Olympiad problems

2010 Federal Competition For Advanced Students, P2, 3

On a circular billiard table a ball rebounds from the rails as if the rail was the tangent to the circle at the point of impact. A regular hexagon with its vertices on the circle is drawn on a circular billiard table. A (point-shaped) ball is placed somewhere on the circumference of the hexagon, but not on one of its edges. Describe a periodical track of this ball with exactly four points at the rails. With how many different directions of impact can the ball be brought onto such a track?

2016 Oral Moscow Geometry Olympiad, 1

Tags: geometry , equal segments , angle , diagonal , hexagon

Angles are equal in a hexagon, three main diagonals are equal and the other six diagonals are also equal. Is it true that it has equal sides?

1997 Croatia National Olympiad, Problem 1

Tags: geometry , hexagon

In a regular hexagon $ABCDEF$ with center $O$, points $M$ and $N$ are the midpoints of the sides $CD$ and $DE$, and $L$ the intersection point of $AM$ and $BN$. Prove that: (a) $ABL$ and $DMLN$ have equal areas; (b) $\angle ALD=\angle OLN=60^\circ$; (c) $\angle OLD=90^\circ$.

1996 Argentina National Olympiad, 3

Tags: geometry , circumcircle , orthocenter , circumcenter , hexagon

The non-regular hexagon $ABCDEF$ is inscribed on a circle of center $O$ and $AB = CD = EF$. If diagonals $AC$ and $BD$ intersect at $M$, diagonals $CE$ and $DF$ intersect at $N$, and diagonals $AE$ and $BF$ intersect at $K$, show that the heights of triangle $MNK$ intersect at $O$.

1967 Putnam, B1

Tags: geometry , euclidean geometry , hexagon

Let $ABCDEF$ be a hexagon inscribed in a circle of radius $r.$ Show that if $AB=CD=EF=r,$ then the midpoints of $BC, DE$ and $FA$ are the vertices of an equilateral triangle.

2013 Sharygin Geometry Olympiad, 4

Tags: concyclic , hexagon , geometry

A point $F$ inside a triangle $ABC$ is chosen so that $\angle AFB = \angle BFC = \angle CFA$. The line passing through $F$ and perpendicular to $BC$ meets the median from $A$ at point $A_1$. Points $B_1$ and $C_1$ are defined similarly. Prove that the points $A_1, B_1$, and $C_1$ are three vertices of some regular hexagon, and that the three remaining vertices of that hexagon lie on the sidelines of $ABC$.

2018 Malaysia National Olympiad, A1

Tags: hexagon , equilateral , geometry

Hassan has a piece of paper in the shape of a hexagon. The interior angles are all $120^o$, and the side lengths are $1$, $2$, $3$, $4$, $5$, $6$, although not in that order. Initially, the paper is in the shape of an equilateral triangle, then Hassan has cut off three smaller equilateral triangle shapes, one at each corner of the paper. What is the minimum possible side length of the original triangle?

1997 Pre-Preparation Course Examination, 2

Tags: trigonometry , circumcircle , geometry , hexagon

An acute triangle $ ABC$ is given. Points $ A_1$ and $ A_2$ are taken on the side $ BC$ (with $ A_2$ between $ A_1$ and $ C$), $ B_1$ and $ B_2$ on the side $ AC$ (with $ B_2$ between $ B_1$ and $ A$), and $ C_1$ and $ C_2$ on the side $ AB$ (with $ C_2$ between $ C_1$ and $ B$) so that \[ \angle AA_1A_2 \equal{} \angle AA_2A_1 \equal{} \angle BB_1B_2 \equal{} \angle BB_2B_1 \equal{} \angle CC_1C_2 \equal{} \angle CC_2C_1.\] The lines $ AA_1,BB_1,$ and $ CC_1$ bound a triangle, and the lines $ AA_2,BB_2,$ and $ CC_2$ bound a second triangle. Prove that all six vertices of these two triangles lie on a single circle.

2003 IMO, 3

Tags: geometry , hexagon , equal angles

Each pair of opposite sides of a convex hexagon has the following property: the distance between their midpoints is equal to $\dfrac{\sqrt{3}}{2}$ times the sum of their lengths. Prove that all the angles of the hexagon are equal.

2014 Junior Regional Olympiad - FBH, 5

Tags: combinatorics , hexagon

Let $ABCDEF$ be a hexagon. Sides and diagonals of hexagon are colored in two colors: blue and yellow. Prove that there exist a triangle with vertices from set $\{A,B,C,D,E,F\}$ which sides are all same colour

2018 Czech-Polish-Slovak Junior Match, 2

Tags: perimeter , hexagon , parallel , diagonal , geometry

A convex hexagon $ABCDEF$ is given whose sides $AB$ and $DE$ are parallel. Each of the diagonals $AD, BE, CF$ divides this hexagon into two quadrilaterals of equal perimeters. Show that these three diagonals intersect at one point.

1995 IMO Shortlist, 5

Tags: inequalities , geometry , hexagon , geometric inequality

Let $ ABCDEF$ be a convex hexagon with $ AB \equal{} BC \equal{} CD$ and $ DE \equal{} EF \equal{} FA$, such that $ \angle BCD \equal{} \angle EFA \equal{} \frac {\pi}{3}$. Suppose $ G$ and $ H$ are points in the interior of the hexagon such that $ \angle AGB \equal{} \angle DHE \equal{} \frac {2\pi}{3}$. Prove that $ AG \plus{} GB \plus{} GH \plus{} DH \plus{} HE \geq CF$.

