Olympiad problems

1987 Yugoslav Team Selection Test, Problem 3

Let there be given lines $a,b,c$ in the space, no two of which are parallel. Suppose that there exist planes $\alpha,\beta,\gamma$ which contain $a,b,c$ respectively, which are perpendicular to each other. Construct the intersection point of these three planes. (A space construction permits drawing lines, planes and spheres and translating objects for any vector.)

Ukrainian TYM Qualifying - geometry, 2010.12

Tags: geometry , construction , ruler , tangent

On the plane is drawn a triangle $ABC$ and a circle $\omega$ passing through the vertex $C$, the midpoints of the sides $AC$ and $BC$ and the point of intersection of the medians of the triangle $ABC$. The point $K$ lies on the circle $\omega$ such that $\angle AKB=90^o$. Using only with a ruler, draw a tangent to the circle $\omega$ at point $K$.

III Soros Olympiad 1996 - 97 (Russia), 10.5

Tags: geometry , construction , circles

A circle is drawn on a plane, the center of which is not indicated. On this circle, point $A$ is marked and a second circle with center at $A$ is constructed. The second circle has a radius greater than the radius of the first and intersects the first at two points. Construct the center of the first circle using only a compass, drawing no more than five more circles.

2019 All-Russian Olympiad, 3

Tags: combinatorics , construction

An interstellar hotel has $100$ rooms with capacities $101,102,\ldots, 200$ people. These rooms are occupied by $n$ people in total. Now a VIP guest is about to arrive and the owner wants to provide him with a personal room. On that purpose, the owner wants to choose two rooms $A$ and $B$ and move all guests from $A$ to $B$ without exceeding its capacity. Determine the largest $n$ for which the owner can be sure that he can achieve his goal no matter what the initial distribution of the guests is.

1939 Eotvos Mathematical Competition, 3

Tags: construction , semicircle , equal segments , geometry

$ABC$ 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 $CA'= CB'$.

2001 Czech And Slovak Olympiad IIIA, 2

Tags: construction , right triangle , geometric construction , incircle , geometry

Given a triangle $PQX$ in the plane, with $PQ = 3, PX = 2.6$ and $QX = 3.8$. Construct a right-angled triangle $ABC$ such that the incircle of $\vartriangle ABC$ touches $AB$ at $P$ and $BC$ at $Q$, and point $X$ lies on the line $AC$.

1955 Polish MO Finals, 5

Tags: construction , geometric inequalities , geometry

In the plane, a straight line $ m $ is given and points $ A $ and $ B $ lie on opposite sides of the straight line $ m $. Find a point $ M $ on the line $ m $ such that the difference in distances of this point from points $ A $ and $ B $ is as large as possible.

1963 Poland - Second Round, 2

Tags: geometry , construction , parallelogram

In the plane there is a quadrilateral $ ABCD $ and a point $ M $. Construct a parallelogram with center $ M $ and its vertices lying on the lines $ AB $, $ BC $, $ CD $, $ DA $.

Kyiv City MO Juniors Round2 2010+ geometry, 2014.89.3

Tags: construction , bisects segment , geometry

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)

1976 Czech and Slovak Olympiad III A, 3

Tags: construction , geometry , trapezoid , geometric construction

Consider a half-plane with the boundary line $p$ and two points $M,N$ in it such that the distances $Mp$ and $Np$ are different. Construct a trapezoid $MNPQ$ with area $MN^2$ such that $P,Q\in p.$ Discuss conditions of solvability.

2019 India Regional Mathematical Olympiad, 2

Tags: geometry , construction , constructive geometry

Given a circle $\tau$, let $P$ be a point in its interior, and let $l$ be a line through $P$. Construct with proof using ruler and compass, all circles which pass through $P$, are tangent to $\tau$ and whose center lies on line $l$.

1968 Poland - Second Round, 2

Tags: construction , geometry , geometric construction , orthocenter

Given a circle $ k $ and a point inside it $ H $. Inscribe a triangle in the circle such that this point $ H $ is the point of intersection of the triangle's altitudes.

1983 Bundeswettbewerb Mathematik, 2

Tags: geometry , inradius , circumradius , right triangle , construction

The radii of the circumcircle and the incircle of a right triangle are given. Cconstruct that triangle with compass and ruler, describe the construction and justify why it is correct.

2024 Middle European Mathematical Olympiad, 1

Tags: construction , algebra , function , monotone , sequence , functional inequality

Let $\mathbb{N}_0$ denote the set of non-negative integers. Determine all non-negative integers $k$ for which there exists a function $f: \mathbb{N}_0 \to \mathbb{N}_0$ such that $f(2024) = k$ and $f(f(n)) \leq f(n+1) - f(n)$ for all non-negative integers $n$.

2021 Yasinsky Geometry Olympiad, 5

Tags: geometry , incenter , bisects segment , construction , ruler

Circle $\omega$ is inscribed in the $\vartriangle ABC$, with center $I$. Using only a ruler, divide segment $AI$ in half. (Grigory Filippovsky)

2011 Sharygin Geometry Olympiad, 5

Tags: geometry , angle bisector , construction

It is possible to compose a triangle from the altitudes of a given triangle. Can we conclude that it is possible to compose a triangle from its bisectors?

2025 India National Olympiad, P6

Tags: quadratic residue , base system , construction

Let $b \geqslant 2$ be a positive integer. Anu has an infinite collection of notes with exactly $b-1$ copies of a note worth $b^k-1$ rupees, for every integer $k\geqslant 1$. A positive integer $n$ is called payable if Anu can pay exactly $n^2+1$ rupees by using some collection of her notes. Prove that if there is a payable number, there are infinitely many payable numbers. [i]Proposed by Shantanu Nene[/i]

1969 Dutch Mathematical Olympiad, 4

Tags: geometric inequality , geometry , perimeter , construction

An angle $< 45^o$ is given in the plane of the drawing. Furthermore, the projection $P_1$ of a point $P$ lying above the plane of the drawing and the distance from $P$ to $P_1$ are given. $P_1$ lies within the given angle. On the legs of the given angle, construct points $A$ and $B$, respectively, such that the triangle $PAB$ has a minimal perimeter.

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.

2021 Yasinsky Geometry Olympiad, 5

Tags: geometry , construction

A circle is circumscribed around an isosceles triangle $ABC$ with base $BC$. The bisector of the angle $C$ and the bisector of the angles $A$ intersect the circle at the points $E$ and $D$, respectively, and the segment $DE$ intersects the sides $BC$ and $AB$ at the points $P$ and $Q$, respectively. Reconstruct $\vartriangle ABC$ given points $D, P, Q$, if it is known in which half-plane relative to the line $DQ$ lies the vertex $A$. (Maria Rozhkova)

Ukrainian TYM Qualifying - geometry, XI.4

Tags: geometry , construction , arc , ratio

Chords $AB$ and $CD$, which do not intersect, are drawn in a circle. On the chord $AB$ or on its extension is taken the point $E$. Using a compass and construct the point $F$ on the arc $AB$ , such that $\frac{PE}{EQ} = \frac{m}{n}$, where $m,n$ are given natural numbers, $P$ is the point of intersection of the chord $AB$ with the chord $FC$, $Q$ is the point of intersection of the chord $AB$ with the chord $FD$. Consider cases where $E\in PQ$ and $E \notin PQ$.

2022 Kyiv City MO Round 1, Problem 1

Tags: algebra , construction , fraction

Represent $\frac{1}{2021}$ as a difference of two irreducible fractions with smaller denominators. [i](Proposed by Bogdan Rublov)[/i]