Olympiad problems

1990 Tournament Of Towns, (259) 3

A cake is prepared for a dinner party to which only $p$ or $q$ persons will come ($p$ and $q$ are given co-prime integers). Find the minimum number of pieces (not necessarily equal) into which the cake must be cut in advance so that the cake may be equally shared between the persons in either case. (D. Fomin, Leningrad)

1995 Poland - Second Round, 6

Tags: combinatorial geometry , tiling , combinatorics

Determine all positive integers $n$ for which the square $n \times n$ can be cut into squares $2\times 2$ and $3\times3$ (with the sides parallel to the sides of the big square).

1996 Swedish Mathematical Competition, 6

Tags: rectangle , combinatorics , combinatorial geometry , tiles , tiling , geometry

A rectangle is tiled with rectangles of size $6\times 1$. Prove that one of its side lengths is divisible by $6$.

1987 Brazil National Olympiad, 4

Tags: minimum , combinatorial geometry , algebra , convex hull

Given points $A_1 (x_1, y_1, z_1), A_2 (x_2, y_2, z_2), .., A_n (x_n, y_n, z_n)$ let $P (x, y, z)$ be the point which minimizes $\Sigma ( |x - x_i| + |y -y_i| + |z -z_i| )$. Give an example (for each $n > 4$) of points $A_i $ for which the point $P$ lies outside the convex hull of the points $A_i$.

1989 All Soviet Union Mathematical Olympiad, 488

Tags: combinatorial geometry , tiling , combinatorics

Can $77$ blocks each $3 \times 3 \times1$ be assembled to form a $7 \times 9 \times 11$ block?

2020 Ukrainian Geometry Olympiad - April, 5

Tags: geometry , combinatorial geometry , point , coloring , combinatorics

On the plane painted $101$ points in brown and another $101$ points in green so that there are no three lying on one line. It turns out that the sum of the lengths of all $5050$ segments with brown ends equals the length of all $5050$ segments with green ends equal to $1$, and the sum of the lengths of all $10201$ segments with multicolored equals $400$. Prove that it is possible to draw a straight line so that all brown points are on one side relative to it and all green points are on the other.

2016 Thailand Mathematical Olympiad, 4

Tags: combinatorial geometry , combinatorics , isosceles triangle , coloring , isosceles

Each point on the plane is colored either red, green, or blue. Prove that there exists an isosceles triangle whose vertices all have the same color.

1995 Tournament Of Towns, (479) 3

Tags: geometric inequality , combinatorial geometry , geometry , combinatorics , rectangle

A rectangle with sides of lengths $a$ and $b$ ($a > b$) is cut into rightangled triangles so that any two of these triangles either have a common side, a common vertex or no common points. Moreover, any common side of two triangles is a leg of one of them and the hypotenuse of the other. Prove that $a > 2b$. (A Shapovalov)

2012 Kyiv Mathematical Festival, 1

Tags: geometry , combinatorial geometry , circles

Is it possible to place $2012$ distinct circles with the same diameter on the plane, such that each circle touches at least three others circles?

1991 Mexico National Olympiad, 6

Tags: geometry , combinatorial geometry , set

Given an $n$-gon ($n\ge 4$), consider a set $T$ of triangles formed by vertices of the polygon having the following property: Every two triangles in T have either two common vertices, or none. Prove that $T$ contains at most $n$ triangles.

2004 All-Russian Olympiad, 1

Tags: rotation , geometry , combinatorics , combinatorial geometry , analytic geometry , geometric transformation

Each grid point of a cartesian plane is colored with one of three colors, whereby all three colors are used. Show that one can always find a right-angled triangle, whose three vertices have pairwise different colors.

2020 Thailand TST, 6

Tags: combinatorial geometry , angle , combinatorics

Let $n>1$ be an integer. Suppose we are given $2n$ points in the plane such that no three of them are collinear. The points are to be labelled $A_1, A_2, \dots , A_{2n}$ in some order. We then consider the $2n$ angles $\angle A_1A_2A_3, \angle A_2A_3A_4, \dots , \angle A_{2n-2}A_{2n-1}A_{2n}, \angle A_{2n-1}A_{2n}A_1, \angle A_{2n}A_1A_2$. We measure each angle in the way that gives the smallest positive value (i.e. between $0^{\circ}$ and $180^{\circ}$). Prove that there exists an ordering of the given points such that the resulting $2n$ angles can be separated into two groups with the sum of one group of angles equal to the sum of the other group.

1981 All Soviet Union Mathematical Olympiad, 314

Tags: combinatorial geometry , coloring , table , square , combinatorics

Is it possible to fill a rectangular table with black and white squares (only) so, that the number of black squares will equal to the number of white squares, and each row and each column will have more than $75\%$ squares of the same colour?

