Olympiad problems

2018 Lusophon Mathematical Olympiad, 6

In a $3 \times 25$ board, $1 \times 3$ pieces are placed (vertically or horizontally) so that they occupy entirely $3$ boxes on the board and do not have a common point. What is the maximum number of pieces that can be placed, and for that number, how many configurations are there? [hide=original formulation] Num tabuleiro 3 × 25 s˜ao colocadas pe¸cas 1 × 3 (na vertical ou na horizontal) de modo que ocupem inteiramente 3 casas do tabuleiro e n˜ao se toquem em nenhum ponto. Qual ´e o n´umero m´aximo de pe¸cas que podem ser colocadas, e para esse n´umero, quantas configura¸c˜oes existem? [url=https://www.obm.org.br/content/uploads/2018/09/Provas_OMCPLP_2018.pdf]source[/url][/hide]

1987 Brazil National Olympiad, 4

Tags: algebra , minimum , convex hull , combinatorial geometry

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 $A_i$.

2015 Sharygin Geometry Olympiad, P22

Tags: combinatorics , combinatorial geometry , coloring

The faces of an icosahedron are painted into $5$ colors in such a way that two faces painted into the same color have no common points, even a vertices. Prove that for any point lying inside the icosahedron the sums of the distances from this point to the red faces and the blue faces are equal.

2001 All-Russian Olympiad Regional Round, 10.1

Tags: algebra , geometry , combinatorial geometry

The lengths of the sides of the polygon are $a_1$, $a_2$,. $..$ ,$a_n$. The square trinomial $f(x)$ is such that $f(a_1) = f(a_2 +...+ a_n)$. Prove that if $A$ is the sum of the lengths of several sides of a polygon, $B$ is the sum of the lengths of its remaining sides, then $f(A) = f(B)$.

Kvant 2023, M2777

Tags: combinatorial geometry , homothety , combinatorics , geometry

A convex polygon $\mathcal{P}$ with a center of symmetry $O{}$ is drawn in the plane. Prove that it is possible to place a rhombus in $\mathcal{P}$ whose image following a homothety of factor two centered at $O$ contains $\mathcal{P}$. [i]Proposed by I. Bogdanov, S. Gerdzhikov and N. Nikolov[/i]

1956 Moscow Mathematical Olympiad, 344

Tags: graph , acute , geometry , combinatorial geometry

* Let $A, B, C$ be three nodes of a graph paper. Prove that if $\vartriangle ABC$ is an acute one, then there is at least one more node either inside $\vartriangle ABC$ or on one of its sides.

1997 USAMO, 4

Tags: geometry , modular arithmetic , geometric transformation , homothety , combinatorial geometry

To [i]clip[/i] a convex $n$-gon means to choose a pair of consecutive sides $AB, BC$ and to replace them by the three segments $AM, MN$, and $NC$, where $M$ is the midpoint of $AB$ and $N$ is the midpoint of $BC$. In other words, one cuts off the triangle $MBN$ to obtain a convex $(n+1)$-gon. A regular hexagon ${\cal P}_6$ of area 1 is clipped to obtain a heptagon ${\cal P}_7$. Then ${\cal P}_7$ is clipped (in one of the seven possible ways) to obtain an octagon ${\cal P}_8$, and so on. Prove that no matter how the clippings are done, the area of ${\cal P}_n$ is greater than $\frac 13$, for all $n \geq 6$.

2011 Bundeswettbewerb Mathematik, 1

Tags: square , hexagon , tiling , combinatorial geometry , tiles , angle , geometry

Prove that you can't split a square into finitely many hexagons, whose inner angles are all less than $180^o$.

1999 IMO Shortlist, 2

Tags: geometry , point set , circles , combinatorial geometry

A circle is called a [b]separator[/b] for a set of five points in a plane if it passes through three of these points, it contains a fourth point inside and the fifth point is outside the circle. Prove that every set of five points such that no three are collinear and no four are concyclic has exactly four separators.

2002 Portugal MO, 3

Tags: combinatorial geometry , geometry , coloring , combinatorics

Daniel painted a rectangular painting measuring $2$ meters by $4$ meters with four colors. Knowing that he used more than two colors to paint the four corners of the painting, prove that he painted of the same color two points that are at least $\sqrt5$ meters

1982 IMO Longlists, 10

Tags: 3d geometry , sphere , geometric transformation , homothety , combinatorial geometry , geometry

Let $r_1, \ldots , r_n$ be the radii of $n$ spheres. Call $S_1, S_2, \ldots , S_n$ the areas of the set of points of each sphere from which one cannot see any point of any other sphere. Prove that \[\frac{S_1}{r_1^2} + \frac{S_2}{r_2^2}+\cdots+\frac{S_n}{r_n^2} = 4 \pi.\]

1996 Vietnam Team Selection Test, 1

Tags: rectangle , combinatorial geometry , area of a triangle , combinatorics , geometry

In the plane we are given $3 \cdot n$ points ($n>$1) no three collinear, and the distance between any two of them is $\leq 1$. Prove that we can construct $n$ pairwise disjoint triangles such that: The vertex set of these triangles are exactly the given 3n points and the sum of the area of these triangles $< 1/2$.

