Olympiad problems

1997 Tournament Of Towns, (527) 4

A square is cut into 25 smaller squares, exactly 24 of which are unit squares. Find the area of the original square. (V Proizvolov)

1974 All Soviet Union Mathematical Olympiad, 193

Tags: vector , combinatorial geometry , combinatorics

Given $n$ vectors of unit length in the plane. The length of their total sum is less than one. Prove that you can rearrange them to provide the property: [i]for every[/i] $k, k\le n$[i], the length of the sum of the first[/i] $k$ [i]vectors is less than[/i] $2$.

2022 China Team Selection Test, 6

Tags: combinatorial geometry , complex number , geometry

(1) Prove that, on the complex plane, the area of the convex hull of all complex roots of $z^{20}+63z+22=0$ is greater than $\pi$. (2) Let $a_1,a_2,\ldots,a_n$ be complex numbers with sum $1$, and $k_1<k_2<\cdots<k_n$ be odd positive integers. Let $\omega$ be a complex number with norm at least $1$. Prove that the equation \[ a_1 z^{k_1}+a_2 z^{k_2}+\cdots+a_n z^{k_n}=w \] has at least one complex root with norm at most $3n|\omega|$.

2016 NZMOC Camp Selection Problems, 1

Tags: combinatorial geometry , combinatorics , coloring , equilateral

Suppose that every point in the plane is coloured either black or white. Must there be an equilateral triangle such that all of its vertices are the same colour?

1968 All Soviet Union Mathematical Olympiad, 111

Tags: combinatorial geometry , rectangle , combinatorics

The city is a rectangle divided onto squares by $m$ streets coming from the West to the East and $n$ streets coming from the North to the South. There are militioners (policemen) on the streets but not on the crossroads. They watch the certain automobile, moving along the closed route, marking the time and the direction of its movement. Its trace is not known in advance, but they know, that it will not pass over the same segment of the way twice. What is the minimal number of the militioners providing the unique determination of the route according to their reports?

KoMaL A Problems 2017/2018, A. 710

Tags: combinatorial geometry , combinatorics

For which $n{}$ can we partition a regular $n{}$-gon into finitely many triangles such that no two triangles share a side? [i]Based on a problem of the 2017 Miklós Schweitzer competition[/i]

1971 Poland - Second Round, 1

Tags: combinatorics , combinatorial geometry , chessboard

In how many ways can you choose $ k $ squares on a chessboard $ n \times n $ ( $ k \leq n $) so that no two of the chosen squares lie in the same row or column?

1995 Argentina National Olympiad, 6

Tags: combinatorics , combinatorial geometry

The $27$ points $(a,b,c)$ of the space are marked such that $a$, $b$ and $c$ take the values $0$, $1$ or $2$. We will call these points "junctures". Using $54$ rods of length $1$, all the joints that are at a distance of $1$ are joined together. A cubic structure of $2\times 2\times 2$ is thus formed. An ant starts from a juncture $A$ and moves along the rods; When it reaches a juncture it turns $90^\circ$ and changes rod. If the ant returns to $A$ and has not visited any juncture more than once except $A$, which it visited $2$ times, at the beginning of the walk and at the end of it, what is the greatest length that the path of the ant can have?

1947 Moscow Mathematical Olympiad, 129

Tags: square , chessboard , combinatorics , combinatorial geometry

How many squares different in size or location can be drawn on an $8 \times 8$ chess board? Each square drawn must consist of whole chess board’s squares.

1994 Tournament Of Towns, (407) 5

Tags: combinatorial geometry , geometry , pentagon , similar

Does there exist a convex pentagon from which a similar pentagon can be cut off by a straight line? (S Tokarev)

2020/2021 Tournament of Towns, P4

Tags: combinatorial geometry , southern tournament , combinatorics

The $X{}$ pentomino consists of five $1\times1$ squares where four squares are all adjacent to the fifth one. Is it possible to cut nine such pentominoes from an $8\times 8$ chessboard, not necessarily cutting along grid lines? (The picture shows how to cut three such $X{}$ pentominoes.) [i]Alexandr Gribalko[/i]

1997 Slovenia Team Selection Test, 3

Tags: combinatorics , combinatorial geometry , coloring , point

Let $A_1,A_2,...,A_n$ be $n \ge 2$ distinct points on a circle. Find the number of colorings of these points with $p \ge 2$ colors such that every two adjacent points receive different colors

2018 Latvia Baltic Way TST, P7

Tags: combinatorial geometry , combinatorics

Let $n \ge 3$ points be given in the plane, no three of which lie on the same line. Determine whether it is always possible to draw an $n$-gon whose vertices are the given points and whose sides do not intersect. [i]Remark.[/i] The $n$-gon can be concave.

