Olympiad problems

2016 Ecuador NMO (OMEC), 2

Tags: polygon , geometry

All diagonals are plotted in a $2017$-sided convex polygon. A line $\ell$ intersects said polygon but does not pass through any of its vertices. Show that the line $\ell$ intersects an even number of diagonals of said polygon.

1945 Moscow Mathematical Olympiad, 100

Tags: polygon , minimum , combinatorial geometry , geometry

Suppose we have two identical cardboard polygons. We placed one polygon upon the other one and aligned. Then we pierced polygons with a pin at a point. Then we turned one of the polygons around this pin by $25^o 30'$. It turned out that the polygons coincided (aligned again). What is the minimal possible number of sides of the polygons?

1984 All Soviet Union Mathematical Olympiad, 384

Tags: polygon , perimeter , geometry , area , circles , disc

The centre of the coin with radius $r$ is moved along some polygon with the perimeter $P$, that is circumscribed around the circle with radius $R$ ($R>r$). Find the coin trace area (a sort of polygon ring).

2023 Novosibirsk Oral Olympiad in Geometry, 5

Tags: polygon , geometry

A circle of length $10$ is inscribed in a convex polygon with perimeter $15$. What part of the area of this polygon is occupied by the resulting circle?

1978 All Soviet Union Mathematical Olympiad, 261

Tags: geometric inequality , geometry , perimeter , polygon , cyclic

Given a circle with radius $R$ and inscribed $n$-gon with area $S$. We mark one point on every side of the given polygon. Prove that the perimeter of the polygon with the vertices in the marked points is not less than $2S/R$.

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.

1977 IMO Shortlist, 12

Tags: complex number , geometry , square , polygon

In the interior of a square $ABCD$ we construct the equilateral triangles $ABK, BCL, CDM, DAN.$ Prove that the midpoints of the four segments $KL, LM, MN, NK$ and the midpoints of the eight segments $AK, BK, BL, CL, CM, DM, DN, AN$ are the 12 vertices of a regular dodecagon.

1992 IMO Longlists, 29

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.

2010 IFYM, Sozopol, 5

Tags: combinatorics , polygon , coloring

Each vertex of a right $n$-gon $(n\geq 3)$ is colored in yellow, blue or red. On each turn are chosen two adjacent vertices in different color and then are recolored in the third. For which $n$ can we get from an arbitrary coloring of the $n$-gon a monochromatic one (in one color)?

2010 IMO Shortlist, 3

Tags: geometry , circumcircle , inequality , polygon

Let $A_1A_2 \ldots A_n$ be a convex polygon. Point $P$ inside this polygon is chosen so that its projections $P_1, \ldots , P_n$ onto lines $A_1A_2, \ldots , A_nA_1$ respectively lie on the sides of the polygon. Prove that for arbitrary points $X_1, \ldots , X_n$ on sides $A_1A_2, \ldots , A_nA_1$ respectively, \[\max \left\{ \frac{X_1X_2}{P_1P_2}, \ldots, \frac{X_nX_1}{P_nP_1} \right\} \geq 1.\] [i]Proposed by Nairi Sedrakyan, Armenia[/i]

2013 Saudi Arabia GMO TST, 2

Tags: combinatorics , combinatorial geometry , geometry , equal angles , polygon , angle

Find all values of $n$ for which there exists a convex cyclic non-regular polygon with $n$ vertices such that the measures of all its internal angles are equal.

1994 IMO Shortlist, 7

Tags: combinatorics , combinatorial geometry , convex polygon , polygon

Let $ n > 2$. Show that there is a set of $ 2^{n-1}$ points in the plane, no three collinear such that no $ 2n$ form a convex $ 2n$-gon.

Kyiv City MO 1984-93 - geometry, 1987.10.1

Tags: tangential , polygon , geometry

Is there a $1987$-gon with consecutive sides lengths $1, 2, 3,..., 1986, 1987$, in which you can fit a circle?

2021 Durer Math Competition (First Round), 3

Tags: geometry , polygon

The floor plan of a contemporary art museum is a (not necessarily convex) polygon and its walls are solid. The security guard guarding the museum has two favourite spots (points $A$ and $B$) because one can see the whole area of the museum standing at either point. Is it true that from any point of the $AB$ section one can see the whole museum?