2006 Hanoi Open Mathematics Competitions, 6

Tags: locus , hexagon , geometry

The figure $ABCDEF$ is a regular hexagon. Find all points $M$ belonging to the hexagon such that Area of triangle $MAC =$ Area of triangle $MCD$.

1958 Kurschak Competition, 3

Tags: equal area , hexagon , geometry

The hexagon $ABCDEF$ is convex and opposite sides are parallel. Show that the triangles $ACE$ and $BDF$ have equal area

1985 All Soviet Union Mathematical Olympiad, 413

Tags: combinatorics , hexagon , combinatorial geometry

Given right hexagon. The lines parallel to all the sides are drawn from all the vertices and midpoints of the sides (consider only the interior, with respect to the hexagon, parts of those lines). Thus the hexagon is divided onto $24$ triangles, and the figure has $19$ nodes. $19$ different numbers are written in those nodes. Prove that at least $7$ of $24$ triangles have the property: the numbers in its vertices increase (from the least to the greatest) counterclockwise.

2015 PAMO, Problem 2

Tags: concurrency , geometry , hexagon

A convex hexagon $ABCDEF$ is such that $$AB=BC \quad CD=DE \quad EF=FA$$ and $$\angle ABC=2\angle AEC \quad \angle CDE=2\angle CAE \quad \angle EFA=2\angle ACE$$ Show that $AD$, $CF$ and $EB$ are concurrent.

Kyiv City MO Juniors 2003+ geometry, 2005.89.5

Tags: hexagon , regular , angle , geometry

Let $ABCDEF $ be a regular hexagon. On the line $AF $ mark the point $X$so that $ \angle DCX = 45^o$ . Find the value of the angle $FXE$. (Vyacheslav Yasinsky)

2008 Oral Moscow Geometry Olympiad, 6

Tags: geometry , hexagon , parallel , cyclic

Opposite sides of a convex hexagon are parallel. Let's call the "height" of such a hexagon a segment with ends on straight lines containing opposite sides and perpendicular to them. Prove that a circle can be circumscribed around this hexagon if and only if its "heights" can be parallelly moved so that they form a triangle. (A. Zaslavsky)

2018 Hanoi Open Mathematics Competitions, 2

Tags: geometry , hexagon , equilateral , maximum , area

What is the largest area of a regular hexagon that can be drawn inside the equilateral triangle of side $3$? A. $3\sqrt7$ B. $\frac{3 \sqrt3}{2}$ C. $2\sqrt5$ D. $\frac{3\sqrt3}{8}$ E. $3\sqrt5$

2017 Sharygin Geometry Olympiad, P21

Tags: geometry , midpoint , convex , hexagon , maximum

A convex hexagon is circumscribed about a circle of radius $1$. Consider the three segments joining the midpoints of its opposite sides. Find the greatest real number $r$ such that the length of at least one segment is at least $r.$

2017 Ecuador NMO (OMEC), 6

Tags: hexagon , concurrency , geometry

Let $ABCDEF$ be a convex hexagon with sides not parallel and tangent to a circle $\Gamma$ at the midpoints $P$, $Q$, $R$ of the sides AB, $CD$, $EF$ respectively. $\Gamma$ is tangent to $BC$, $DE$ and $FA$ at the points $X, Y, Z$ respectively. Line $AB$ intersects lines $EF$ and $CD$ at points $M$ and $N$ respectively. Lines $MZ$ and $NX$ intersect at point $K$. Let $ r$ be the line joining the center of $\Gamma$ and point $K$. Prove that the intersection of $PY$ and $QZ$ lies on the line $ r$.

2024 Belarusian National Olympiad, 8.4

Tags: geometry , hexagon

In a convex hexagon $ABCDEF$ equalities $\angle ABC= \angle CDE= \angle EFA$ hold, and the angle bisectors of angles $ABC$, $CDE$ and $EFA$ intersect in one point. Rays $AB$ and $DC$ intersect at $P$, rays $BC$ and $ED$ - at $Q$, rays $CD$ and $FE$ - at $R$, rays $DE$ and $AF$ - at $S$. Prove that $PR=QS$ [i]M. Zorka[/i]

2007 Oral Moscow Geometry Olympiad, 4

Tags: equal segments , geometry , hexagon , equilateral

The midpoints of the opposite sides of the hexagon are connected by segments. It turned out that the points of pairwise intersection of these segments form an equilateral triangle. Prove that the drawn segments are equal. (M. Volchkevich)

1993 Tournament Of Towns, (395) 3

Tags: geometry , fixed , hexagon , equilateral

Consider the hexagon which is formed by the vertices of two equilateral triangles (not necessarily equal) when the triangles intersect. Prove that the area of the hexagon is unchanged when one of the triangles is translated (without rotating) relative to the other in such a way that the hexagon continues to be defined. (V Proizvolov)

2018 Hanoi Open Mathematics Competitions, 6

Tags: geometry , perimeter , hexagon , square

In the below figure, there is a regular hexagon and three squares whose sides are equal to $4$ cm. Let $M,N$, and $P$ be the centers of the squares. The perimeter of the triangle $MNP$ can be written in the form $a + b\sqrt3$ (cm), where $a, b$ are integers. Compute the value of $a + b$. [img]https://cdn.artofproblemsolving.com/attachments/e/8/5996e994d4bbed8d3b3269d3e38fc2ec5d2f0b.png[/img]