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: 48

2011 Sharygin Geometry Olympiad, 4
Tags: geometry , regular polygon , hexagon , inscribed , combinatorics , circles , chord
Given the circle of radius $1$ and several its chords with the sum of lengths $1$. Prove that one can be inscribe a regular hexagon into that circle so that its sides don’t intersect those chords.

1956 Moscow Mathematical Olympiad, 330
Tags: geometric inequality , geometry , inradius , square , inscribed
A square of side $a$ is inscribed in a triangle so that two of the square’s vertices lie on the base, and the other two lie on the sides of the triangle. Prove that if $r$ is the radius of the circle inscribed in the triangle, then $r\sqrt2 < a < 2r$.

Ukrainian TYM Qualifying - geometry, I.7
Tags: ratio , geometry , polygon , inscribed , area
Given a natural number $n> 3$. On the plane are considered convex $n$ - gons $F_1$ and $F_2$ such that on each side of $F_1$ lies one vertex of $F_2$ and no two vertices $F_1$ and $F_2$ coincide. For each $n$, determine the limits of the ratio of the areas of the polygons $F_1$ and $F_2$. For each $n$, determine the range of the areas of the polygons $F_1$ and $F_2$, if $F_1$ is a regular $n$-gon. Determine the set of values of this value for other partial cases of the polygon $F_1$.

2010 Saudi Arabia BMO TST, 2
Tags: geometry , geometric inequality , inscribed , square , area
Let $ABC$ be an acute triangle and let $MNPQ$ be a square inscribed in the triangle such that $M ,N \in BC$, $P \in AC$, $Q \in AB$. Prove that $area \, [MNPQ] \le \frac12 area\, [ABC]$.

2020 Tournament Of Towns, 3
Tags: cyclic , inscribed , polygon , integer , geometry , combinatorial geometry , combinatorics
Is it possible to inscribe an $N$-gon in a circle so that all the lengths of its sides are different and all its angles (in degrees) are integer, where a) $N = 19$, b) $N = 20$ ? Mikhail Malkin

2021 Iranian Geometry Olympiad, 2
Tags: inscribed , geometry , square , area
Points $K, L, M, N$ lie on the sides $AB, BC, CD, DA$ of a square $ABCD$, respectively, such that the area of $KLMN$ is equal to one half of the area of $ABCD$. Prove that some diagonal of $KLMN$ is parallel to some side of $ABCD$. [i]Proposed by Josef Tkadlec - Czech Republic[/i]

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.

1952 Poland - Second Round, 3
Tags: geometry , rectangle , inscribed , square , rhombus
Are the following statements true? a) if the four vertices of a rectangle lie on the four sides of a rhombus, then the sides of the rectangle are parallel to the diagonals of the rhombus; b) if the four vertices of a square lie on the four sides of a rhombus that is not a square, then the sides of the square are parallel to the diagonals of the rhombus.

Kyiv City MO Seniors 2003+ geometry, 2010.11.3
Tags: congruent , inscribed , orthocenter , geometry
The quadrilateral $ABCD$ is inscribed in a circle and has perpendicular diagonals. Points $K,L,M,Q$ are the points of intersection of the altitudes of the triangles $ABD, ACD, BCD, ABC$, respectively. Prove that the quadrilateral $KLMQ$ is equal to the quadrilateral $ABCD$. (Rozhkova Maria)

Ukrainian TYM Qualifying - geometry, IV.7
Tags: max , min , geometric inequality , inscribed , perimeter , geometry
Let $ABCD$ be the quadrilateral whose area is the largest among the quadrilaterals with given sides $a, b, c, d$, and let $PORS$ be the quadrilateral inscribed in $ABCD$ with the smallest perimeter. Find this perimeter.