2023 Romania Team Selection Test, P4

Tags: geometry , combinatorics , combinatorial geometry

Fix a positive integer $n.{}$ Consider an $n{}$-point set $S{}$ in the plane. An [i]eligible[/i] set is a non-empty set of the form $S\cap D,{}$ where $D$ is a closed disk in the plane. In terms of $n,$ determine the smallest possible number of eligible subsets $S{}$ may contain. [i]Proposed by Cristi Săvescu[/i]

1998 Switzerland Team Selection Test, 4

Tags: square , combinatorial geometry , combinatorics

Find all numbers $n$ for which it is possible to cut a square into $n$ smaller squares.

2002 Junior Balkan Team Selection Tests - Romania, 3

Tags: combinatorics , combinatorial geometry , equilateral

A given equilateral triangle of side $10$ is divided into $100$ equilateral triangles of side $1$ by drawing parallel lines to the sides of the original triangle. Find the number of equilateral triangles, having vertices in the intersection points of parallel lines whose sides lie on the parallel lines.

2013 Tournament of Towns, 5

Tags: polygon , perpendicular , combinatorial geometry , geometry

A $101$-gon is inscribed in a circle. From each vertex of this polygon a perpendicular is dropped to the opposite side or its extension. Prove that at least one perpendicular drops to the side.

1972 IMO Longlists, 20

Tags: combinatorics , partition , point set , geometry , combinatorial geometry

Let $n_1, n_2$ be positive integers. Consider in a plane $E$ two disjoint sets of points $M_1$ and $M_2$ consisting of $2n_1$ and $2n_2$ points, respectively, and such that no three points of the union $M_1 \cup M_2$ are collinear. Prove that there exists a straightline $g$ with the following property: Each of the two half-planes determined by $g$ on $E$ ($g$ not being included in either) contains exactly half of the points of $M_1$ and exactly half of the points of $M_2.$

1997 Tournament Of Towns, (550) 4

Tags: combinatorics , combinatorial geometry , chessboard

We want to draw a number of straight lines such that for each square of a chessboard, at least one of the lines passes through an interior point of the square. What is the smallest number of lines needed for a (a) $3\times 3$; (b) $4\times 4$ chessboard? Use a picture to show that this many lines are enough, and prove that no smaller number would do. (M Vyalyi)

1948 Moscow Mathematical Olympiad, 147

Tags: combinatorial geometry , geometry , line

Consider a circle and a point $A$ outside it. We start moving from $A$ along a closed broken line consisting of segments of tangents to the circle (the segment itself should not necessarily be tangent to the circle) and terminate back at $A$. (On the links of the broken line are solid.) We label parts of the segments with a plus sign if we approach the circle and with a minus sign otherwise. Prove that the sum of the lengths of the segments of our path, with the signs given, is zero. [img]https://cdn.artofproblemsolving.com/attachments/3/0/8d682813cf7dfc88af9314498b9afcecdf77d2.png[/img]

1992 Tournament Of Towns, (344) 2

Tags: geometry , combinatorics , combinatorial geometry

On the plane a square is given, and $1993$ equilateral triangles are inscribed in this square. All vertices of any of these triangles lie on the border of the square. Prove that one can find a point on the plane belonging to the borders of no less than $499$ of these triangles. (N Sendrakyan)

Oliforum Contest V 2017, 2

Tags: square , equilateral , tiling , combinatorics , combinatorial geometry

Find all quadrilaterals which can be covered (without overlappings) with squares with side $ 1$ and equilateral triangles with side $ 1$. (Emanuele Tron)

2006 All-Russian Olympiad Regional Round, 10.8

Tags: combinatorics , combinatorial geometry , geometry , 3d geometry , polyhedron

A convex polyhedron has $2n$ faces ($n\ge 3$), and all faces are triangles. What is the largest number of vertices at which converges exactly $3$ edges at a such a polyhedron ?

2019 Saint Petersburg Mathematical Olympiad, 6

Tags: positive integer , combinatorial geometry , combinatorics , chessboard , infinite chessboard

Is it possible to arrange everything in all cells of an infinite checkered plane all natural numbers (once) so that for each $n$ in each square $n \times n$ the sum of the numbers is a multiple of $n$?

1980 Czech And Slovak Olympiad IIIA, 6

Tags: combinatorial geometry , geometry , 3d geometry , combinatorics

Let $M$ be the set of five points in space, none of which four do not lie in a plane. Let $R$ be a set of seven planes with properties: a) Each plane from the set $R$ contains at least one point of the set$ M$. b) None of the points of the set M lie in the five planes of the set $R$. Prove that there are also two distinct points $P$, $Q$, $ P \in M$, $Q \in M$, that the line $PQ$ is not the intersection of any two planes from the set $R$.

1977 Czech and Slovak Olympiad III A, 1

Tags: geometry , 3d geometry , combinatorial geometry , cube

There are given 2050 points in a unit cube. Show that there are 5 points lying in an (open) ball with the radius 1/9.