There is a table with $n > 2$ cells in the first row, $n-1$ cells in the second row is a cell, $n-2$ in the third row, $\ldots$, $1$ cell in the $n$-th row. The cells are arranged as shown below. [img]https://i.ibb.co/0Z1CR0c/UMO24-8-2.png[/img] In each cell of the top row Petryk writes a number from $1$ to $n$, so that each number is written exactly once. For each other cell, if the cells directly above it contains numbers $a, b$, it contains number $|a-b|$. What is the largest number that can be written in a single cell of the bottom row? [i]Proposed by Bogdan Rublov[/i]

Prove that for three arbitrary infinite sequences, of natural numbers $a_1,a_2,...,a_n,... $ , $b_1,b_2,...,b_n,... $, $c_1,c_2,...,c_n,...$ there exist numbers $p$ and $q$ such, that $a_p \ge a_q$, $b_p \ge b_q$ and $c_p \ge c_q$.

Let $k$ be the circle and $A$ and $B$ points on circle which are not diametrically opposite. On minor arc $AB$ lies point arbitrary point $C$. Let $D$, $E$ and $F$ be foots of perpendiculars from $C$ on chord $AB$ and tangents of circle $k$ in points $A$ and $B$. Prove that $CD= \sqrt {CE \cdot CF}$

A cycling group that has $4n$ members will have several cycling events, such that: a) Two cycling events are done every week; once on Saturday and once on Sunday. b) Exactly $2n$ members participate in any cycling event. c) No member may participate in both cycling events of a week. d) After all cycling events are completed, the number of events where each pair of members meet is the same for all pairs of members. Prove that after all cycling events are completed, the number of events where each group of three members meet is the same value $t$ for all groups of three members, and that for $n \ge 2$, $t$ is divisible by $n-1$.

Prove that for any polygon with all equal angles and for any interior point $A$, the sum of distances from $A$ to the sides of the polygon does not depend on the position of $A$.

$AB$ is a chord of $O$ and $AB$ is not a diameter of $O$. The tangent lines to $O$ at $A$ and $B$ meet at $C$. Let $M$ and $N$ be the midpoint of the segments $AC$ and $BC$, respectively. A circle passing through $C$ and tangent to $O$ meets line $MN$ at $P$ and $Q$. Prove that $\angle PCQ = \angle CAB$.

Let $S = \{x_0, x_1,\dots , x_n\}$ be a finite set of numbers in the interval $[0, 1]$ with $x_0 = 0$ and $x_1 = 1$. We consider pairwise distances between numbers in $S$. If every distance that appears, except the distance $1$, occurs at least twice, prove that all the $x_i$ are rational.

$n$ natural numbers ($n>3$) are written on the circumference. The relation of the two neighbours sum to the number itself is a whole number. Prove that the sum of those relations is a) not less than $2n$ b) less than $3n$

Let $\alpha\geq 1$ be a real number. Define the set $$A(\alpha)=\{\lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor, \lfloor 3\alpha\rfloor,\dots\}$$ Suppose that all the positive integers that [b]does not belong[/b] to the $A(\alpha)$ are exactly the positive integers that have the same remainder $r$ in the division by $2021$ with $0\leq r<2021$. Determine all the possible values of $\alpha$.

Right triangle $XY Z$, with hypotenuse $Y Z$, has an incircle of radius $\frac38$ and one leg of length $3$. Find the area of the triangle.

Find all $f : N \longmapsto N$ that there exist $k \in N$ and a prime $p$ that: $\forall n \geq k \ f(n+p)=f(n)$ and also if $m \mid n$ then $f(m+1) \mid f(n)+1$

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

One day, Rabbit was about to go for a meeting with Donkey, but Winnie the Pooh and Duck unexpectedly came to his home. Being well-bred, Rabbit offered the guests some refreshments. Pooh tied Duck’s mouth by a napkin and ate $10$ pots of honey and $22$ cups of condensed milk alone, whereby he needed two minutes for each pot of honey and $1$ minute for each cup of milk. Knowing that there was nothing sweet left in the house, Pooh released the Duck. Afflicted Rabbit observed that he wouldn’t have been late for the meeting with Donkey if Pooh had shared the refreshments with Duck. Knowing that Duck needs $5$ minutes for a pot of honey and $3$ minutes for a cup of milk, he computed the time the guests would have needed to devastate his supplies. What is that time?

