Denote by $d(n)$ the number of divisors of the positive integer $n$. A positive integer $n$ is called highly divisible if $d(n) > d(m)$ for all positive integers $m < n$. Two highly divisible integers $m$ and $n$ with $m < n$ are called consecutive if there exists no highly divisible integer $s$ satisfying $m < s < n$. (a) Show that there are only finitely many pairs of consecutive highly divisible integers of the form $(a, b)$ with $a\mid b$. (b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.

A rhombus has vertices at $(0,0)$, $(6, 8)$, $(16, 8)$, and $(10, 0)$. A line with slope $m$ passes through the point $(3, 1)$ and splits the rhombus into $2$ regions of equal area. Find $m$.

Find all natural numbers $ n\ge 4 $ that satisfy the property that the affixes of any nonzero pairwise distinct complex numbers $ a,b,c $ that verify the equation $$ (a-b)^n+(b-c)^n+(c-a)^n=0, $$ represent the vertices of an equilateral triangle in the complex plane.

Two circles $k_1$ and $k_2$ with radii $r_1$ and $r_2$ have no common points. The line$ AB$ is a common internal tangent, and the line $CD$ is a common external tangent to these circles, where $A, C \in k_1$ and $B, D \in k_2$. Knowing that $AB=12$ and $CD =16$, find the value of the product $r_1r_2$.

Let $p{}$ be a fixed prime number. Determine the number of ordered $k$-tuples $(a_1,\ldots,a_k)$ of non-negative integers smaller than $p{}$ for which $p\mid a_1^2+\cdots+a_k^2$ where a) $k=3$ and b) $k$ is an arbitrary odd positive integer.

For a given natural number $n > 3$, the real numbers $x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2}$ satisfy the conditions $0 < x_1 < x_2 < \cdots < x_n < x_{n + 1} < x_{n + 2}$. Find the minimum possible value of \[\frac{(\sum _{i=1}^n \frac{x_{i + 1}}{x_i})(\sum _{j=1}^n \frac{x_{j + 2}}{x_{j + 1}})}{(\sum _{k=1}^n \frac{x_{k + 1} x_{k + 2}}{x_{k + 1}^2 + x_k x_{k + 2}})(\sum _{l=1}^n \frac{x_{l + 1}^2 + x_l x_{l + 2}}{x_l x_{l + 1}})}\] and find all $(n + 2)$-tuplets of real numbers $(x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2})$ which gives this value.

Prove that in any acute triangle with sides $a, b, c$ circumscribed in a circle of radius $R$ the following inequality holds: $$\frac{\sqrt2}{4} <\frac{Rp}{2aR + bc} <\frac{1}{2}$$ where $p$ represents the semi-perimeter of the triangle.

Bisectrices $AA_1$ and $BB_1$ of triangle $ABC$ meet in $I$. Segments $A_1I$ and $B_1I$ are the bases of isosceles triangles with opposite vertices $A_2$ and $B_2$ lying on line $AB$. It is known that line $CI$ bisects segment $A_2B_2$. Is it true that triangle $ABC$ is isosceles?

In the plane, $n$ lines are drawn in general position (that is, there are neither two of them parallel nor three of them passing through the same point). Prove that it is possible to put a positive integer in each region (finite or infinite) determined by these lines so that for each line the sum of the numbers in the regions of a sdemiplane is equal to the sum of the numbers in the regions of the other semiplane. Note: A region is a set of points such that the straight line connecting any two of them it does not intersect any of the lines. For example, a line divides the plane into $2$ infinite regions and three lines into general position divide the plane into $7$ regions, some finite(s) and others infinite.

Let $ABC$ be an acute triangle inscribed in circle $(O)$, with orthocenter $H$. Median $AM$ of triangle $ABC$ intersects circle $(O)$ at $A$ and $N$. $AH$ intersects $(O)$ at $A$ and $K$. Three lines $KN, BC$ and line through $H$ and perpendicular to $AN$ intersect each other and form triangle $X Y Z$. Prove that the circumcircle of triangle $X Y Z$ is tangent to $(O)$.

Let $P$ be a point on the plane. Three nonoverlapping equilateral triangles $PA_1A_2$, $PA_3A_4$, $PA_5A_6$ are constructed in a clockwise manner. The midpoints of $A_2A_3$, $A_4A_5$, $A_6A_1$ are $L$, $M$, $N$, respectively. Prove that triangle $LMN$ is equilateral.

Let $a$, $b$ and $c$ be rational numbers for which $a+bc$, $b+ac$ and $a+b$ are all non-zero and for which we have \[\frac{1}{a+bc}+\frac{1}{b+ac}=\frac{1}{a+b}.\] Prove that $\sqrt{(c-3)(c+1)}$ is rational.

