Olympiad problems

1985 Tournament Of Towns, (105) 5

(a) The point $O$ lies inside the convex polygon $A_1A_2A_3...A_n$ . Consider all the angles $A_iOA_j$ where $i, j$ are distinct natural numbers from $1$ to $n$ . Prove that at least $n- 1$ of these angles are not acute . (b) Same problem for a convex polyhedron with $n$ vertices. (V. Boltyanskiy, Moscow)

Ukrainian TYM Qualifying - geometry, VI.18

Tags: geometry , geometric inequality , convex , convex polygon , max , area

The convex polygon $A_1A_2...A_n$ is given in the plane. Denote by $T_k$ $(k \le n)$ the convex $k$-gon of the largest area, with vertices at the points $A_1, A_2, ..., A_n$ and by $T_k(A+1)$ the convex k-gon of the largest area with vertices at the points $A_1, A_2, ..., A_n$ in which one of the vertices is in $A_1$. Set the relationship between the order of arrangement in the sequence $A_1, A_2, ..., A_n$ vertices: 1) $T_3$ and $T_3 (A_2)$ 2) $T_k$ and $T_k (A_1) $ 3) $T_k$ and $T_{k+1}$

2007 IMAC Arhimede, 6

Tags: convex , convex polygon , polygon , geometry

Let $A_1A_2...A_n$ ba a polygon. Prove that there is a convex polygon $B_1B_2...B_n$ such that $B_iB_{i + 1} = A_iA_{i + 1}$ for $i \in \{1, 2,...,n-1\}$ and $B_nB_1 = A_nA_1$ (some of the successive vertices of the polygon $B_1B_2...B_n$ can be colinear).

2009 Tournament Of Towns, 1

Tags: combinatorial geometry , convex polygon , convex , diagonal , 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.

1980 All Soviet Union Mathematical Olympiad, 287

Tags: midpoint , locus , convex , area , geometry

The points $M$ and $P$ are the midpoints of $[BC]$ and $[CD]$ sides of a convex quadrangle $ABCD$. It is known that $|AM| + |AP| = a$. Prove that $ABCD$ has area less than $\frac{a^2}{2}$.

1987 All Soviet Union Mathematical Olympiad, 448

Tags: geometry , combinatorial geometry , combinatorics , odd , convex , broken line

Given two closed broken lines in the plane with odd numbers of edges. All the lines, containing those edges are different, and not a triple of them intersects in one point. Prove that it is possible to chose one edge from each line such, that the chosen edges will be the opposite sides of a convex quadrangle.

1952 Miklós Schweitzer, 1

Tags: geometry , tetrahedron , 3d geometry , pyramid , convex

Find all convex polyhedra which have no diagonals (that is, for which every segment connecting two vertices lies on the boundary of the polyhedron).

2024 Indonesia MO, 6

Tags: polygon , geometry , 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.

2001 Estonia National Olympiad, 1

Tags: polygon , angle , convex , geometry

The angles of a convex $n$-gon are $a,2a, ... ,na$. Find all possible values of $n$ and the corresponding values of $a$.

Durer Math Competition CD 1st Round - geometry, 2018.C5

Tags: geometry , convex polygon , convex

A convex $n$-gon is called [i]nice[/i] if its sides are not all the same length, and the sum of the distances of any interior point to the side lines is $1$. Find all integers $n \ge 4$ such that a nice $n$-gon exists .

2005 Greece JBMO TST, 1

Tags: convex polygon , convex , geometry

Examine if we can place $9$ convex $6$-angled polygons the one next to the other (with common only one side or part of her) to construct a convex $39$-angled polygon.

2004 Thailand Mathematical Olympiad, 4

Tags: geometry , convex , area

Let $ABCD$ be a convex quadrilateral. Prove that area $(ABCD) \le \frac{AB^2 + BC^2 + CD^2 + DA^2}{4}$

2017 Sharygin Geometry Olympiad, P21

Tags: geometry , midpoint , convex , hexagon , maximum

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 $r.$

1947 Moscow Mathematical Olympiad, 126

Tags: pentagon , combinatorial geometry , convex , line , combinatorics

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.

2002 Austrian-Polish Competition, 2

Tags: geometry , convex polygon , convex

Let $P_{1}P_{2}\dots P_{2n}$ be a convex polygon with an even number of corners. Prove that there exists a diagonal $P_{i}P_{j}$ which is not parallel to any side of the polygon.

1998 Bundeswettbewerb Mathematik, 4

Tags: geometry , convex , polyhedron , combinatorial geometry

Let $3(2^n -1)$ points be selected in the interior of a polyhedron $P$ with volume $2^n$, where n is a positive integer. Prove that there exists a convex polyhedron $U$ with volume $1$, contained entirely inside $P$, which contains none of the selected points.

2008 Swedish Mathematical Competition, 1

Tags: geometry , rhombus , convex , 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 $d_2$.

1976 Czech and Slovak Olympiad III A, 5

Tags: polygon , geometry , inequalities , perimeter , geometric inequalities , convex

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 $o_1\le o_2.$

1987 Brazil National Olympiad, 2

Tags: geometry , convex , polyhedron

Given a point $p$ inside a convex polyhedron $P$. Show that there is a face $F$ of $P$ such that the foot of the perpendicular from $p$ to $F$ lies in the interior of $F$.

2005 Switzerland - Final Round, 5

Tags: geometry , combinatorial geometry , geometric inequalities , convex

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.

2010 Mathcenter Contest, 4

Tags: geometry , functional , convex

Let $P$ be a plane. Prove that there is no function $f :P\rightarrow P$ where, for any convex quadrilateral $ABCD$, the points $f(A)$, $f(B)$, $f(C)$, $f (D)$ are the vertices of a concave quadrilateral. [i](tatari/nightmare)[/i]

Estonia Open Senior - geometry, 2004.1.3

Tags: geometry , radii , equal circles , convex

a) Does there exist a convex quadrangle $ABCD$ satisfying the following conditions (1) $ABCD$ is not cyclic; (2) the sides $AB, BC, CD$ and $DA$ have pairwise different lengths; (3) the circumradii of the triangles $ABC, ADC, BAD$ and $BCD$ are equal? b) Does there exist such a non-convex quadrangle?

1995 Bundeswettbewerb Mathematik, 3

Tags: geometry , pentagon , convex , parallel , diagonal

Each diagonal of a convex pentagon is parallel to one side of the pentagon. Prove that the ratio of the length of a diagonal to that of its corresponding side is the same for all five diagonals, and compute this ratio.

1994 Tuymaada Olympiad, 4

Tags: geometry , convex , polyhedron , sphere

Let a convex polyhedron be given with volume $V$ and full surface $S$. Prove that inside a polyhedron it is possible to arrange a ball of radius $\frac{V}{S}$.

2005 Estonia National Olympiad, 4

Tags: parallelogram , geometry , circumcenter , convex

In a fixed plane, consider a convex quadrilateral $ABCD$. Choose a point $O$ in the plane and let $K, L, M$, and $N$ be the circumcentres of triangles $AOB, BOC, COD$, and $DOA$, respectively. Prove that there exists exactly one point $O$ in the plane such that $KLMN$ is a parallelogram.