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

2023 Turkey MO (2nd round), 5
Tags: construction , set , algebra
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 $2422$?

2023 4th Memorial "Aleksandar Blazhevski-Cane", P4
Tags: construction , number theory
Does the equation $$z(y-x)(x+y)=x^3$$ have finitely many solutions in the set of positive integers? [i]Proposed by Nikola Velov[/i]

Kvant 2022, M2707
Tags: construction , algebra
Prove that infinitely many positive integers can be represented as $(a-1)/b + (b-1)/c + (c-1)/a$, where $a$, $b$ and $c$ are pairwise distinct positive integers greater than 1.

2022 Taiwan TST Round 2, 4
Tags: construction , number theory , combinatorics
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.

Swiss NMO - geometry, 2015.4
Tags: construction , circles , geometry
Given a circle $k$ and two points $A$ and $B$ outside the circle. Specify how to can construct a circle with a compass and ruler, so that $A$ and $B$ lie on that circle and that circle is tangent to $k$.

2004 BAMO, 2
Tags: geometry , straightedge , perpendicular , construction
A given line passes through the center $O$ of a circle. The line intersects the circle at points $A$ and $B$. Point $P$ lies in the exterior of the circle and does not lie on the line $AB$. Using only an unmarked straightedge, construct a line through $P$, perpendicular to the line $AB$. Give complete instructions for the construction and prove that it works.

2019 Yasinsky Geometry Olympiad, p5
Tags: right triangle , construction , geometry
In a right triangle $ABC$ with a hypotenuse $AB$, the angle $A$ is greater than the angle $B$. Point $N$ lies on the hypotenuse $AB$ , such that $BN = AC$. Construct this triangle $ABC$ given the point $N$, point $F$ on the side $AC$ and a straight line $\ell$ containing the bisector of the angle $A$ of the triangle $ABC$. (Grigory Filippovsky)

2012 Sharygin Geometry Olympiad, 3
Tags: angle bisector , construction , geometry
In triangle $ABC$, the bisector $CL$ was drawn. The incircles of triangles $CAL$ and $CBL$ touch $AB$ at points $M$ and $N$ respectively. Points $M$ and $N$ are marked on the picture, and then the whole picture except the points $A, L, M$, and $N$ is erased. Restore the triangle using a compass and a ruler. (V.Protasov)

2017 Czech-Polish-Slovak Junior Match, 4
Tags: construction , geometry , trapezoid
Bolek draw a trapezoid $ABCD$ trapezoid ($AB // CD$) on the board, with its midsegment line $EF$ in it. Point intersection of his diagonal $AC, BD$ denote by $P,$ and his rectangular projection on line $AB$ denote by $Q$. Lolek, wanting to tease Bolek, blotted from the board everything except segments $EF$ and $PQ$. When Bolek saw it, wanted to complete the drawing and draw the original trapezoid, but did not know how to do it. Can you help Bolek?

Ukrainian TYM Qualifying - geometry, 2018.16
Tags: geometry , construction
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 $M$.

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

1957 Polish MO Finals, 5
Tags: trisection , construction , geometry
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.

1959 IMO, 4
Tags: construction , median , trigonometry , geometry
Construct a right triangle with given hypotenuse $c$ such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle.

1975 Chisinau City MO, 106
Tags: construction , quadrilateral construction , geometry , square
Construct a square from four points, one on each side.

1949-56 Chisinau City MO, 34
Tags: construction , geometry
Construct a triangle by its altitude , median and angle bisector originating from one vertex.

1995 Tournament Of Towns, (476) 4
Tags: construction , geometry , 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)

2006 Oral Moscow Geometry Olympiad, 4
Tags: construction , equal circles , inradius , geometry
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)

1958 Poland - Second Round, 6
Tags: construction , geometry
In a plane, two circles $ C_1 $ and $ C_2 $ and a line $ m $ are given. Find a point on the line $ m $ from which one can draw tangents to the circles $ C_1 $ and $ C_2 $ with equal inclination to the line $ m $.

2017 Oral Moscow Geometry Olympiad, 3
Tags: construction , ruler , circumcircle , geometry
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.

1968 IMO Shortlist, 18
Tags: 3d geometry , construction , concurrency , geometry
If an acute-angled triangle $ABC$ is given, construct an equilateral triangle $A'B'C'$ in space such that lines $AA',BB', CC'$ pass through a given point.

Kyiv City MO Juniors 2003+ geometry, 2004.9.7
Tags: construction , geometry
The board depicts the triangle $ABC$, the altitude $AH$ and the angle bisector $AL$ which intersectthe inscribed circle in the triangle at the points $M$ and $N, P$ and $Q$, respectively. After that, the figure was erased, leaving only the points $H, M$ and $Q$. Restore the triangle $ABC$. (Bogdan Rublev)

1951 Polish MO Finals, 6
Tags: construction , similar , geometry , similar triangles
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 $.

Durer Math Competition CD 1st Round - geometry, 2009.C3
Tags: construction , altitude , geometry
We know the lengths of the $3$ altitudes of a triangle. Construct the triangle.

2016 Sharygin Geometry Olympiad, P11
Tags: symmedian , construction , circumcircle , geometry
Restore a triangle $ABC$ by vertex $B$, the centroid and the common point of the symmedian from $B$ with the circumcircle.

2023 Novosibirsk Oral Olympiad in Geometry, 6
Tags: construction , geometric construction , area , 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]