Olympiad problems

1951 Polish MO Finals, 6

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 $.

1960 IMO Shortlist, 4

Tags: altitude , geometry , construction

Construct triangle $ABC$, given $h_a$, $h_b$ (the altitudes from $A$ and $B$), and $m_a$, the median from vertex $A$.

2020 Indonesia MO, 8

Tags: number theory , construction , bounding , bait

Determine the smallest natural number $n > 2$, or show that no such natural numbers $n$ exists, that satisfy the following condition: There exists natural numbers $a_1, a_2, \dots, a_n$ such that \[ \gcd(a_1, a_2, \dots, a_n) = \sum_{k = 1}^{n - 1} \underbrace{\left( \frac{1}{\gcd(a_k, a_{k + 1})} + \frac{1}{\gcd(a_k, a_{k + 2})} + \dots + \frac{1}{\gcd(a_k, a_n)} \right)}_{n - k \ \text{terms}} \]

2025 International Zhautykov Olympiad, 3

Tags: number theory , construction , prime

A pair of positive integers $(x, y)$ is [i] good [/i] if they satisfy $\text{rad}(x) = \text{rad}(y)$ and they do not divide each-other. Given coprime positive integers $a$ and $b$, show that there exist infinitely many $n$ for which there exists a positive integer $m$ such that $(a^n + bm, b^n + am)$ is [i] good[/i]. (Here, $\text{rad}(x)$ denotes the product of $x$'s prime divisors, as usual.)

Ukrainian TYM Qualifying - geometry, II.18

Tags: geometry , construction , tangent

Inside an acute angle is a circle. Investigate the possibility of constructing with only a compass and a ruler, a tangent to this circle that the point of contact will bisect the segment of the tangent that is cut off by the sides of the angle.

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.

2010 Sharygin Geometry Olympiad, 8

Tags: construction , geometry , concurrency

Triangle $ABC$ is inscribed into circle $k$. Points $A_1,B_1, C_1$ on its sides were marked, after this the triangle was erased. Prove that it can be restored uniquely if and only if $AA_1, BB_1$ and $CC_1$ concur.

1979 Poland - Second Round, 5

Tags: number theory , construction , coprime

Prove that among every ten consecutive natural numbers there is one that is coprime to each of the other nine.

1967 Swedish Mathematical Competition, 2

Tags: geometry , angle bisector , construction , geometric construction , ruler , perpendicular bisector

You are given a ruler with two parallel straight edges a distance $d$ apart. It may be used (1) to draw the line through two points, (2) given two points a distance $\ge d$ apart, to draw two parallel lines, one through each point, (3) to draw a line parallel to a given line, a distance d away. One can also (4) choose an arbitrary point in the plane, and (5) choose an arbitrary point on a line. Show how to construct : (A) the bisector of a given angle, and (B) the perpendicular to the midpoint of a given line segment.

1951 Poland - Second Round, 6

Tags: geometry , construction , circles

The given points are $ A $ and $ B $ and the circle $ k $. Draw a circle passing through the points $ A $ and $ B $ and defining, at the intersection with the circle $ k $, a common chord of a given length $ d $.

2017 India Regional Mathematical Olympiad, 1

Tags: construction , geometry

Let $AOB$ be a given angle less than $180^{\circ}$ and let $P$ be an interior point of the angular region determined by $\angle AOB$. Show, with proof, how to construct, using only ruler and compass, a line segment $CD$ passing through $P$ such that $C$ lies on the way $OA$ and $D$ lies on the ray $OB$, and $CP:PD=1:2$.

2005 Tournament of Towns, 2

Tags: construction , geometry

A segment of length $\sqrt2 + \sqrt3 + \sqrt5$ is drawn. Is it possible to draw a segment of unit length using a compass and a straightedge? [i](3 points)[/i]

Kyiv City MO 1984-93 - geometry, 1989.7.3

Tags: construction , geometry

The student drew a triangle $ABC$ on the board, in which $AB>BC$. On the side $AB$ is taken point $D$ such that $BD = AC$. Let points $E$ and $F$ be the midpoints of the segments $AD$ and $BC$ respectively. Then the whole picture was erased, leaving only dots $E$ and $F$. Restore triangle $ABC$.

2005 Sharygin Geometry Olympiad, 9.5

Tags: angle bisector , median , construction , altitude , angle , geometry

It is given that for no side of the triangle from the height drawn to it, the bisector and the median it is impossible to make a triangle. Prove that one of the angles of the triangle is greater than $135^o$

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).

2022 Argentina National Olympiad Level 2, 3

Tags: geometry , construction

Let $A$, $X$ and $Y$ be three non-collinear points on the plane. Construct with a straightedge and compass a square $ABCD$ such that $X$ is on the line $BC$ and $Y$ is on the line $CD$.

Kvant 2024, M2809

Tags: conditional geometry , angle measure , construction , right angle , beautiful geometry

Given is a triangle $ABC$ and the points $M, P$ lie on the segments $AB, BC$, respectively, such that $AM=BC$ and $CP=BM$. If $AP$ and $CM$ meet at $O$ and $2\angle AOM=\angle ABC$, find the measure of $\angle ABC$.

Kyiv City MO Juniors Round2 2010+ geometry, 2014.89.3

Tags: construction , geometry , bisects segment

Given a triangle $ABC$, on the side $BC$ which marked the point $E$ such that $BE \ge CE$. Construct on the sides $AB$ and $AC$ the points $D$ and $F$, respectively, such that $\angle DEF = 90 {} ^ \circ$ and the segment $BF$ is bisected by the segment $DE $. (Black Maxim)

2006 Oral Moscow Geometry Olympiad, 4

Tags: geometry , equal circles , inradius , construction

An arbitrary triangle $ABC$ is given. Construct a straight line passing through vertex $B$ and dividing it into two triangles, the radii of the inscribed circles of which are equal. (M. Volchkevich)

2003 Czech And Slovak Olympiad III A, 4

Tags: construction , triangle , geometry

Let be given an obtuse angle $AKS$ in the plane. Construct a triangle $ABC$ such that $S$ is the midpoint of $BC$ and $K$ is the intersection point of $BC$ with the bisector of $\angle BAC$.

2003 Oral Moscow Geometry Olympiad, 1

Tags: geometry , construction

Construct a triangle given an angle, the side opposite the angle and the median to the other side (researching the number of solutions is not required).

2012 Oral Moscow Geometry Olympiad, 6

Tags: geometry , construction , orthocenter , median , angle bisector

Restore the triangle with a compass and a ruler given the intersection point of altitudes and the feet of the median and angle bisectors drawn to one side. (No research required.)

2020 Tuymaada Olympiad, 5

Tags: parabola , construction , geometry , algebra , analytic geometry , quadratic , conic

Coordinate axes (without any marks, with the same scale) and the graph of a quadratic trinomial $y = x^2 + ax + b$ are drawn in the plane. The numbers $a$ and $b$ are not known. How to draw a unit segment using only ruler and compass?

2021 Kyiv City MO Round 1, 10.2

Tags: permutation , construction , combinatorics

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]

1971 IMO Longlists, 27

Tags: combinatorics , graph coloring , construction

Let $n \geq 2$ be a natural number. Find a way to assign natural numbers to the vertices of a regular $2n$-gon such that the following conditions are satisfied: (1) only digits $1$ and $2$ are used; (2) each number consists of exactly $n$ digits; (3) different numbers are assigned to different vertices; (4) the numbers assigned to two neighboring vertices differ at exactly one digit.