Olympiad problems

Ukraine Correspondence MO - geometry, 2019.7

Given a triangle $ABC$. Construct a point $D$ on the side $AB$ and point $E$ on the side $AC$ so that $BD = CE$ and $\angle ADC = \angle BEC$

2021 Kyiv City MO Round 1, 10.2

Tags: construction , combinatorics , permutation

The $1 \times 1$ cells located around the perimeter of a $4 \times 4$ square are filled with the numbers $1, 2, \ldots, 12$ so that the sums along each of the four sides are equal. In the upper left corner cell is the number $1$, in the upper right - the number $5$, and in the lower right - the number $11$. [img]https://i.ibb.co/PM0ry1D/Kyiv-City-MO-2021-Round-1-10-2.png[/img] Under these conditions, what number can be located in the last corner cell? [i]Proposed by Mariia Rozhkova[/i]

2008 Oral Moscow Geometry Olympiad, 1

Tags: analytic geometry , geometry , construction

A coordinate system was drawn on the board and points $A (1,2)$ and $B (3,1)$ were marked. The coordinate system was erased. Restore it by the two marked points.

2022 Yasinsky Geometry Olympiad, 1

Tags: geometry , construction , area

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)

1982 Spain Mathematical Olympiad, 5

Tags: geometry , construction , square

Construct a square knowing the sum of the diagonal and the side.

2023 Yasinsky Geometry Olympiad, 3

Tags: geometry , construction , excenter

Let $I$ be the center of the inscribed circle of the triangle $ABC$. The inscribed circle is tangent to sides $BC$ and $AC$ at points $K_1$ and $K_2$ respectively. Using a ruler and a compass, find the center of excircle for triangle $CK_1K_2$ which is tangent to side $CK_2$, in at most $4$ steps (each step is to draw a circle or a line). (Hryhorii Filippovskyi, Volodymyr Brayman)

2011 Sharygin Geometry Olympiad, 12

Tags: geometry , equal angles , construction , altitude

Let $AP$ and $BQ$ be the altitudes of acute-angled triangle $ABC$. Using a compass and a ruler, construct a point $M$ on side $AB$ such that $\angle AQM = \angle BPM$.

1939 Eotvos Mathematical Competition, 3

Tags: geometry , semicircle , construction , equal segments

$ABC$ is an acute triangle. Three semicircles are constructed outwardly on the sides $BC$, $CA$ and $AB$ respectively. Construct points $A'$ , $B'$ and $C' $ on these semicìrcles respectively so that $AB' = AC'$, $BC' = BA'$ and $CA'= CB'$.

1951 Polish MO Finals, 6

Tags: similar , similar triangles , construction , geometry

Given a circle and a segment $ MN $. Find a point $ C $ on the circle such that the triangle $ ABC $, where $ A $ and $ B $ are the intersection points of the lines $ MC $ and $ NC $ with the circle, is similar to the triangle $ MNC $.

2023 Sharygin Geometry Olympiad, 8

Tags: geometry , construction , length

A triangle $ABC$ $(a>b>c)$ is given. Its incenter $I$ and the touching points $K, N$ of the incircle with $BC$ and $AC$ respectively are marked. Construct a segment with length $a-c$ using only a ruler and drawing at most three lines.

2022 Taiwan TST Round 2, 4

Tags: number theory , combinatorics , construction

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.

2023 Yasinsky Geometry Olympiad, 5

Tags: geometry , circumcenter , construction

Let $ABC$ be a scalene triangle. Given the center $I$ of the inscribe circle and the points $K_1$, $K_2$ and $K_3$ where the inscribed circle is tangent to the sides $BC$, $AC$ and $AB$. Using only a ruler, construct the center of the circumscribed circle of triangle $ABC$. (Hryhorii Filippovskyi)

Kyiv City MO Juniors Round2 2010+ geometry, 2019.7.31

Tags: geometry , construction , analytic geometry , coordinate geometry , point

The teacher drew a coordinate plane on the board and marked some points on this plane. Unfortunately, Vasya's second-grader, who was on duty, erased almost the entire drawing, except for two points $A (1, 2)$ and $B (3,1)$. Will the excellent Andriyko be able to follow these two points to construct the beginning of the coordinate system point $O (0, 0)$? Point A on the board located above and to the left of point $B$.