Suppose a function $f(x)$ is continuous on $[-1,\ 1]$ and satisfies the condition : 1) $f(-1)\geq f(1).$ 2) $x+f(x)$ is non decreasing function. 3) $\int_{-1}^ 1 f(x)\ dx=0.$ Show that $\int_{-1}^1 f(x)^2dx\leq \frac 23.$

We can obviously put 100 unit balls in a $10\times10\times1$ box. How can one put $105$ unit balls in? How can we put $106$ unit balls in?

If an integer $n$ is such that $7n$ is the form $a^2 +3b^2$, prove that $n$ is also of that form.

$6$ cats can eat $6$ fish in $1$ day, and $c$ cats can eat $91$ fish in $d$ days. Given that $c$ and $d$ are both whole numbers, and the number of cats, $c$, is more than $1$ but less than $10$, find $c + d$.

Without using a calculator, find out which number is greater: $$29^{200}\cdot 2^{151} \,\,\, or \,\,\, 5^{279} \cdot 3^{300}$$

Brennan chooses a set $A = \{a, b,c, d, e \}$ of five real numbers with $a \leq b \leq c \leq d \leq e.$ Delaney determines the subsets of $A$ containing three numbers and adds up the numbers in these subsets. She obtains the sums $0, 3; 4, 8; 9, 10, 11, 12, 14, 19.$ What are the five numbers in Brennan's set?

Let $OX, OY$ and $OZ$ be three rays in the space, and $G$ a point "[i]between these rays[/i]" (i. e. in the interior of the part of the space bordered by the angles $Y OZ, ZOX$ and $XOY$). Consider a plane passing through $G$ and meeting the rays $OX, OY$ and $OZ$ in the points $A, B, C$, respectively. There are infinitely many such planes; construct the one which minimizes the volume of the tetrahedron $OABC$.

A rod moves freely between the horizontal floor and the slanted wall. When the end in contact with the floor is moving at v, what is the speed of the end in contact with the wall? $\textbf{(A)} v\frac{\sin{\theta}}{\cos(\alpha-\theta)}$ $\textbf{(B)}v\frac{\sin(\alpha - \theta)}{\cos(\alpha + \theta)} $ $\textbf{(C)}v\frac{\cos(\alpha - \theta)}{\sin(\alpha + \theta)}$ $\textbf{(D)}v\frac{\cos(\theta)}{\cos(\alpha - \theta)}$ $\textbf{(E)}v\frac{\sin(\theta)}{\cos(\alpha + \theta)}$

An [i]exterior [/i] angle is the supplementary angle to an interior angle in a polygon. What is the sum of the exterior angles of a triangle and dodecagon ($12$-gon), in degrees?

Consider the parallelogram $ABCD$. A circle is drawn that passes through $A$ and intersects side $AD$ at $N$, side $AB$ at $M$ and diagonal $AC$ in $P$ such that points $A, M, N, P$ are different. Prove that $$AP\cdot AC = AM \cdot AB + AN \cdot AD.$$

Let $ABC$ be an isosceles triangle with $AB = AC$. Suppose $P,Q,R$ are points on segments $AC, AB, BC$ respectively such that $AP = QB$, $\angle PBC = 90^\circ - \angle BAC$ and $RP = RQ$. Let $O_1, O_2$ be the circumcenters of $\triangle APQ$ and $\triangle CRP$. Prove that $BR = O_1O_2$. [i]Proposed by Atul Shatavart Nadig[/i]

Do there exist two polynomials $P$ and $Q$ with integer coefficient such that i) both $P$ and $Q$ have a coefficient with absolute value bigger than $2021$, ii) all coefficients of $P \cdot Q$ by absolute value are at most $1$.