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: 85

2014 Sharygin Geometry Olympiad, 3
Tags: diagonal , convex , polyhedron , geometry
Do there exist convex polyhedra with an arbitrary number of diagonals (a diagonal is a segment joining two vertices of a polyhedron and not lying on the surface of this polyhedron)? (A. Blinkov)

1947 Moscow Mathematical Olympiad, 126
Tags: combinatorial geometry , combinatorics , convex , pentagon , line
Given a convex pentagon $ABCDE$, prove that if an arbitrary point $M$ inside the pentagon is connected by lines with all the pentagon’s vertices, then either one or three or five of these lines cross the sides of the pentagon opposite the vertices they pass. Note: In reality, we need to exclude the points of the diagonals, because that in this case the drawn lines can pass not through the internal points of the sides, but through the vertices. But if the drawn diagonals are not considered or counted twice (because they are drawn from two vertices), then the statement remains true.

1989 Bundeswettbewerb Mathematik, 3
Tags: polygon , geometry , convex quadrilateral , convex
A convex polygon is divided into finitely many quadrilaterals. Prove that at least one of these quadrilaterals must also be convex.

2017 Sharygin Geometry Olympiad, P21
Tags: geometry , convex , hexagon , maximum , midpoint
A convex hexagon is circumscribed about a circle of radius $1$. Consider the three segments joining the midpoints of its opposite sides. Find the greatest real number $r$ such that the length of at least one segment is at least $r.$

2006 Miklós Schweitzer, 10
Tags: convex , compactness , real analysis , euclidean space
Let $K_1,...,K_d$ be convex, compact sets in $R^d$ with non-empty interior. Suppose they are strongly separated, which means for any choice of $x_1 \in K_1, x_2 \in K_2, ...$, their affine hull is a hyperplane in $R^d$. Also let $0< \alpha_i <1$. A half-space H is called an $\alpha$-cut if $vol(K_i \cap H) = \alpha_i\cdot vol(K_i)$ for all i. How many $\alpha$-cuts are there?

2011 Bundeswettbewerb Mathematik, 3
Tags: pentagon , geometry , convex , trisector , diagonal
The diagonals of a convex pentagon divide each of its interior angles into three equal parts. Does it follow that the pentagon is regular?

1988 All Soviet Union Mathematical Olympiad, 464
Tags: equal area , geometry , convex , area
$ABCD$ is a convex quadrilateral. The midpoints of the diagonals and the midpoints of $AB$ and $CD$ form another convex quadrilateral $Q$. The midpoints of the diagonals and the midpoints of $BC$ and $CA$ form a third convex quadrilateral $Q'$. The areas of $Q$ and $Q'$ are equal. Show that either $AC$ or $BD$ divides $ABCD$ into two parts of equal area.

1976 Czech and Slovak Olympiad III A, 5
Tags: geometry , perimeter , geometric inequalities , convex , polygon , inequalities
Let $\mathbf{P}_1,\mathbf{P}_2$ be convex polygons with perimeters $o_1,o_2,$ respectively. Show that if $\mathbf P_1\subseteq\mathbf P_2,$ then $o_1\le o_2.$

2009 Tournament Of Towns, 1
Tags: convex polygon , convex , diagonal , combinatorial geometry , even , geometry
In a convex $2009$-gon, all diagonals are drawn. A line intersects the $2009$-gon but does not pass through any of its vertices. Prove that the line intersects an even number of diagonals.

2007 Sharygin Geometry Olympiad, 5
Tags: combinatorial geometry , convex , polyhedron , 3-dimensional geometry , geometry
Each edge of a convex polyhedron is shifted such that the obtained edges form the frame of another convex polyhedron. Are these two polyhedra necessarily congruent?

2004 Thailand Mathematical Olympiad, 4
Tags: convex , geometry , area
Let $ABCD$ be a convex quadrilateral. Prove that area $(ABCD) \le \frac{AB^2 + BC^2 + CD^2 + DA^2}{4}$

2005 Switzerland - Final Round, 5
Tags: geometric inequalities , combinatorial geometry , convex , geometry
Tweaking a convex $n$-gon means the following: choose two adjacent sides $AB$ and $BC$ and replaces them with the line segment $AM$, $MN$, $NC$, where $M \in AB$ and $N \in BC$ are arbitrary points inside these segments. In other words, you cut off a corner and get an $(n+1)$-corner. Starting from a regular hexagon $P_6$ with area $1$, by continuous Tweaks a sequence $P_6,P_7,P_8, ...$ convex polygons. Show that Area of $​​P_n$ for all $n\ge 6$ greater than $\frac1 2$ is, regardless of how tweaks takes place.

