Prove there is an integer $ k$ for which $ k^3 \minus{} 36 k^2 \plus{} 51 k \minus{} 97$ is a multiple of $ 3^{2008.}$

The longer base of trapezoid $ABCD$ is $AB$, while the shorter base is $CD$. Diagonal $AC$ bisects the interior angle at $A$. The interior bisector at $B$ meets diagonal $AC$ at $E$. Line $DE$ meets segment $AB$ at $F$. Suppose that $AD = FB$ and $BC = AF$. Find the interior angles of quadrilateral $ABCD$, if we know that $\angle BEC = 54^o$.

A square board is divided by lines parallel to the board sides ($7$ lines in each direction, not necessarily equidistant ) into $64$ rectangles. Rectangles are colored into white and black in alternating order. Assume that for any pair of white and black rectangles the ratio between area of white rectangle and area of black rectangle does not exceed $2.$ Determine the maximal ratio between area of white and black part of the board. White (black) part of the board is the total sum of area of all white (black) rectangles.

In a convex quadrilateral $ ABCD$, let $ E$ be the intersection point of the diagonals, and let $ F_1,F_2,$ and $ F$ be the areas of $ ABE,CDE,$ and $ ABCD,$ respectively. Prove that: $ \sqrt {F_1}\plus{}\sqrt {F_2} \le \sqrt {F}.$

Fedir and Mykhailo have three piles of stones: the first contains $100$ stones, the second $101$, the third $102$. They are playing a game, going in turns, Fedir makes the first move. In one move player can select any two piles of stones, let's say they have $a$ and $b$ stones left correspondently, and remove $gcd(a, b)$ stones from each of them. The player after whose move some pile becomes empty for the first time wins. Who has a winning strategy? As a reminder, $gcd(a, b)$ denotes the greatest common divisor of $a, b$. [i](Proposed by Oleksii Masalitin)[/i]

Let \( AD \) and \( BE \) be altitudes of triangle \( \triangle ABC \) that meet at the orthocenter \( H \). The midpoints of segments \( AB \) and \( CH \) are \( X \) and \( Y \), respectively. Prove that the line \( XY \) is perpendicular to line \( DE \).

Let $ABC$ be a triangle and $H$ be its orthocenter. If it is given that $B$ is $(0,0)$, $C$ is $(1,2)$ and $H$ is $(5,0)$, find $A$.

In the quadrilateral $PQRS, A, B, C$ and $D$ are the midpoints of the sides $PQ, QR, RS$ and $SP$ respectively, and $M$ is the midpoint of $CD$. Suppose $H$ is the point on the line $AM$ such that $HC = BC$. Prove that $\angle BHM = 90^o$.

Let $P$ be a point inside triangle $ABC$ and let $R$ denote the circumradius of triangle $ABC$. Prove that \[ \frac{PA}{AB\cdot AC}+\frac{PB}{BC\cdot BA}+\frac{PC}{CA\cdot CB}\ge\frac{1}{R}.\]

Is it possible to write the numbers from $1$ to $100$ in the cells of a of a $10 \times 10$ square so that: 1. Each cell contains exactly one number; 2. Each number is written exactly once; 3. For any two cells that are symmetrical with respect to any of the perpendicular bisectors of sides of the original $10 \times 10$ square, the numbers in them must have the same parity. The figure below shows examples of such pairs of cells, in which the numbers must have the same parity. [img]https://i.ibb.co/b3P8t36/Kyiv-MO-2024-7-2.png[/img] [i]Proposed by Mykhailo Shtandenko[/i]

In a group of $n$ persons, (i) each person is acquainted to exactly $k$ others, (ii) any two acquainted persons have exactly $l$ common acquaintances, (iii) any two non-acquainted persons have exactly $m$ common acquaintances. Prove that $m(n-k -1) = k(k -l -1)$.

Suppose $ABC$ is a triangle, and define $B_1$ and $C_1$ such that $\triangle AB_1C$ and $\triangle AC_1B$ are isosceles right triangles on the exterior of $\triangle ABC$ with right angles at $B_1$ and $C_1$, respectively. Let $M$ be the midpoint of $\overline{B_1C_1}$; if $B_1C_1 = 12$, $BM = 7$ and $CM = 11$, what is the area of $\triangle ABC$?

Find the natural numbers $ n\ge 2 $ which have the property that the ring of integers modulo $ n $ has exactly an element that is not a sum of two squares.

Let $a$ be an odd digit and $b$ an even digit. Prove that for every positive integer $n$ there exists a positive integer, divisible by $2^n$, whose decimal representation contains no digits other than $a$ and $b$.

