the locus of $H$ if $M$ is permitted to range over (a) the side $AB$, (b) the interior of $\triangle OAB$?

In a plane a set of $n$ points ($n \geq 3$) is given. Each pair of points is connected by a segment. Let $d$ be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length $d$. Prove that the number of diameters of the given set is at most $n$.

(a) Find all positive integers $n$ for which $2^n - 1$ is divisible by 7.

(b) Prove that there is no positive integer $n$ for which $2^n + 1$ is divisible by 7.

Suppose $a, b, c$ are the sides of a triangle. Prove that $$a^2(b + c - a) + b^2(c + a - b) + c^2(a + b - c) \leq 3abc.$$

A circle is inscribed in triangle $ABC$ with sides $a, b, c$. Tangents to the circle parallel to the sides of the triangle are constructed. Each of these tangents cuts off a triangle from $\triangle ABC$. In each of these triangles, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of $a, b, c$).

Seventeen people correspond by mail with one another - each one with all the rest. In their letters only three different topics are discussed. Each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.

Suppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maximum number of intersections that these perpendiculars can have.

In tetrahedron $ABCD$, vertex $D$ is connected with $D_0$ the centroid of $\triangle ABC$. Lines parallel to $DD_0$ are drawn through $A, B$ and $C$. These lines intersect the planes $BCD, CAD$ and $ABD$ in points $A_1, B_1$ and $C_1$, respectively. Prove that the volume of $ABCD$ is one third the volume of $A_1B_1C_1D_0$. Is the result true if point $D_0$ is selected anywhere within $\triangle ABC$?

Find all real roots of the equation $$\sqrt{x^2 - p} + 2\sqrt{x^2 - 1} = x,$$ where $p$ is a real parameter.

Point $A$ and segment $BC$ are given. Determine the locus of points in space which are vertices of right angles with one side passing through $A$, and the other side intersecting the segment $BC$.

In an $n$-gon all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation $$a_1 \geq a_2 \geq \cdots \geq a_n.$$ Prove that $a_1 = a_2 = \cdots = a_n$.

Find all solutions $x_1, x_2, x_3, x_4, x_5$ of the system $$\begin{aligned} x_5 + x_2 &= yx_1\\ x_1 + x_3 &= yx_2\\ x_2 + x_4 &= yx_3\\ x_3 + x_5 &= yx_4\\ x_4 + x_1 &= yx_5, \end{aligned}$$ where $y$ is a parameter.

Prove that $\cos\frac{\pi}{7} - \cos\frac{2\pi}{7} + \cos\frac{3\pi}{7} = \frac{1}{2}$.

Five students, $A, B, C, D, E$, took part in a contest. One prediction was that the contestants would finish in the order $ABCDE$. This prediction was very poor. In fact no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had the contestants finishing in the order $DAECB$. This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.

Find the smallest natural number $n$ which has the following properties:

(a) Its decimal representation has 6 as the last digit.

(b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number $n$.

Determine all real numbers $x$ which satisfy the inequality: $$\sqrt{3-x}-\sqrt{x+1}>\frac{1}{2}.$$

Consider the cube $ABCDA'B'C'D'$ ($ABCD$ and $A'B'C'D'$ are the upper and lower bases, respectively, and edges $AA',BB',CC',DD'$ are parallel). The point $X$ moves at constant speed along the perimeter of the square $ABCD$ in the direction $ABCDA$, and the point $Y$ moves at the same rate along the perimeter of the square $B'C'CB$ in the direction $B'C'CBB'$. Points $X$ and $Y$ begin their motion at the same instant from the starting positions $A$ and $B'$, respectively. Determine and draw the locus of the midpoints of the segments $XY$.

Solve the equation $\cos^2 x + \cos^2 2x + \cos^2 3x = 1$.

On the circle $K$ there are given three distinct points $A,B,C.$ Construct (using only straightedge and compasses) a fourth point $D$ on $K$ such that a circle can be inscribed in the quadrilateral thus obtained.

Consider an isosceles triangle. Let $r$ be the radius of its circumscribed circle and $\rho$ the radius of its inscribed circle. Prove that the distance $d$ between the centers of these two circles is $$d = \sqrt{r(r-2\rho)}.$$

The tetrahedron $SABC$ has the following property: there exist five spheres, each tangent to the edges $SA,SB,SC,BC,CA,AB$, or to their extensions.

(a) Prove that the tetrahedron $SABC$ is regular.

(b) Prove conversely that for every regular tetrahedron five such spheres exist.

Solve the system of equations: $$\begin{aligned} x + y + z &= a \\ x^2 + y^2 + z^2 &= b^2 \\ xy &= z^2 \end{aligned}$$ where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisfy so that $x, y, z$ (the solutions of the system) are distinct positive numbers.

Let $a, b, c$ be the sides of a triangle, and $T$ its area. Prove: $a^2 + b^2 + c^2 \geq 4\sqrt{3}T$. In what case does equality hold?

Solve the equation $\cos^n x - \sin^n x = 1$, where $n$ is a natural number.