Find all positive integers $n$ such that $-5^4 + 5^5 + 5^n$ is a perfect square. Do the same for $2^4 + 2^7 + 2^n.$

Find all nonnegative integers $(x,y,z,u)$ with satisfy the following equation: $2^x + 3^y + 5^z = 7^u.$

In convex hexagon $ ABCDEF$ all sides have equal length and $ \angle A\plus{}\angle C\plus{}\angle E\equal{}\angle B\plus{}\angle D\plus{}\angle F$. Prove that the diagonals $ AD,BE,CF$ are concurrent.

Find all real polynomial functions $f : R \to R$ such that $f(\sin x) = f(\cos x)$.

Given a positive integer $n\ge 3$, colour each cell of an $n\times n$ square array with one of $\lfloor (n+2)^2/3\rfloor$ colours, each colour being used at least once. Prove that there is some $1\times 3$ or $3\times 1$ rectangular subarray whose three cells are coloured with three different colours. [i](Russia) Ilya Bogdanov, Grigory Chelnokov, Dmitry Khramtsov[/i]

A positive integer $n\geqslant 3$ is [i]almost squarefree[/i] if there exists a prime number $p\equiv 1\bmod 3$ such that $p^2\mid n$ and $n/p$ is squarefree. Prove that for any almost squarefree positive integer $n$ the ratio $2\sigma(n)/d(n)$ is an integer.

Numbers $-1011, -1010, \ldots, -1, 1, \ldots, 1011$ in some order form the sequence $a_1,a_2,\ldots, a_{2022}$. Find the maximum possible value of the sum $$|a_1|+|a_1+a_2|+\ldots+|a_1+\ldots+a_{2022}|$$

Let $ABC$ be an acute triangle with $\angle BAC = 30^{\circ}$. The internal and external angle bisectors of $\angle ABC$ meet the line $AC$ at $B_1$ and $B_2$ , respectively, and the internal and external angle bisectors of $\angle ACB$ meet the line $AB$ at $C_1$ and $C_2$ , respectively. Suppose that the circles with diameters $B_1B_2$ and $C_1C_2$ meet inside the triangle $ABC$ at point $P$ . Prove that $\angle BPC = 90^{\circ}$.

Point $P_1$ is located $600$ miles West of point $P_2$. At $7:00\text{AM}$ a car departs from $P_1$ and drives East at a speed of $50$mph. At $8:00\text{AM}$ another car departs from $P_2$ and drives West at a constant speed of $x$ miles per hour. If the cars meet each other exactly halfway between $P_1$ and $P_2$, what is the value of $x$?

Air Michael and Air Patrick operate direct flights connecting Belfast, Cork, Dublin, Galway, Limerick, and Waterord. For each pair of cities exactly one of the airlines operates the route (in both directions) connecting the cities. Prove that there are four cities for which one of the airlines operates a round trip. (Note that a round trip of four cities $ P,Q,R,$ and $ S$, is a journey that follows the path $ P \rightarrow Q \rightarrow R \rightarrow S \rightarrow P$.)

A $3n$-table is a table with three rows and $n$ columns containing all the numbers $1, 2, …, 3n$. Such a table is called [i]tidy [/i] if the $n$ numbers in the first row appear in ascending order from left to right, and the three numbers in each column appear in ascending order from top to bottom. How many tidy $3n$-tables exist?

At the vertices $A, B, C, D, E, F, G, H$ of a cube, $2001, 2002, 2003, 2004, 2005, 2008, 2007$ and $2006$ stones respectively are placed. It is allowed to move a stone from a vertex to each of its three neighbours, or to move a stone to a vertex from each of its three neighbours. Which of the following arrangements of stones at $A, B, \ldots , H$ can be obtained? $(\text{a})\quad 2001, 2002, 2003, 2004, 2006, 2007, 2008, 2005;$ $(\text{b})\quad 2002, 2003, 2004, 2001, 2006, 2005, 2008, 2007;$ $(\text{c})\quad 2004, 2002, 2003, 2001, 2005, 2008, 2007, 2006.$

Which one of the following rigid transformations (isometries) maps the line segment $\overline{AB}$ onto the line segment $\overline{A'B'}$ so that the image of $A(-2,1)$ is $A'(2,-1)$ and the image of $B(-1,4)$ is $B'(1,-4)?$ $\textbf{(A) } $ reflection in the $y$-axis $\textbf{(B) } $ counterclockwise rotation around the origin by $90^{\circ}$ $\textbf{(C) } $ translation by 3 units to the right and 5 units down $\textbf{(D) } $ reflection in the $x$-axis $\textbf{(E) } $ clockwise rotation about the origin by $180^{\circ}$