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: convex , polyhedron , diagonal , 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)

2006 Miklós Schweitzer, 10
Tags: real analysis , convex , compactness , 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?

Estonia Open Junior - geometry, 2017.1.5
Tags: polygon , geometry , convex , acute
Find all possibilities: how many acute angles can there be in a convex polygon?

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

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

1996 Austrian-Polish Competition, 5
Tags: geometry , 3d geometry , polyhedron , convex , sphere
A sphere $S$ divides every edge of a convex polyhedron $P$ into three equal parts. Show that there exists a sphere tangent to all the edges of $P$.

2009 Estonia Team Selection Test, 3
Tags: geometry , polyhedron , regular polygon , convex , 3d geometry
Find all natural numbers $n$ for which there exists a convex polyhedron satisfying the following conditions: (i) Each face is a regular polygon. (ii) Among the faces, there are polygons with at most two different numbers of edges. (iii) There are two faces with common edge that are both $n$-gons.

2009 Oral Moscow Geometry Olympiad, 5
Tags: geometry , convex , polyhedron
Prove that any convex polyhedron has three edges that can be used to form a triangle. (Barbu Bercanu, Romania)

KoMaL A Problems 2018/2019, A. 737
Tags: graph theory , convex , polyhedron , point set , 3-dimensional geometry , combinatorics
$100$ points are given in space such that no four of them lie in the same plane. Consider those convex polyhedra with five vertices that have all vertices from the given set. Prove that the number of such polyhedra is even.

2004 Estonia National Olympiad, 3
Tags: geometry , bisects area , convex , parallel
On the sides $AB , BC$ of the convex quadrilateral $ABCD$ lie points $M$ and $N$ such that $AN$ and $CM$ each divide the quadrilateral $ABCD$ into two equal area parts. Prove that the line $MN$ and $AC$ are parallel.