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: 51

1996 German National Olympiad, 5
Tags: construction , geometric construction , geometry
Given two non-intersecting chords $AB$ and $CD$ of a circle $k$ and a length $a <CD$. Determine a point $X$ on $k$ with the following property: If lines $XA$ and $XB$ intersect $CD$ at points $P$ and $Q$ respectively, then $PQ = a$. Show how to construct all such points $X$ and prove that the obtained points indeed have the desired property.

1976 Czech and Slovak Olympiad III A, 3
Tags: geometry , trapezoid , construction , 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.

1951 Moscow Mathematical Olympiad, 191
Tags: geometry , trapezoid , geometric construction , isosceles , isosceles trapezoid
Given an isosceles trapezoid $ABCD$ and a point $P$. Prove that a quadrilateral can be constructed from segments $PA, PB, PC, PD$. Note: It is allowed that the vertices of a quadrilateral lie not only not only on the sides of the trapezoid, but also on their extensions.

1940 Moscow Mathematical Olympiad, 060
Tags: equidistant , circle construction , geometric construction , geometry
Construct a circle equidistant from four points on a plane. How many solutions are there?

1935 Moscow Mathematical Olympiad, 002
Tags: geometric construction , geometry
Given the lengths of two sides of a triangle and that of the bisector of the angle between these sides, construct the triangle.

1962 All Russian Mathematical Olympiad, 018
Tags: geometry , geometric construction , triangle construction
Given two sides of the triangle. Construct that triangle, if medians to those sides are orthogonal.

IV Soros Olympiad 1997 - 98 (Russia), 9.10
Tags: angle bisector , geometric construction , ruler , geometry
On the plane there is an image of a circle with a marked center. Let an arbitrary angle be drawn on this plane. Using one ruler, construct the bisector of this angle.

1951 Moscow Mathematical Olympiad, 198
Tags: protractor , geometry , geometric construction , point construction
* On a plane, given points $A, B, C$ and angles $\angle D, \angle E, \angle F$ each less than $180^o$ and the sum equal to $360^o$, construct with the help of ruler and protractor a point $O$ such that $\angle AOB = \angle D, \angle BOC = \angle E$ and $\angle COA = \angle F.$

1935 Moscow Mathematical Olympiad, 005
Tags: geometry , geometric construction , square
Given three parallel straight lines. Construct a square three of whose vertices belong to these lines.

1994 Tournament Of Towns, (421) 2
Tags: ratio , fixed , geometry , geometric construction , point construction
Two circles, one inside the other, are given in the plane. Construct a point $O$, inside the inner circle, such that if a ray from $O$ cuts the circles at $A$ and $B$ respectively, then the ratio $OA/OB$ is constant. (Folklore)

1965 Czech and Slovak Olympiad III A, 2
Tags: ratio , geometry , geometric construction , right triangle , construction
Line segment $AM=d>0$ is given in the plane. Furthermore, a positive number $v$ is given. Construct a right triangle $ABC$ with hypotenuse $AB$, altitude to the hypotenuse of the length $v$ and the leg $BC$ being divided by $M$ in ration $MB/MC=2/3$. Discuss conditions of solvability in terms of $d, v$.

Novosibirsk Oral Geo Oly VIII, 2023.6
Tags: area , construction , geometric construction , geometry
Let's call a convex figure, the boundary of which consists of two segments and an arc of a circle, a mushroom-gon (see fig.). An arbitrary mushroom-gon is given. Use a compass and straightedge to draw a straight line dividing its area in half. [img]https://cdn.artofproblemsolving.com/attachments/d/e/e541a83a7bb31ba14b3637f82e6a6d1ea51e22.png[/img]

1939 Moscow Mathematical Olympiad, 050
Tags: find point , geometric construction , circles , geometry
Given two points $A$ and $B$ and a circle, find a point $X$ on the circle so that points $C$ and $D$ at which lines $AX$ and $BX$ intersect the circle are the endpoints of the chord $CD$ parallel to a given line $MN$.

