Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children? $ \textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$

There are $2019$ coins on a table. Some are placed with head up and others tail up. A group of $2019$ persons perform the following operations: the first person chooses any one coin and then turns it over, the second person choses any two coins and turns them over and so on and the $2019$-th person turns over all the coins. Prove that no matter which sides the coins are up initially, the $2019$ persons can come up with a procedure for turning the coins such that all the coins have smae side up at the end of the operations.

The yearly changes in the population census of a town for four consecutive years are, respectively, $25\%$ increase, $25\%$ increase, $25\%$ decrease, $25\%$ decrease. The net change over the four years, to the nearest percent, is: ${{ \textbf{(A)}\ -12 \qquad\textbf{(B)}\ -1 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 1}\qquad\textbf{(E)}\ 12} $

If $xyz=1$ find the value of $\left(\frac{1}{1+x+\frac{1}{y}}+\frac{1}{1+y+\frac{1}{z}}+\frac{1}{1+z+\frac{1}{x}}\right)^2$.

Consider an equilateral triangle with every side divided by $n$ points into $n+1$ equal parts. We put a marker on every of the $3n$ division points. We draw lines parallel to the sides of the triangle through the division points, and this way divide the triangle into $(n+1)^2$ smaller ones. Consider the following game: if there is a small triangle with exactly one vertex unoccupied, we put a marker on it and simultaneously take markers from the two its occupied vertices. We repeat this operation as long as it is possible. (a) If $n\equiv1\pmod3$, show that we cannot manage that only one marker remains. (b) If $n\equiv0$ or $n\equiv2\pmod3$, prove that we can finish the game leaving exactly one marker on the triangle.

Po picks $100$ points $P_1,P_2,\cdots, P_{100}$ on a circle independently and uniformly at random. He then draws the line segments connecting $P_1P_2,P_2P_3,\ldots,P_{100}P_1.$ Find the expected number of regions that have all sides bounded by straight lines.

In a $5 \times 9$ checkered rectangle, the middle row and middle column are colored gray. You leave the corner cell and move to the cell adjacent to the side with each move. For each transition from a gray cell to a gray one you need to pay a ruble. What is the smallest number of rubles you need to pay to go around all the squares of the board exactly once (it is not necessary to return to the starting square)?

Prove that for all polynomials $P \in \mathbb{R}[x]$ and positive integers $n$, $P(x)-x$ divides $P^n(x)-x$ as polynomials.

In a 2025 by 2025 grid, every cell initially contains a `1'. Every minute, we simultaneously replace the number in each cell with the sum of numbers in the cells that share an edge with it. (For example, after the first minute, the number 2 is written in each of the four corner cells.) After 2025 minutes, we colour the board in checkerboard fashion, such that the top left corner is black. Find the difference between the sum of numbers in black cells and the sum of numbers in white cells. [i]Proposed by chorn[/i]

Let $D=\{0,1,\dots,9\}$. A direction function for $D$ is a function $f:D \times D \to \{0,1\}.$ A real $r\in [0,1]$ is compatible with $f$ if it can be written in the form $$r = \sum_{j=1}^{\infty} \frac{d_j}{10^j}$$ with $d_j \in D$ and $f(d_j,d_{j+1})=1$ for every positive integer $j$. Determine the least integer $k$ such that for any direction fuction $f$, if there are $k$ compatible reals with $f$ then there are infinite reals compatible with $f$.

Mykhailo drew a triangular grid with side \( n \) for \( n \geq 2 \). It is formed from an equilateral triangle \( T \) with side length \( n \), by dividing each side into \( n \) equal parts. Then lines are drawn parallel to the sides of triangle \( T \), dividing it into \( n^2 \) equilateral triangles with side length \( 1 \), which we will call \textbf{cells}. Next, Oleksii writes some positive integer into each cell. Mykhailo receives 1 candy for each cell, where the number written is equal to the sum of all the numbers in the adjacent cells. Oleksii wants to arrange the numbers in such a way that Mykhailo receives the maximum number of candies. How many candies can Mykhailo receive under such conditions? In the figure below, an example is shown for \( n = 4 \) with 16 cells and numbers written inside them. For the numbers arranged as in the figure, Mykhailo receives 5 candies for the numbers \( 2 \) (the topmost cell), \( 8 \), \( 13 \), \( 12 \), and \( 11 \). [img]https://i.ibb.co/LrLks9q/Kyiv-MO-2025-R2-7-1.png[/img] [i]Proposed by Mykhailo Shtandenko[/i]

Let $n$ be a positive integer, $h$ a real number and $f(x)$ a polynomial (whole rational function) with real coefficients of degree n, which has no real zeros. Prove that then also the polynomial $$F(x) = f(x) + h f'(x) + h^2 f''(x) +... + h^n f^{(n)}(x)$$ has no real zeros.