2006 Tournament of Towns, 5
Tags: 3d geometry , octahedron , cube , inscribed , geometry
Can a regular octahedron be inscribed in a cube in such a way that all vertices of the octahedron are on cube's edges? (4)

1984 Tournament Of Towns, (070) T4
Tags: geometry , rectangle , perimeter , geometric inequality , diagonal , inscribed
Inside a rectangle is inscribed a quadrilateral, which has a vertex on each side of the rectangle. Prove that the perimeter of the inscribed quadrilateral is not smaller than double the length of a diagonal of the rectangle. (V. V . Proizvolov , Moscow)

1979 All Soviet Union Mathematical Olympiad, 269
Tags: isosceles , inscribed , area , geometry
What is the least possible ratio of two isosceles triangles areas, if three vertices of the first one belong to three different sides of the second one?

2014 Belarusian National Olympiad, 3
Tags: geometry , perimeter , inscribed , geometric inequality
The angles at the vertices $A$ and $C$ in the convex quadrilateral $ABCD$ are not acute. Points $K, L, M$ and $N$ are marked on the sides $AB, BC, CD$ and $DA$ respectively. Prove that the perimeter of $KLMN$ is not less than the double length of the diagonal $AC$.

1999 All-Russian Olympiad Regional Round, 8.3
Tags: geometry , inscribed , ratio , median
On sides $BC$, $CA$, $AB$ of triangle $ABC$, points $A_1$, $B_1$, $C_1$ are chosen, respectively, so that the medians $A_1A_2$, $B_1B_2$, $C_1C_2$ of the triangle $A_1B_1C_1$ are respectively parallel to straight lines $AB$, $BC$, $CA$. Determine in what ratio points $A_1$, $B_1$, $C_1$ divide the sides of the triangle $ABC$.

1975 Kurschak Competition, 2
Tags: geometry , rhombus , inscribed , geometric inequalities
Prove or disprove: given any quadrilateral inscribed in a convex polygon, we can find a rhombus inscribed in the polygon with side not less than the shortest side of the quadrilateral.

2006 Bosnia and Herzegovina Team Selection Test, 2
Tags: geometry , rectangle , inscribed , locus
It is given a triangle $\triangle ABC$. Determine the locus of center of rectangle inscribed in triangle $ABC$ such that one side of rectangle lies on side $AB$.

1949 Moscow Mathematical Olympiad, 170
Tags: geometry , symmetry , inscribed , polygon
What is a centrally symmetric polygon of greatest area one can inscribe in a given triangle?

2005 Sharygin Geometry Olympiad, 8
Tags: geometry , rectangle , inscribed
Around the convex quadrilateral $ABCD$, three rectangles are circumscribed . It is known that two of these rectangles are squares. Is it true that the third one is necessarily a square? (A rectangle is circumscribed around the quadrilateral $ABCD$ if there is one vertex $ABCD$ on each side of the rectangle).

2010 Oral Moscow Geometry Olympiad, 5
Tags: 3d geometry , pyramid , inscribed , cylinder , geometry
All edges of a regular right pyramid are equal to $1$, and all vertices lie on the side surface of a (infinite) right circular cylinder of radius $R$. Find all possible values of $R$.

2008 Swedish Mathematical Competition, 1
Tags: convex , geometry , rhombus , inscribed
A rhombus is inscribed in a convex quadrilateral. The sides of the rhombus are parallel with the diagonals of the quadrilateral, which have the lengths $d_1$ and $d_2$. Calculate the length of side of the rhombus , expressed in terms of $d_1$ and $d_2$.

2005 Thailand Mathematical Olympiad, 1
Tags: geometry , trapezoid , inscribed , isosceles trapezoid
Let $ABCD$ be a trapezoid inscribed in a unit circle with diameter $AB$. If $DC = 4AD$, compute $AD$.

1985 Polish MO Finals, 6
Tags: 3d geometry , geometry , polyhedron , sphere , inscribed
There is a convex polyhedron with $k$ faces. Show that if more than $k/2$ of the faces are such that no two have a common edge, then the polyhedron cannot have an inscribed sphere.