2003 Poland - Second Round, 5

Tags: geometry , triangle , construction , similar triangles

Point $A$ lies outside circle $o$ of center $O$. From point $A$ draw two lines tangent to a circle $o$ in points $B$ and $C$. A tangent to a circle $o$ cuts segments $AB$ and $AC$ in points $E$ and $F$, respectively. Lines $OE$ and $OF$ cut segment $BC$ in points $P$ and $Q$, respectively. Prove that from line segments $BP$, $PQ$, $QC$ can construct triangle similar to triangle $AEF$.

1977 Bundeswettbewerb Mathematik, 2

Tags: geometry , construction , parallelogram , disks

On a plane are given three non-collinear points $A, B, C$. We are given a disk of diameter different from that of the circle passing through $A, B, C$ large enough to cover all three points. Construct the fourth vertex of the parallelogram $ABCD$ using only this disk (The disk is to be used as a circular ruler, for constructing a circle passing through two given points).

Ukrainian TYM Qualifying - geometry, XII.2

Tags: geometry , construction , ruler , incenter

The figure shows a triangle, a circle circumscribed around it and the center of its inscribed circle. Using only one ruler (one-sided, without divisions), construct the center of the circumscribed circle.

1962 Polish MO Finals, 2

Tags: geometry , construction , area

Inside a given convex quadrilateral, find a point such that the segments connecting this point with the midpoints of the quadrilateral's sides divide the quadrilateral into four parts with equal areas.

Kyiv City MO Juniors 2003+ geometry, 2007.9.3

Tags: geometry , circumcircle , incircle , construction

On a straight line $4$ points are successively set , $A, P, Q,W $, which are the points of intersection of the bisector $AL $ of the triangle $ABC$ with the circumscribed and inscribed circle. Knowing only these points, construct a triangle $ABC $.

2007 Sharygin Geometry Olympiad, 2

Tags: geometry , angle bisector , construction

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 $B$.

IV Soros Olympiad 1997 - 98 (Russia), 10.5

Tags: geometry , construction

Three rays with a common origin are drawn on the plane, dividing the plane into three angles. One point is marked inside each corner. Using one ruler, construct a triangle whose vertices lie on the given rays and whose sides contain the given points.

1999 Cono Sur Olympiad, 2

Tags: geometry , construction , right triangle , rectangle , area

Let $ABC$ be a triangle right in $A$. Construct a point $P$ on the hypotenuse $BC$ such that if $Q$ is the foot of the perpendicular drawn from $P$ to side $AC$, then the area of the square of side $PQ$ is equal to the area of the rectangle of sides $PB$ and $PC$. Show construction steps.

1977 Czech and Slovak Olympiad III A, 2

Tags: geometry , rectangle , construction , geometric construction

The numbers $p,q>0$ are given. Construct a rectangle $ABCD$ with $AE=p,AF=q$ where $E,F$ are midpoints of $BC,CD,$ respectively. Discuss conditions of solvability.

2005 Oral Moscow Geometry Olympiad, 5

Tags: geometry , circumcircle , equilateral , construction

The triangle $ABC$ is inscribed in the circle. Construct a point $P$ such that the points of intersection of lines $AP, BP$ and $CP$ with this circle are the vertices of an equilateral triangle. (A. Zaslavsky)

1970 Dutch Mathematical Olympiad, 1

Tags: geometry , construction , square

Four different points $A,B,C$ and $D$ lie in a plane. No three of these points lie on a single straight line. Describe the construction of a square $PQRS$ such that on each of the sides of $PQRS$, or the extensions , lies one of the points $A, B, C$ and $D$.

2011 Sharygin Geometry Olympiad, 2

Tags: geometry , rectangle , construction , paper

Peter made a paper rectangle, put it on an identical rectangle and pasted both rectangles along their perimeters. Then he cut the upper rectangle along one of its diagonals and along the perpendiculars to this diagonal from two remaining vertices. After this he turned back the obtained triangles in such a way that they, along with the lower rectangle form a new rectangle. Let this new rectangle be given. Restore the original rectangle using compass and ruler.