2009 Postal Coaching, 3

Tags: inequalities , polygon , cyclic , geometric inequalities

Let $\Omega$ be an $n$-gon inscribed in the unit circle, with vertices $P_1, P_2, ..., P_n$. (a) Show that there exists a point $P$ on the unit circle such that $PP_1 \cdot PP_2\cdot ... \cdot PP_n \ge 2$. (b) Suppose for each $P$ on the unit circle, the inequality $PP_1 \cdot PP_2\cdot ... \cdot PP_n \le 2$ holds. Prove that $\Omega$ is regular.

2004 IMO Shortlist, 6

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

Let $P$ be a convex polygon. Prove that there exists a convex hexagon that is contained in $P$ and whose area is at least $\frac34$ of the area of the polygon $P$. [i]Alternative version.[/i] Let $P$ be a convex polygon with $n\geq 6$ vertices. Prove that there exists a convex hexagon with [b]a)[/b] vertices on the sides of the polygon (or) [b]b)[/b] vertices among the vertices of the polygon such that the area of the hexagon is at least $\frac{3}{4}$ of the area of the polygon. [i]Proposed by Ben Green and Edward Crane, United Kingdom[/i]

2001 Estonia National Olympiad, 1

Tags: convex , polygon , geometry , angle

The angles of a convex $n$-gon are $a,2a, ... ,na$. Find all possible values of $n$ and the corresponding values of $a$.

1968 IMO Shortlist, 5

Tags: inequalities , geometry , polygon , geometric inequality

Let $h_n$ be the apothem (distance from the center to one of the sides) of a regular $n$-gon ($n \geq 3$) inscribed in a circle of radius $r$. Prove the inequality \[(n + 1)h_n+1 - nh_n > r.\] Also prove that if $r$ on the right side is replaced with a greater number, the inequality will not remain true for all $n \geq 3.$

2010 Germany Team Selection Test, 2

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

Let $P$ be a polygon that is convex and symmetric to some point $O$. Prove that for some parallelogram $R$ satisfying $P\subset R$ we have \[\frac{|R|}{|P|}\leq \sqrt 2\] where $|R|$ and $|P|$ denote the area of the sets $R$ and $P$, respectively. [i]Proposed by Witold Szczechla, Poland[/i]

1985 All Soviet Union Mathematical Olympiad, 398

Tags: coloring , polygon

You should paint all the sides and diagonals of the regular $n$-gon so, that every pair of segments, having the common point, would be painted with different colours. How many colours will you require?

1967 IMO Longlists, 27

Tags: geometry , 3d geometry , polygon , cube , intersection

Which regular polygon can be obtained (and how) by cutting a cube with a plane ?

2016 Turkey Team Selection Test, 8

Tags: geometry , circumcircle , polygon

All angles of the convex $n$-gon $A_1A_2\dots A_n$ are obtuse, where $n\ge5$. For all $1\le i\le n$, $O_i$ is the circumcenter of triangle $A_{i-1}A_iA_{i+1}$ (where $A_0=A_n$ and $A_{n+1}=A_1$). Prove that the closed path $O_1O_2\dots O_n$ doesn't form a convex $n$-gon.

2015 Middle European Mathematical Olympiad, 2

Tags: combinatorics , polygon , combinatorial geometry , diagonal

Let $n\ge 3$ be an integer. An [i]inner diagonal[/i] of a [i]simple $n$-gon[/i] is a diagonal that is contained in the $n$-gon. Denote by $D(P)$ the number of all inner diagonals of a simple $n$-gon $P$ and by $D(n)$ the least possible value of $D(Q)$, where $Q$ is a simple $n$-gon. Prove that no two inner diagonals of $P$ intersect (except possibly at a common endpoint) if and only if $D(P)=D(n)$. [i]Remark:[/i] A simple $n$-gon is a non-self-intersecting polygon with $n$ vertices. A polygon is not necessarily convex.

2010 Oral Moscow Geometry Olympiad, 1

Tags: geometry , combinatorial geometry , combinatorics , cyclic , polygon , convex polygon

Convex $n$-gon $P$, where $n> 3$, is cut into equal triangles by diagonals that do not intersect inside it. What are the possible values of $n$ if the $n$-gon is cyclic?

1986 All Soviet Union Mathematical Olympiad, 434

Tags: vector , polygon , geometry

Given a regular $n$-gon $A_1A_2...A_n$. Prove that if a) $n$ is even number, than for the arbitrary point $M$ in the plane, it is possible to choose signs in an expression $$\pm \overrightarrow{MA_1} \pm \overrightarrow{MA_2} \pm ... \pm \overrightarrow{MA_n}$$to make it equal to the zero vector . b) $n$ is odd, than the abovementioned expression equals to the zero vector for the finite set of $M$ points only.