This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 297

2001 Czech And Slovak Olympiad IIIA, 2
Tags: geometric construction , construction , right triangle , geometry , incircle
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 $AC$.

1970 Dutch Mathematical Olympiad, 1
Tags: construction , square , geometry
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 $D$.

2016 Sharygin Geometry Olympiad, 7
Tags: construction , symmedian point , circumcenter , circumcircle , geometric transformation , reflection , geometry
Restore a triangle by one of its vertices, the circumcenter and the Lemoine's point. [i](The Lemoine's point is the intersection point of the reflections of the medians in the correspondent angle bisectors)[/i]

1980 Brazil National Olympiad, 2
Tags: number theory , reciprocal sum , construction
Show that for any positive integer $n > 2$ we can find $n$ distinct positive integers such that the sum of their reciprocals is $1$.

1996 Tuymaada Olympiad, 4
Tags: compass , construction , geometry
Given a segment of length $7\sqrt3$ . Is it possible to use only compass to construct a segment of length $\sqrt7$?

Kyiv City MO 1984-93 - geometry, 1991.8.4
Tags: geometry , square , construction
Construct a square, if you know its center and two points that lie on adjacent sides.

1962 Dutch Mathematical Olympiad, 1
Tags: geometry , parallel , construction , right triangle , tangent
Given a triangle $ABC$ with $\angle C = 90^o$. a) Construct the circle with center $C$, so that one of the tangents from $A$ to that circle is parallel to one of the tangents from $B$ to that circle. b) A circle with center $C$ has two parallel tangents passing through A and go respectively. If $AC = b$ and $BC = a$, express the radius of the circle in terms of $a$ and $b$.

2020 Princeton University Math Competition, A2
Tags: geometry , rectangle , construction
Helen has a wooden rectangle of unknown dimensions, a straightedge, and a pencil (no compass). Is it possible for her to construct a line segment on the rectangle connecting the midpoints of two opposite sides, where she cannot draw any lines or points outside the rectangle? Note: Helen is allowed to draw lines between two points she has already marked, and mark the intersection of any two lines she has already drawn, if the intersection lies on the rectangle. Further, Helen is allowed to mark arbitrary points either on the rectangle or on a segment she has previously drawn. Assume that only the four vertices of the rectangle have been marked prior to the beginning of this process.

2025 Bangladesh Mathematical Olympiad, P9
Tags: combinatorics , construction , game
Suppose there are several juice boxes, one of which is poisoned. You have $n$ guinea pigs to test the boxes. The testing happens in the following way: [list] [*] At each round, you can have the guinea pigs taste any number of juice boxes. [*] Conversely, a juice box can be tasted by any number of guinea pigs. [*] After the round ends, any guinea pigs who tasted the poisoned juice die. [/list] Suppose you have to find the poisoned juice box in at most $k$ rounds. What is the maximum number of juice boxes such that it is possible?

2019 Yasinsky Geometry Olympiad, p6
Tags: geometry , construction , angle , angle bisector , circumcircle
In an acute triangle $ABC$ , the bisector of angle $\angle A$ intersects the circumscribed circle of the triangle $ABC$ at the point $W$. From point $W$ , a parallel is drawn to the side $AB$, which intersects this circle at the point $F \ne W$. Describe the construction of the triangle $ABC$, if given are the segments $FA$ , $FW$ and $\angle FAC$. (Andrey Mostovy)

1995 Tournament Of Towns, (476) 4
Tags: geometry , construction , geometric inequality
Three different points $A$, $B$ and $C$ are placed in the plane. Construct a line $m$ through $C$ so that the product of the distances from $A$ and $B$ to $m$ has the maximal value. Is $m$ unique for every triple $A$, $B$ and $C$? (NB Vassiliev)

2022 Yasinsky Geometry Olympiad, 3
Tags: geometry , construction
Reconstruct the triangle$ ABC$, in which $\angle B - \angle C = 90^o$ , by the orthocenter $H$ and points $M_1$ and $L_1$ the feet of the median and angle bisector drawn from vertex $A$, respectively. (Gryhoriy Filippovskyi)

2004 Oral Moscow Geometry Olympiad, 2
Tags: geometry , construction
Construct a triangle $ABC$ given angle $A$ and the medians drawn from vertices $B$ and $C$.

Ukrainian TYM Qualifying - geometry, 2018.16
Tags: construction , geometry
Let $K, T$ be the points of tangency of inscribed and exscribed circles to the side $BC$ triangle $ABC$, $M$ is the midpoint of the side $BC$. Using a compass and a ruler, construct triangle ABC given rays $AK$ and $AT$ (points $K, T$ are not marked on them) and point $M$.

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)

2011 Sharygin Geometry Olympiad, 8
Tags: geometry , construction , division
Using only the ruler, divide the side of a square table into $n$ equal parts. All lines drawn must lie on the surface of the table.

2017 Yasinsky Geometry Olympiad, 4
Tags: geometry , construction , distance
Three points are given on the plane. With the help of compass and ruler construct a straight line in this plane, which will be equidistant from these three points. Explore how many solutions have this construction.

1974 Spain Mathematical Olympiad, 6
Tags: geometry , construction , trisect , chord
Two chords are drawn in a circle of radius equal to unit, $AB$ and $AC$ of equal length. a) Describe how you can construct a third chord $DE$ that is divided into three equal parts by the intersections with $AB$ and $AC$. b) If $AB = AC =\sqrt2$, what are the lengths of the two segments that the chord $DE$ determines in $AB$?

2011 Sharygin Geometry Olympiad, 3
Tags: isosceles , construction , isosceles triangle , geometry
Restore the isosceles triangle $ABC$ ($AB = AC$) if the common points $I, M, H$ of bisectors, medians and altitudes respectively are given.

1959 Czech and Slovak Olympiad III A, 1
Tags: median , construction , triangle
Construct a triangle $ABC$ with the right angle at vertex $C$ given lengths of its medians $m_a$, $m_b$. Discuss conditions of solvability.

1957 Polish MO Finals, 5
Tags: geometry , trisection , construction
Given a line $ m $ and a segment $ AB $ parallel to it. Divide the segment $ AB $ into three equal parts using only a ruler, i.e. drawing only the lines.

2023 Turkey MO (2nd round), 5
Tags: algebra , construction , set
Is it possible that a set consisting of $23$ real numbers has a property that the number of the nonempty subsets whose product of the elements is rational number is exactly $2422$?

2017 Oral Moscow Geometry Olympiad, 3
Tags: geometry , circumcircle , construction , ruler
On the plane, a non-isosceles triangle is given, a circle circumscribed around it and the center of its inscribed circle are marked. Using only a ruler without tick marks and drawing no more than seven lines, construct the diameter of the circumcircle.

2019 Oral Moscow Geometry Olympiad, 3
Tags: geometry , circumcircle , construction
Restore the acute triangle $ABC$ given the vertex $A$, the foot of the altitude drawn from the vertex $B$ and the center of the circle circumscribed around triangle $BHC$ (point $H$ is the orthocenter of triangle $ABC$).

Kyiv City MO 1984-93 - geometry, 1989.8.5
Tags: geometry , right triangle , equilateral , inscribed , construction
The student drew a right triangle $ABC$ on the board with a right angle at the vertex $B$ and inscribed in it an equilateral triangle $KMP$ such that the points $K, M, P$ lie on the sides $AB, BC, AC$, respectively, and $KM \parallel AC$. Then the picture was erased, leaving only points $A, P$ and $C$. Restore erased points and lines.