In a triangle $ABC$ the bisectors $BB'$ and $CC'$ are drawn. After that, the whole picture except the points $A, B'$, and $C'$ is erased. Restore the triangle using a compass and a ruler. (A.Karlyuchenko)

Three lines $k, l, m$ are drawn through a point $P$ inside a triangle $ABC$ such that $k$ meets $AB$ at $A_1$ and $AC$ at $A_2 \ne A_1$ and $PA_1 = PA_2$, $l $ meets $BC$ at $B_1$ and $BA$ at $B_2 \ne B_1$ and $PB_1 = PB_2$, $m$ meets $CA$ at $C_1$ and $CB$ at $C_2\ne C_1$ and $PC_1=PC_2$. Prove that the lines $k,l,m$ are uniquely determined by these conditions. Find point $P$ for which the triangles $AA_1A_2, BB_1B_2, CC_1C_2$ have the same area and show that this point is unique.

Find the smallest positive integer $n$ for which one can select $n$ distinct real numbers such that each of them is equal to the sum of some two other selected numbers. [i]Proposed by Anton Trygub[/i]

In a plane, we are given line $\ell$, two points $A$ and $B$ neither of which lies on line $\ell$, and the reflection $A_1$ of point $A$ across line $\ell$. Using only a straightedge, construct the reflection $B_1$ of point $B$ across line $\ell$. Prove that your construction works. Note: “Using only a straightedge” means that you can perform only the following operations: (a) Given two points, you can construct the line through them. (b) Given two intersecting lines, you can construct their intersection point. (c) You can select (mark) points in the plane that lie on or off objects already drawn in the plane. (The only facts you can use about these points are which lines they are on or not on.)

Given a segment $ AB $ and a line $ m $ parallel to this segment. Find the midpoint of the segment $ AB $ using only a ruler, i.e. drawing only straight lines.

Construct a right triangle given the lengths of segments of the medians $m_a,m_b$ corresponding on its legs.

In an acute triangle $ABC$, altitudes at vertices $A$ and $B$ and bisector line at angle $C$ intersect the circumcircle again at points $A_1, B_1$ and $C_0$. Using the straightedge and compass, reconstruct the triangle by points $A_1, B_1$ and $C_0$.

Construct a point $N$ inside a given right triangle $ABC$ such that the angles $\angle NBC$, $\angle NCA$ and $\angle NAB$ are equal.

If possible, construct an equilateral triangle whose three vertices are on three given circles.

Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \in \mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x$ is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$. Find all $r$ for which Alice can move the red bead to $1$ in at most $2021$ moves.

Show how to construct (by a ruler and a compass) a right-angled triangle, given its inradius and circumradius.

Construct a triangle given the radii of the excircles.

Let $I$ be the incenter of triangle $ABC$. $K_1$ and $K_2$ are the points on $BC$ and $AC$ respectively, at which the inscribed circle is tangent. Using a ruler and a compass, find the center of the inscribed circle for triangle $CK_1K_2$ in the minimal possible number of steps (each step is to draw a circle or a line). (Hryhorii Filippovskyi)

From the triangle $ABC$, are gicen only the incenter $I$, the touchpoint $K$ of the inscribed circle with the side $AB$, as well as the center $I_a$ of the exscribed circle, that touches the side $BC$ . Construct a triangle equal in size to triangle $ABC$. (Gryhoriy Filippovskyi)

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]

In connection with the formation of a stable government, the President invited all $240$ Members of Parliament to three separate consultations, where each member participated in exactly one consultation, and at each consultation there has been at least one member present. Discussions between pairs of members are to take place to discuss the consultations. Is it possible for these discussions to occur in such a way that there exists a non-negative integer $k$, such that for every two members who participated in different consultations, there are exactly $k$ members who participated in the remaining consultation, with whom each of the two members has a conversation, and exactly $k$ members who participated in the remaining consultation, with whom neither of the two has a conversation? If yes, then find all possible values of $k$.

There is a ruler and a "rusty" compass, with which you can construct a circle of radius $R$. The point $K$ is from the line $\ell$ at a distance greater than $R$. How to use this ruler and this compass to draw a line passing through the point $K$ and perpendicular to line $\ell$? (Misha Sidorenko, Katya Sidorenko, Rodion Osokin)

Restore the isosceles triangle $ABC$ ($AB = AC$) if the common points $I, M, H$ of bisectors, medians and altitudes respectively are given.

Construct the quadrilateral $ ABCD $ given the lengths of the sides $ AB $ and $ CD $ and the angles of the quadrilateral.

By straightedge and compass, reconstruct a right triangle $ABC$ ($\angle C = 90^o$), given the vertices $A, C$ and a point on the bisector of angle $B$.

In the plane we have a line $r$ and two points $A$ and $B$ outside the line and in the same half plane. Determine a point $M$ on the line such that the angle of $r$ with $AM$ is double that of $r$ with $BM$. (Consider the smaller angle of two lines of the angles they form).

A convex quadrilateral is given. Using a compass and a ruler construct a point such that its projections to the sidelines of this quadrilateral are the vertices of a parallelogram. (A. Zaslavsky)

Using a compass and a ruler, construct a triangle $ABC$ given the sides $b, c$ and the segment $AI$, where$ I$ is the center of the inscribed circle of this triangle.

A $2\times 2$ square was cut from a squared sheet of paper. Using only a ruler without divisions and without going beyond the square, divide the diagonal of the square into $6$ equal parts.

Given two circles of radii $r$ and $r'$ exterior to each other, construct a line parallel to a given line and intersecting the two circles in chords with the sum of lengths $\ell$.