Prove that the following inequality holds with the exception of finitely many positive integers $n$: $\sum^{n}_{i=1}\sum^{n}_{j=1}gcd(i,j)>4n^2$.

In how many ways we can paint a $N \times N$ chessboard using 4 colours such that squares with a common side are painted with distinct colors and every $2 \times 2$ square (formed with 4 squares in consecutive lines and columns) is painted with the four colors?

A student has $100$ cards on which the integers $1$ to $100$ are printed, as well as a sufficiently large number of cards on which the symbols $+$ and $=$ are printed. What is the maximal number of correct equalities the student can construct, if each card is used at most once? (R Zhenodarov)

Find the smallest positive integer $n$ such that [list][*] $n$ has exactly $144$ distinct positive divisors, [*] there are ten consecutive integers among the positive divisors of $n$. [/list]

Prove that every positive rational number can be represented in the form $\dfrac{a^{3}+b^{3}}{c^{3}+d^{3}}$ where a,b,c,d are positive integers.

Does there exist an infinite sequence of real numbers ${a}_{1},{a}_{2},{a}_{3},\ldots$ such that ${a}_{1} = 1$ and for all positive integers $k$ we have the equality $$ {a}_{k} = {a}_{2k} + {a}_{3k} + {a}_{4k} + \ldots ? $$ Ilya Lobatsky

Compute the number of ways of tiling the $2\times 10$ grid below with the three tiles shown. There is an infinite supply of each tile, and rotating or reflecting the tiles is not allowed. [img]https://cdn.artofproblemsolving.com/attachments/5/a/bb279c486fc85509aa1bcabcda66a8ea3faff8.png[/img]

What mass of $\ce{NaHCO3}$ $(\text{M=84})$ is required to completely neutralize $25.0$ mL of $0.125$ M $\ce{H2SO4}$? $ \textbf{(A) }\text{0.131 g}\qquad\textbf{(B) }\text{0.262 g}\qquad\textbf{(C) }\text{0.525 g}\qquad\textbf{(D) }\text{1.05 g}\qquad$

Suppose that $n$ is a positive integer number. Consider a regular polygon with $2n$ sides such that one of its largest diagonals is parallel to the $x$-axis. Find the smallest integer $d$ such that there is a polynomial $P$ of degree $d$ whose graph intersects all sides of the polygon on points other than vertices. Proposed by Mohammad Ahmadi

What is the area of the triangle formed by the lines $y=5$, $y=1+x$, and $y=1-x$? $\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16$

Let $ABCD$ be a unit square in the plane. Points $X$ and $Y$ are chosen independently and uniformly at random on the perimeter of $ABCD$. If the expected value of the area of triangle $\triangle AXY$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$. [i]Proposed by Nathan Xiong[/i]

Here $G_{n}$ denotes a simple undirected graph with $n$ vertices, $K_{n}$ denotes the complete graph with $n$ vertices, $K_{n,m}$ the complete bipartite graph whose components have $m$ and $n$ vertices, and $C_{n}$ a circuit with $n$ vertices. The number of edges in the graph $G_{n}$ is denoted $e(G_{n})$. (a) If $G_{n}$ does not contain $K_{2,3}$ , prove that $e(G_{n}) \leq\frac{n\sqrt{n}}{\sqrt{2}}+n$. (b) Given $n \geq 16$ distinct points $P_{1}, . . . , P_{n}$ in the plane, prove that at most $n\sqrt{n}$ of the segments $P_{i}P_{j}$ have unit length.

Ali puts $5$ points on the plane such that no three of them are collinear. Ramtin adds a sixth point that is not collinear with any two of the former points.Ali wants to eventually construct two triangles from the six points such that one can be placed inside another. Can Ali put the 5 points in such a manner so that he would always be able to construct the desired triangles? (We say that triangle $T_1$ can be placed inside triangle $T_2$ if $T_1$ is congruent to a triangle that is located completely inside $T_2$.)