2019 Iran Team Selection Test, 5

Tags: combinatorics , combinatorial geometry , perimeter

Let $P$ be a simple polygon completely in $C$, a circle with radius $1$, such that $P$ does not pass through the center of $C$. The perimeter of $P$ is $36$. Prove that there is a radius of $C$ that intersects $P$ at least $6$ times, or there is a circle which is concentric with $C$ and have at least $6$ common points with $P$. [i]Proposed by Seyed Reza Hosseini[/i]

May Olympiad L2 - geometry, 2019.5

Tags: combinatorics , combinatorial geometry , regular polygon

We consider the $n$ vertices of a regular polygon with $n$ sides. There is a set of triangles with vertices at these $n$ points with the property that for each triangle in the set, the sides of at least one are not the side of any other triangle in the set. What is the largest amount of triangles that can have the set? [hide=original wording]Consideramos los n vértices de un polígono regular de n lados. Se tiene un conjunto de triángulos con vértices en estos n puntos con la propiedad que para cada triángulo del conjunto, al menos uno de sus lados no es lado de ningún otro triángulo del conjunto. ¿Cuál es la mayor cantidad de triángulos que puede tener el conjunto?[/hide]

2023 Chile National Olympiad, 4

Tags: geometry , combinatorics , combinatorial geometry

Inside a square with side $60$, $121$ points are drawn. Prove them are three points that are vertices of a triangle of area not exceeding $30$.

1987 Tournament Of Towns, (145) 2

Tags: combinatorial geometry , cover , disc , distance

Α disk of radius $1$ is covered by seven identical disks. Prove that their radii are not less than $\frac12$ .

1983 Polish MO Finals, 1

Tags: polygon , combinatorial geometry , triangle , combinatorics

On the plane are given a convex $n$-gon $P_1P_2....P_n$ and a point $Q$ inside it, not lying on any of its diagonals. Prove that if $n$ is even, then the number of triangles $P_iP_jP_k$ containing the point $Q$ is even.

1976 Yugoslav Team Selection Test, Problem 1

Tags: combinatorial geometry , combinatorics , geometry

Prove that for a given convex polygon of area $A$ and perimeter $P$ there exists a circle of radius $\frac AP$ that is contained in the interior of the polygon.

2018 Romania Team Selection Tests, 3

Tags: combinatorics , combinatorial geometry , ratio

Divide the plane into $1$x$1$ squares formed by the lattice points. Let$S$ be the set-theoretic union of a finite number of such cells, and let $a$ be a positive real number less than or equal to 1/4.Show that S can be covered by a finite number of squares satisfying the following three conditions: 1) Each square in the cover is an array of $1$x$1$ cells 2) The squares in the cover have pairwise disjoint interios and 3)For each square $Q$ in the cover the ratio of the area $S \cap Q$ to the area of Q is at least $a$ and at most $a {(\lfloor a^{-1/2} \rfloor)} ^2$

2025 China National Olympiad, 4

Tags: combinatorial geometry , combinatorics

The [i]fractional distance[/i] between two points $(x_1,y_1)$ and $(x_2,y_2)$ is defined as \[ \sqrt{ \left\| x_1 - x_2 \right\|^2 + \left\| y_1 - y_2 \right\|^2},\]where $\left\| x \right\|$ denotes the distance between $x$ and its nearest integer. Find the largest real $r$ such that there exists four points on the plane whose pairwise fractional distance are all at least $r$.

KoMaL A Problems 2022/2023, A. 851

Tags: algebra , polynomial , counting , combinatorial geometry , combinatorics

Let $k$, $\ell $ and $m$ be positive integers. Let $ABCDEF$ be a hexagon that has a center of symmetry whose angles are all $120^\circ$ and let its sidelengths be $AB=k$, $BC=\ell$ and $CD=m$. Let $f(k,\ell,m)$ denote the number of ways we can partition hexagon $ABCDEF$ into rhombi with unit sides and an angle of $120^\circ$. Prove that by fixing $\ell$ and $m$, there exists polynomial $g_{\ell,m}$ such that $f(k,\ell,m)=g_{\ell,m}(k)$ for every positive integer $k$, and find the degree of $g_{\ell,m}$ in terms of $\ell$ and $m$. [i]Submitted by Zoltán Gyenes, Budapest[/i]

1945 Moscow Mathematical Olympiad, 094

Tags: geometry , combinatorial geometry , impossible , congruent triangles , divide

Prove that it is impossible to divide a scalene triangle into two equal triangles.

ICMC 5, 3

Tags: symmetry , combinatorial geometry , geometry , 3d geometry

A set of points has [i]point symmetry[/i] if a reflection in some point maps the set to itself. Let $\cal P$ be a solid convex polyhedron whose orthogonal projections onto any plane have point symmetry. Prove that $\cal P$ has point symmetry. [i]Proposed by Ethan Tan[/i]

1996 North Macedonia National Olympiad, 4

Tags: geometry , polygon , parallel , combinatorial geometry

A polygon is called [i]good [/i] if it satisfies the following conditions: (i) All its angles are in $(0,\pi)$ or in $(\pi ,2\pi)$, (ii) It is not self-intersecing, (iii) For any three sides, two are parallel and equal. Find all $n$ for which there exists a [i]good [/i] $n$-gon.