Olympiad problems

1995 Mexico National Olympiad, 3

$A, B, C, D$ are consecutive vertices of a regular $7$-gon. $AL$ and $AM$ are tangents to the circle center $C$ radius $CB$. $N$ is the intersection point of $AC$ and $BD$. Show that $L, M, N$ are collinear.

2015 Irish Math Olympiad, 2

Tags: combinatorial geometry , coloring , regular polygon , combinatorics

A regular polygon with $n \ge 3$ sides is given. Each vertex is coloured either red, green or blue, and no two adjacent vertices of the polygon are the same colour. There is at least one vertex of each colour. Prove that it is possible to draw certain diagonals of the polygon in such a way that they intersect only at the vertices of the polygon and they divide the polygon into triangles so that each such triangle has vertices of three different colours.

2022 Iran Team Selection Test, 7

Tags: algebra , regular polygon

Suppose that $n$ is a positive integer number. Consider a regular polygon with $2n$ sides such that one of its largest diagonals is parallel to the $x$-axis. Find the smallest integer $d$ such that there is a polynomial $P$ of degree $d$ whose graph intersects all sides of the polygon on points other than vertices. Proposed by Mohammad Ahmadi

2006 Sharygin Geometry Olympiad, 10

Tags: diagonal , regular polygon , cut , isosceles , geometry , combinatorial geometry

At what $n$ can a regular $n$-gon be cut by disjoint diagonals into $n- 2$ isosceles (including equilateral) triangles?

2005 iTest, 22

Tags: regular polygon , angle , geometry

A regular $n$-gon has $135$ diagonals. What is the measure of its exterior angle, in degrees? (An exterior angle is the supplement of an interior angle.)

2023 Israel TST, P1

Tags: combinatorics , regular polygon

A regular polygon with $20$ vertices is given. Alice colors each vertex in one of two colors. Bob then draws a diagonal connecting two opposite vertices. Now Bob draws perpendicular segments to this diagonal, each segment having vertices of the same color as endpoints. He gets a fish from Alice for each such segment he draws. How many fish can Bob guarantee getting, no matter Alice's goodwill?

2021 Israel TST, 1

Tags: geometry , regular polygon

Let $ABCDEFGHIJ$ be a regular $10$-gon. Let $T$ be a point inside the $10$-gon, such that the $DTE$ is isosceles: $DT = ET$ , and its angle at the apex is $72^\circ$. Prove that there exists a point $S$ such that $FTS$ and $HIS$ are both isosceles, and for both of them the angle at the apex is $72^\circ$.

2024 TASIMO, 3

Tags: combinatorics , regular polygon

$Abdulqodir$ cut out $2024$ congruent regular $n-$gons from a sheet of paper and placed these $n-$gons on the table such that some parts of each of these $n-$gons may be covered by others. We say that a vertex of one of the afore-mentioned $n-$gons is $visible$ if it is not in the interior of another $n-$gon that is placed on top of it. For any $n>2$ determine the minimum possible number of visible vertices. \\ Proposed by David Hrushka, Slovakia

Ukrainian TYM Qualifying - geometry, III.13

Tags: geometry , regular polygon , tangent circle

Inside the regular $n$ -gon $M$ with side $a$ there are $n$ equal circles so that each touches two adjacent sides of the polygon $M$ and two other circles. Inside the formed "star", which is bounded by arcs, these $n$ equal circles are reconstructed so that each touches the two adjacent circles built in the previous step, and two more newly built circles. This process will take $k$ steps. Find the area $S_n (k)$ of the "star", which is formed in the center of the polygon $M$. Consider the spatial analogue of this problem.

2000 Tournament Of Towns, 2

Tags: geometry , 3d geometry , regular polygon , cube , combinatorics , combinatorial geometry

What is the largest integer $n$ such that one can find $n$ points on the surface of a cube, not all lying on one face and being the vertices of a regular $n$-gon? (A Shapovalov)

2012 Bundeswettbewerb Mathematik, 4

Tags: combinatorial geometry , regular polygon , trapezoid , isosceles triangle , geometry , isosceles , combinatorics

From the vertices of a regular 27-gon, seven are chosen arbitrarily. Prove that among these seven points there are three points that form an isosceles triangle or four points that form an isosceles trapezoid.

2017 JBMO Shortlist, C1

Tags: combinatorics , regular polygon , coloring

Consider a regular $2n + 1$-gon $P$ in the plane, where n is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$, if the line segment $SE$ contains no other points that lie on the sides of $P$ except $S$. We want to color the sides of $P$ in $3$ colors, such that every side is colored in exactly one color, and each color must be used at least once. Moreover, from every point in the plane external to $P$, at most $2$ different colors on $P$ can be seen (ignore the vertices of $P$, we consider them colorless). Find the largest positive integer for which such a coloring is possible.