2015 Ukraine Team Selection Test, 9
Tags: point , heptagon , pentagon , convex , combinatorics , combinatorial geometry
The set $M$ consists of $n$ points on the plane and satisfies the conditions: $\bullet$ there are $7$ points in the set $M$, which are vertices of a convex heptagon, $\bullet$ for arbitrary five points with $M$, which are vertices of a convex pentagon, there is a point that also belongs to $M$ and lies inside this pentagon. Find the smallest possible value that $n$ can take .

2011 Tournament of Towns, 3
Tags: geometry , convex , quadrilateral , angle
In a convex quadrilateral $ABCD, AB = 10, BC = 14, CD = 11$ and $DA = 5$. Determine the angle between its diagonals.

2024 Indonesia MO, 6
Tags: geometry , polygon , inequality , convex , inequalities
Suppose $A_1 A_2 \ldots A_n$ is an $n$-sided polygon with $n \geq 3$ and $\angle A_j \leq 180^{\circ}$ for each $j$ (in other words, the polygon is convex or has fewer than $n$ distinct sides). For each $i \leq n$, suppose $\alpha_i$ is the smallest possible value of $\angle{A_i A_j A_{i+1}}$ where $j$ is neither $i$ nor $i+1$. (Here, we define $A_{n+1} = A_1$.) Prove that \[ \alpha_1 + \alpha_2 + \cdots + \alpha_n \leq 180^{\circ} \] and determine all equality cases.

1999 German National Olympiad, 4
Tags: perimeter , geometric inequalities , inequalities , convex , polygon , geometry
A convex polygon $P$ is placed inside a unit square $Q$. Prove that the perimeter of $P$ does not exceed $4$.

2002 Regional Competition For Advanced Students, 3
Tags: geometry , perimeter , inequalities , convex , hexagon , geometric inequality
In the convex $ABCDEF$ (has all interior angles less than $180^o$) with the perimeter $s$ the triangles $ACE$ and $BDF$ have perimeters $u$ and $v$ respectively. a) Show the inequalities $\frac{1}{2} \le \frac{s}{u+v}\le 1$ b) Check whether $1$ is replaced by a smaller number or $1/2$ by a larger number can the inequality remains valid for all convex hexagons.

2003 All-Russian Olympiad Regional Round, 9.8
Tags: geometry , combinatorics , combinatorial geometry , convex , convex polygon , acute
Prove that a convex polygon can be cut by disjoint diagonals into acute triangles in at least one way.

1995 Romania Team Selection Test, 4
Tags: combinatorial geometry , combinatorics , coloring , convex
A convex set $S$ on a plane, not lying on a line, is painted in $p$ colors. Prove that for every $n \ge 3$ there exist infinitely many congruent $n$-gons whose vertices are of the same color.

2001 Regional Competition For Advanced Students, 3
Tags: convex , pentagon , area of a triangle , area , geometry
In a convex pentagon $ABCDE$, the area of the triangles $ABC, ABD, ACD$ and $ADE$ are equal and have the value $F$. What is the area of the triangle $BCE$ ?

2006 Sharygin Geometry Olympiad, 8.5
Tags: convex polygon , diagonal , side , geometry , convex
Is there a convex polygon with each side equal to some diagonal, and each diagonal equal to some side?

1962 Putnam, A1
Tags: point , geometry , convex
Consider $5$ points in the plane, such that there are no $3$ of them collinear. Prove that there is a convex quadrilateral with vertices at $4$ points.

2014 Oral Moscow Geometry Olympiad, 3
Tags: geometry , convex , pentagon , diagonal
Is there a convex pentagon in which each diagonal is equal to a side?

1955 Moscow Mathematical Olympiad, 295
Tags: combinatorial geometry , convex , line , geometry
Which convex domains (figures) on a plane can contain an entire straight line? It is assumed that the figure is flat and does not degenerate into a straight line and is closed, that is, it contains all its boundary points.

2004 Swedish Mathematical Competition, 6
Tags: geometric inequalities , convex , area , geometry
Prove that every convex $n$-gon of area $1$ contains a quadrilateral of area at least $\frac12 $. .