The real-valued function $f(x)$ is defined on the reals. It satisfies $|f(x)| \le A$, $|f''(x)| \le B$ for some positive $A, B$ (and all $x$). Show that $|f'(x)| \le C$, for some fixed$ C$, which depends only on $A$ and $B$. What is the smallest possible value of $C$?

A sequence of positive integers $a_0, a_1, a_2, \dots$ satisfies $a_0 = 83$ and $a_n = (a_{n-1})^{a_{n-1}}$ for all positive integers $n$. Compute the remainder when $a_{2021}$ is divided by $60$.

There are relatively prime positive integers $m$ and $n$ so that \[\frac{m}{n}=\log_4\left(32^{\log_927}\right).\] Find $m+n$.

Luka is making lemonade to sell at a school fundraiser. His recipe requires $4$ times as much water as sugar and twice as much sugar as lemon juice. He uses $3$ cups of lemon juice. How many cups of water does he need? $\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 24$

Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of $ 3$ kilometers per hour, bikes at a rate of $ 20$ kilometers per hour, and runs at a rate of $ 10$ kilometers per hour. Which of the following is closest to the triathlete's average speed, in kilometers per hour, for the entire race? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$

For each real number $ r$, $ [r]$ denotes the largest integer less than or equal to $ r$, e.g. $ [6] \equal{} 6, [\pi] \equal{} 3, [\minus{}1.5] \equal{} \minus{}2$. Indicate on the $ (x,y)$-plane the set of all points $ (x,y)$ for which $ [x]^2 \plus{} [y]^2 \equal{} 4$.

In a triangle $ABC$, $D$ is the orthogonal projection of the incenter $I$ onto $BC$. Line $DI$ meets the incircle again at $E$. Line $AE$ intersects side $BC$ at point $F$. Suppose that the segment IO is parallel to $BC$, where $O$ is the circumcenter of $\triangle ABC$. If $R$ is the circumradius and $r$ the inradius of the triangle, prove that $EF=2(R-2r)$.

Find all continuous functions $f:[1,8]\to\mathbb R$, such that $$\int^2_1f(t^3)^2dt+2\int^2_1f(t^3)dt=\frac23\int^8_1f(t)dt-\int^2_1(t^2-1)^2dt.$$

[b]7.1[/b] Construct a trapezoid given four sides. [b]7.2[/b] Prove that $$(1 + x + x^2 + ...+ x^{100})(1 + x^{102}) - 102x^{101} \ge 0 .$$ [b]7.3 [/b] In a quadrilateral $ABCD$, $M$ is the midpoint of AB, $N$ is the midpoint of $CD$. Lines $AD$ and BC intersect $MN$ at points $P$ and $Q$, respectively. Prove that if $\angle BQM = \angle APM$ , then $BC=AD$. [img]https://cdn.artofproblemsolving.com/attachments/a/2/1c3cbc62ee570a823b5f3f8d046da9fbb4b0f2.png[/img] [b]7.4 / 6.4[/b] Each of the eight given different natural numbers less than $16$. Prove that among their pairwise differences there is at least at least three are the same. [b]7.5 / 8.4[/b] An entire arc of circle is drawn through the vertices $A$ and $C$ of the rectangle $ABCD$ lying inside the rectangle. Draw a line parallel to $AB$ intersecting $BC$ at point $P$, $AD$ at point $Q$, and the arc $AC$ at point $R$ so that the sum of the areas of the figures $AQR$ and $CPR$ is the smallest. [img]https://cdn.artofproblemsolving.com/attachments/1/4/9b5a594f82a96d7eff750e15ca6801a5fc0bf1.png[/img] [b]7.6 / 6.5 [/b]The distance AB is 100 km. From A and B , cyclists simultaneously ride towards each other at speeds of 20 km/h and 30 km/hour accordingly. Together with the first A, a fly flies out with speed 50 km/h, she flies until she meets the cyclist from B, after which she turns around and flies back until she meets the cyclist from A, after which turns around, etc. How many kilometers will the fly fly in the direction from A to B until the cyclists meet? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here[/url].

Consider the sequence of natural numbers $a_n$ defined as $a_0=4$ and $a_{n+1}=\frac{a_n(a_n-1)}{2}$ for each $n\geq 0$. Define a new sequence $b_n$ as follows: $b_n=0$ if $a_n$ is even, and $b_n=1$ if $a_n$ is odd. Prove that for each natural $m$, the sequence \[b_m, b_{m+1}, b_{m+2},b_{m+3}, \dots\] is not periodic.