1984 Brazil National Olympiad, 3

Tags: dodecahedron , regular polygon , 3d geometry , geometry , rectangle

Given a regular dodecahedron of side $a$. Take two pairs of opposite faces: $E, E' $ and $F, F'$. For the pair $E, E'$ take the line joining the centers of the faces and take points $A$ and $C$ on the line each a distance $m$ outside one of the faces. Similarly, take $B$ and $D$ on the line joining the centers of $F, F'$ each a distance $m$ outside one of the faces. Show that $ABCD$ is a rectangle and find the ratio of its side lengths.

2010 Sharygin Geometry Olympiad, 8

Tags: combinatorics , point , polygon , regular polygon , minimum , perimeter , geometry

Given is a regular polygon. Volodya wants to mark $k$ points on its perimeter so that any another regular polygon (maybe having a different number of sides) doesn’t contain all marked points on its perimeter. Find the minimal $k$ sufficient for any given polygon.

1996 Spain Mathematical Olympiad, 6

Tags: geometry , regular polygon , volume , 3-dimensional geometry

A regular pentagon is constructed externally on each side of a regular pentagon of side $1$. The figure is then folded and the two edges of the external pentagons meeting at each vertex of the original pentagon are glued together. Find the volume of water that can be poured into the obtained container.

2012 Tournament of Towns, 6

Tags: combinatorial geometry , geometry , centroid , regular polygon , perpendicular

(a) A point $A$ is marked inside a circle. Two perpendicular lines drawn through $A$ intersect the circle at four points. Prove that the centre of mass of these four points does not depend on the choice of the lines. (b) A regular $2n$-gon ($n \ge 2$) with centre $A$ is drawn inside a circle (A does not necessarily coincide with the centre of the circle). The rays going from $A$ to the vertices of the $2n$-gon mark $2n$ points on the circle. Then the $2n$-gon is rotated about $A$. The rays going from $A$ to the new locations of vertices mark new $2n$ points on the circle. Let $O$ and $N$ be the centres of gravity of old and new points respectively. Prove that $O = N$.

1983 Austrian-Polish Competition, 9

Tags: symmetry , combinatorics , combinatorial geometry , rectangle , coloring , regular polygon , geometry

To each side of the regular $p$-gon of side length $1$ there is attached a $1 \times k$ rectangle, partitioned into $k$ unit cells, where $k$ and $p$ are given positive integers and p an odd prime. Let $P$ be the resulting nonconvex star-like polygonal figure consisting of $kp + 1$ regions ($kp$ unit cells and the $p$-gon). Each region is to be colored in one of three colors, adjacent regions having different colors. Furthermore, it is required that the colored figure should not have a symmetry axis. In how many ways can this be done?

2012 IFYM, Sozopol, 3

Tags: regular polygon , geometry

In a circle with radius 1 a regular n-gon $A_1 A_2...A_n$ is inscribed. Calculate the product: $A_1 A_2.A_1 A_3 \dots A_1 A_{n-1} .A_1 A_n$.

2025 AIME, 11

Tags: regular polygon , geometry

Let $S$ be the set of vertices of a regular $24$-gon. Find the number of ways to draw $12$ segments of equal lengths so that each vertex in $S$ is an endpoint of exactly one of the $12$ segments.

1985 Czech And Slovak Olympiad IIIA, 1

Tags: geometry , combinatorial geometry , regular polygon , combinatorics

A regular $1985$-gon is given in the plane. Let's pass a straight line through each side of it. Determine the number of parts into which these lines divide the plane.

2000 Austrian-Polish Competition, 5

Tags: combinatorics , coloring , regular polygon

For which integers $n \ge 5$ is it possible to color the vertices of a regular$ n$-gon using at most $6$ colors in such a way that any $5$ consecutive vertices have different colors?

2020 Junior Balkan Team Selection Tests - Moldova, 3

Tags: combinatorics , regular polygon

Let there be a regular polygon of $n$ sides with center $O$. Determine the highest possible number of vertices $k$ $(k \geq 3)$, which can be coloured in green, such that $O$ is strictly outside of any triangle with $3$ vertices coloured green. Determine this $k$ for $a) n=2019$ ; $b) n=2020$.

2007 Sharygin Geometry Olympiad, 7

Tags: geometry , convex polygon , inscribed circle , regular polygon

A convex polygon is circumscribed around a circle. Points of contact of its sides with the circle form a polygon with the same set of angles (the order of angles may differ). Is it true that the polygon is regular?

1984 Bundeswettbewerb Mathematik, 2

Tags: combinatorial geometry , regular polygon , geometry

Given is a regular $n$-gon with circumradius $1$. $L$ is the set of (different) lengths of all connecting segments of its endpoints. What is the sum of the squares of the elements of $L$?

2000 Switzerland Team Selection Test, 11

Tags: combinatorics , regular polygon

The vertices of a regular $2n$-gon ($n \ge 3$) are labelled with the numbers $1,2,...,2n$ so that the sum of the numbers at any two adjacent vertices equals the sum of the numbers at the vertices diametrically opposite to them. Show that this is only possible if $n$ is odd.