Olympiad problems

2022 Baltic Way, 8

For a natural number $n \ge 3$, we draw $n - 3$ internal diagonals in a non self-intersecting, but not necessarily convex, n-gon, cutting the $n$-gon into $n - 2$ triangles. It is known that the value (in degrees) of any angle in any of these triangles is a natural number and no two of these angle values are equal. What is the largest possible value of $n$?

1970 IMO Longlists, 8

Tags: trigonometry , polygon , trigonometric identities , geometry

Consider a regular $2n$-gon and the $n$ diagonals of it that pass through its center. Let $P$ be a point of the inscribed circle and let $a_1, a_2, \ldots , a_n$ be the angles in which the diagonals mentioned are visible from the point $P$. Prove that \[\sum_{i=1}^n \tan^2 a_i = 2n \frac{\cos^2 \frac{\pi}{2n}}{\sin^4 \frac{\pi}{2n}}.\]

1963 IMO Shortlist, 3

Tags: inequalities , geometry , polygon , geometric inequality , angle

In an $n$-gon $A_{1}A_{2}\ldots A_{n}$, all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation \[a_{1}\geq a_{2}\geq \dots \geq a_{n}. \] Prove that $a_{1}=a_{2}= \ldots= a_{n}$.

2016 Ukraine Team Selection Test, 1

Tags: combinatorics , game , polygon

Consider a regular polygon $A_1A_2\ldots A_{6n+3}$. The vertices $A_{2n+1}, A_{4n+2}, A_{6n+3}$ are called [i]holes[/i]. Initially there are three pebbles in some vertices of the polygon, which are also vertices of equilateral triangle. Players $A$ and $B$ take moves in turn. In each move, starting from $A$, the player chooses pebble and puts it to the next vertex clockwise (for example, $A_2\rightarrow A_3$, $A_{6n+3}\rightarrow A_1$). Player $A$ wins if at least two pebbles lie in holes after someone's move. Does player $A$ always have winning strategy? [i]Proposed by Bohdan Rublov [/i]

1989 IMO Shortlist, 7

Tags: geometry , angle , geometric inequality , polygon

Show that any two points lying inside a regular $ n\minus{}$gon $ E$ can be joined by two circular arcs lying inside $ E$ and meeting at an angle of at least $ \left(1 \minus{} \frac{2}{n} \right) \cdot \pi.$

1978 Austrian-Polish Competition, 9

Tags: diagonal , combinatorial geometry , polygon , combinatorics

In a convex polygon $P$ some diagonals have been drawn, without intersections inside $P$. Show that there exist at least two vertices of $P$, neither one of them being an endpoint of any one of those diagonals.

Kvant 2020, M2608

Tags: geometry , polygon

A hinged convex quadrilateral was made of four slats. Then, two points on its opposite sides were connected with another slat, but the structure remained non-rigid. Does it follow from this that this quadrilateral is a parallelogram? [i]Proposed by A. Zaslavsky[/i] [center][img width="40"]https://i.ibb.co/dgqSvLQ/Screenshot-2023-03-09-231327.png[/img][/center]

1971 IMO Longlists, 45

Tags: combinatorics , length , point set , polygon

A broken line $A_1A_2 \ldots A_n$ is drawn in a $50 \times 50$ square, so that the distance from any point of the square to the broken line is less than $1$. Prove that its total length is greater than $1248.$

2014 Czech-Polish-Slovak Junior Match, 5

Tags: polygon , square , sum , combinatorial geometry , combinatorics , angle

A square is given. Lines divide it into $n$ polygons. What is he the largest possible sum of the internal angles of all polygons?

1969 IMO Shortlist, 20

Tags: polygon , lattice points , area , geometry

$(FRA 3)$ A polygon (not necessarily convex) with vertices in the lattice points of a rectangular grid is given. The area of the polygon is $S.$ If $I$ is the number of lattice points that are strictly in the interior of the polygon and B the number of lattice points on the border of the polygon, find the number $T = 2S- B -2I + 2.$

1992 IMO Shortlist, 8

Tags: algebra , convex polygon , incircle , polygon , additive combinatorics

Show that in the plane there exists a convex polygon of 1992 sides satisfying the following conditions: [i](i)[/i] its side lengths are $ 1, 2, 3, \ldots, 1992$ in some order; [i](ii)[/i] the polygon is circumscribable about a circle. [i]Alternative formulation:[/i] Does there exist a 1992-gon with side lengths $ 1, 2, 3, \ldots, 1992$ circumscribed about a circle? Answer the same question for a 1990-gon.

KoMaL A Problems 2019/2020, A. 764

Tags: combinatorics , polygon

We call a diagonal of a polygon [i]nice[/i], if it is entirely inside the polygon or entirely outside the polygon. Let $P$ be an $n$–gon with no three of its vertices being on the same line. Prove that $P$ has at least $3(n-3)/2$ nice diagonals. [i]Proposed by Bálint Hujter, Budapest and Gábor Szűcs, Szikszó[/i]

2021/2022 Tournament of Towns, P4

Tags: geometry , polygon

A convex $n{}$-gon with $n > 4$ is such that if a diagonal cuts a triangle from it then this triangle is isosceles. Prove that there are at least 2 equal sides among any 4 sides of the $n{}$-gon. [i]Maxim Didin[/i]

1969 IMO Shortlist, 52

Tags: geometry , polygon , dissection

Prove that a regular polygon with an odd number of edges cannot be partitioned into four pieces with equal areas by two lines that pass through the center of polygon.

2020 Tournament Of Towns, 3

Tags: cyclic , inscribed , polygon , integer , geometry , combinatorial geometry , combinatorics

Is it possible to inscribe an $N$-gon in a circle so that all the lengths of its sides are different and all its angles (in degrees) are integer, where a) $N = 19$, b) $N = 20$ ? Mikhail Malkin

2009 Kyiv Mathematical Festival, 3

Tags: combinatorics , polygon

Let $AB$ be a segment of a plane. Is it possible to paint the plane in $2009$ colors in such a way that both of the following conditions are satisfied? 1) Every two points of the same color can be connected by a polygonal line. 2) For any point $C$ of $AB$, every $n \in N$ and every $k\in \{1,2,3,...,2009\}$ , there exists a point $D$, painted in $k$-th color such that the length of $CD$ is less than $0,0...01$, where all the zeros after the decimal point are exactly $n$.

2016 Peru Cono Sur TST, P2

Tags: geometry , geometric inequality , inequalities , polygon , area

Let $\omega$ be a circle. For each $n$, let $A_n$ be the area of a regular $n$-sided polygon circumscribed to $\omega$ and $B_n$ the area of a regular $n$-sided polygon inscribed in $\omega$ . Try that $3A_{2015} + B_{2015}> 4A_{4030}$

2021 Lotfi Zadeh Olympiad, 4

Tags: polygon , angle

Find the number of sequences of $0, 1$ with length $n$ satisfying both of the following properties: [list] [*] There exists a simple polygon such that its $i$-th angle is less than $180$ degrees if and only if the $i$-th element of the sequence is $1$. [*] There exists a convex polygon such that its $i$-th angle is less than $90$ degrees if and only if the $i$-th element of the sequence is $1$. [/list]

2006 IMO Shortlist, 2

Tags: combinatorics , polygon , extremal combinatorics

Let $P$ be a regular $2006$-gon. A diagonal is called [i]good[/i] if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$. The sides of $P$ are also called [i]good[/i]. Suppose $P$ has been dissected into triangles by $2003$ diagonals, no two of which have a common point in the interior of $P$. Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.

1989 IMO Longlists, 16