1940 Moscow Mathematical Olympiad, 057
Tags: tangent , geometry , circle construction , geometric construction , tangent circle
Draw a circle that has a given radius $R$ and is tangent to a given line and a given circle. How many solutions does this problem have?

1994 Tournament Of Towns, (399) 1
Tags: geometry , geometric construction , quadrilateral construction , construction
Construct a convex quadrilateral given the lengths of all its sides and the length of the segment between the midpoints of its diagonals. (Folklore)

1960 Czech and Slovak Olympiad III A, 2
Tags: geometry , 3d geometry , sphere , symmetry , geometric construction
Consider a cube $ABCDA'B'C'D'$ (where $ABCD$ is a square and $AA' \parallel BB' \parallel CC' \parallel DD'$) and a point $P$ on the line $AA'$. Construct center $S$ of a sphere which has plane $ABB'$ as a plane of symmetry, $P$ lies on the sphere and $p = AB$, $q = A'D'$ are its tangent lines. Discuss conditions of solvability with respect to different position of the point $P$ (on line $AA'$).

1961 Czech and Slovak Olympiad III A, 2
Tags: geometry , construction , square , isosceles right triangle , geometric construction
Let a right isosceles triangle $APQ$ with the hypotenuse $AP$ be given in plane. Construct such a square $ABCD$ that the lines $BC, CD$ contain points $P, Q,$ respectively. Compute the length of side $AB = b$ in terms of $AQ=a$.

1993 Czech And Slovak Olympiad IIIA, 3
Tags: geometry , isosceles trapezoid , trapezoid , geometric construction , construction , concurrency
Let $AKL$ be a triangle such that $\angle ALK > 90^o +\angle LAK$. Construct an isosceles trapezoid $ABCD$ with $AB \parallel CD$ such that $K$ lies on the side $BC, L$ on the diagonal $AC$ and the lines $AK$ and $BL$ intersect at the circumcenter of the trapezoid.

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 $AC$.

1936 Moscow Mathematical Olympiad, 023
Tags: rectangle , triangle construction , geometric construction , isosceles
All rectangles that can be inscribed in an isosceles triangle with two of their vertices on the triangle’s base have the same perimeter. Construct the triangle.

1982 All Soviet Union Mathematical Olympiad, 339
Tags: geometric construction , parabola , geometry , conic
There is a parabola $y = x^2$ drawn on the coordinate plane. The axes are deleted. Can you restore them with the help of compass and ruler?

1941 Moscow Mathematical Olympiad, 086
Tags: geometry , triangle construction , geometric construction
Given three points $H_1, H_2, H_3$ on a plane. The points are the reflections of the intersection point of the heights of the triangle $\vartriangle ABC$ through its sides. Construct $\vartriangle ABC$.

1997 Tournament Of Towns, (552) 2
Tags: geometry , geometric construction , line construction
$M$ is the midpoint of the side $BC$ of a triangle $ABC$. Construct a line $\ell$ intersecting the triangle and parallel to $BC$ such that the segment of $\ell$ between the sides $AB$ and $AC$ is the hypotenuse of a right-angled triangle with $M$ being its third vertex. (Folklore)

1960 Czech and Slovak Olympiad III A, 3
Tags: geometry , rhombus , construction , geometric construction
Two different points $A, M$ are given in a plane, $AM = d > 0$. Let a number $v > 0$ be given. Construct a rhombus $ABCD$ with the height of length $v$ and $M$ being a midpoint of $BC$. Discuss conditions of solvability and determine number of solutions. Can the resulting quadrilateral $ABCD$ be a square?

1940 Moscow Mathematical Olympiad, 068
Tags: triangle construction , geometric construction , circumcenter , geometry
The center of the circle circumscribing $\vartriangle ABC$ is mirrored through each side of the triangle and three points are obtained: $O_1, O_2, O_3$. Reconstruct $\vartriangle ABC$ from $O_1, O_2, O_3$ if everything else is erased.