IV Soros Olympiad 1997 - 98 (Russia), 11.9

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

Cut pyramid $ABCD$ into $8$ equal and similar pyramids, if: a) $AB = BC = CD$, $\angle ABC =\angle BCD = 90^o$, dihedral angle at edge $BC$ is right b) all plane angles at vertex $B$ are right and $AB = BC = BD\sqrt2$. Note. Whether there are other types of triangular pyramids that can be cut into any number similar to the original pyramids (their number is not necessarily $8$ and the pyramids are not necessarily equal to each other) is currently unknown

2001 All-Russian Olympiad Regional Round, 10.4

Tags: geometry , combinatorics , combinatorial geometry

Three families of parallel lines are drawn,$10$ lines each, are drawn. What is the greatest number of triangles they can cut from plane?

1991 China Team Selection Test, 3

Tags: invariant , induction , combinatorial geometry , combinatorics

$5$ points are given in the plane, any three non-collinear and any four non-concyclic. If three points determine a circle that has one of the remaining points inside it and the other one outside it, then the circle is said to be [i]good[/i]. Let the number of good circles be $n$; find all possible values of $n$.

2000 Harvard-MIT Mathematics Tournament, 13

Tags: combinatorics , combinatorial geometry , geometry

Let $P_1, P_2,..., P_n$ be a convex $n$-gon. If all lines $P_iP_j$ are joined, what is the maximum possible number of intersections in terms of $n$ obtained from strictly inside the polygon?

2016 Thailand Mathematical Olympiad, 4

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

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.

1951 Kurschak Competition, 3

Tags: combinatorics , combinatorial geometry

An open half-plane is the set of all points lying to one side of a line, but excluding the points on the line itself. If four open half-planes cover the plane, show that one can select three of them which still cover the plane.

1970 Kurschak Competition, 1

Tags: combinatorics , combinatorial geometry , angle

What is the largest possible number of acute angles in an $n$-gon which is not selfintersecting (no two non-adjacent edges interesect)?

2010 IMAC Arhimede, 1

Tags: point , combinatorics , combinatorial geometry , collinear

$3n$ points are given ($n\ge 1$) in the plane, each $3$ of them are not collinear. Prove that there are $n$ distinct triangles with the vertices those points.

1981 Czech and Slovak Olympiad III A, 2

Tags: combinatorial geometry , geometry , closed , segment

Let $n$ be a positive integer. Consider $n^2+1$ (closed, i.e. including endpoints) segments on a single line. Show that at least one of the following statements holds: a) there are $n+1$ segments with non-empty intersection, b) there are $n+1$ segments among which two of them are disjoint.

2015 Postal Coaching, Problem 4

Tags: combinatorics , combinatorial geometry

For an integer $n \geq 5,$ two players play the following game on a regular $n$-gon. Initially, three consecutive vertices are chosen, and one counter is placed on each. A move consists of one player sliding one counter along any number of edges to another vertex of the $n$-gon without jumping over another counter. A move is legal if the area of the triangle formed by the counters is strictly greater after the move than before. The players take turns to make legal moves, and if a player cannot make a legal move, that player loses. For which values of $n$ does the player making the first move have a winning strategy?

1983 Swedish Mathematical Competition, 5

Tags: geometry , combinatorial geometry , combinatorics , disks , geometric inequalities , min , disc

Show that a unit square can be covered with three equal disks with radius less than $\frac{1}{\sqrt{2}}$. What is the smallest possible radius?

2016 Latvia Baltic Way TST, 11

Tags: square , combinatorics , combinatorial geometry , rectangle , geometry

Is it possible to cut a square with side $\sqrt{2015}$ into no more than five pieces so that these pieces can be rearranged into a rectangle with sides of integer length? (The cuts should be made using straight lines, and flipping of the pieces is disallowed.)