Consider triangle $P_1P_2P_3$ and a point $P$ within the triangle. Lines $P_1P, P_2P, P_3P$ intersect the opposite sides in points $Q_1, Q_2, Q_3$ respectively. Prove that, of the numbers $$\frac{P_1P}{PQ_1}, \frac{P_2P}{PQ_2}, \frac{P_3P}{PQ_3}$$ at least one is $\leq 2$ and at least one is $\geq 2$.

Construct triangle $ABC$ if $AC = b$, $AB = c$ and $\measuredangle AMB = \omega$, where $M$ is the midpoint of segment $BC$ and $\omega < 90°$. Prove that a solution exists if and only if $$b \tan \frac{\omega}{2} \leq c < b.$$ In what case does the equality hold?

Consider a plane $\varepsilon$ and three non-collinear points $A, B, C$ on the same side of $\varepsilon$; suppose the plane determined by these three points is not parallel to $\varepsilon$. In plane $\varepsilon$ take three arbitrary points $A', B', C'$. Let $L, M, N$ be the midpoints of segments $AA', BB', CC'$; let $G$ be the centroid of triangle $LMN$. (We will not consider positions of the points $A', B', C'$ such that the points $L, M, N$ do not form a triangle.) What is the locus of point $G$ as $A', B', C'$ range independently over the plane $\varepsilon$?

Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $N/11$ is equal to the sum of the squares of the digits of $N$.

For what values of the variable $x$ does the following inequality hold: $$\frac{4x^{2}}{(1-\sqrt{1+2x})^{2}}<2x+9?$$

In a given right triangle $ABC$, the hypotenuse $BC$, of length $a$, is divided into $n$ equal parts ($n$ an odd integer). Let $\alpha$ be the acute angle subtending, from $A$, that segment which contains the midpoint of the hypotenuse. Let $h$ be the length of the altitude to the hypotenuse of the triangle. Prove: $$\tan\alpha=\frac{4nh}{(n^{2}-1)a}.$$

Construct triangle $ABC$, given $h_{a}, h_{b}$ (the altitudes from $A$ and $B$) and $m_{a}$, the median from vertex $A$.

Consider the cube $ABCDA^{\prime}B^{\prime}C^{\prime}D^{\prime}$ (with face $ABCD$ directly above face $A^{\prime}B^{\prime}C^{\prime}D^{\prime}$).

(a) Find the locus of the midpoints of segments $XY$, where $X$ is any point of $AC$ and $Y$ is any point of $B^{\prime}D^{\prime}$.

(b) Find the locus of points $Z$ which lie on the segments $XY$ of part (a) with $ZY=2XZ$.

Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. Let $V_{1}$ be the volume of the cone and $V_{2}$ the volume of the cylinder.

(a) Prove that $V_{1}\neq V_{2}$.

(b) Find the smallest number $k$ for which $V_{1}=kV_{2}$, for this case, construct the angle subtended by a diameter of the base of the cone at the vertex of the cone.

An isosceles trapezoid with bases $a$ and $c$ and altitude $h$ is given.

(a) On the axis of symmetry of this trapezoid, find all points $P$ such that both legs of the trapezoid subtend right angles at $P$.

(b) Calculate the distance of $P$ from either base.

(c) Determine under what conditions such points $P$ actually exist. (Discuss various cases that might arise.)

Prove that the fraction $\frac{21n+4}{14n+3}$ is irreducible for every natural number $n$.

For what real values of $x$ is

$$\sqrt{(x+\sqrt{2x-1})}+\sqrt{(x-\sqrt{2x-1})}=A,$$

given (a) $A=\sqrt{2}$, (b) $A=1$, (c) $A=2$, where only non-negative real numbers are admitted for square roots?

Let $a,b,c$ be real numbers. Consider the quadratic equation in $\cos x$:

$$a\cos^{2}x+b\cos x+c=0.$$

Using the numbers $a,b,c,$ form a quadratic equation in $\cos 2x$, whose roots are the same as those of the original equation. Compare the equations in $\cos x$ and $\cos 2x$ for $a=4,b=2,c=-1$.

Construct a right triangle with given hypotenuse $c$ such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle.

An arbitrary point $M$ is selected in the interior of the segment $AB$. The squares $AMCD$ and $MBEF$ are constructed on the same side of $AB$, with the segments $AM$ and $MB$ as their respective bases. The circles circumscribed about these squares, with centers $P$ and $Q$, intersect at $M$ and also at another point $N$. Let $N'$ denote the point of intersection of the straight lines $AF$ and $BC$.

(a) Prove that the points $N$ and $N'$ coincide.

(b) Prove that the straight lines $MN$ pass through a fixed point $S$ independent of the choice of $M$.

(c) Find the locus of the midpoints of the segments $PQ$ as $M$ varies between $A$ and $B$.

Two planes, $P$ and $Q$, intersect along the line $p$. The point $A$ is given in the plane $P$, and the point $C$ in the plane $Q$; neither of these points lies on the straight line $p$. Construct an isosceles trapezoid $ABCD$ (with $AB$ parallel to $CD$) in which a circle can be inscribed, and with vertices $B$ and $D$ lying in the planes $P$ and $Q$ respectively.