Olympiad problems

Cono Sur Shortlist - geometry, 2018.G3

Consider the pentagon $ABCDE$ such that $AB = AE = x$, $AC = AD = y$, $\angle BAE = 90^o$ and $\angle ACB = \angle ADE = 135^o$. It is known that $C$ and $D$ are inside the triangle $BAE$. Determine the length of $CD$ in terms of $x$ and $y$.

1997 Estonia National Olympiad, 3

Tags: geometry , ratio , pentagon , diagonal , parallel

Each diagonal of a convex pentagon is parallel to one of its sides. Prove that the ratio of the length of each diagonal to the length of the corresponding parallel side is the same, and find this ratio.

1947 Moscow Mathematical Olympiad, 126

Tags: pentagon , combinatorial geometry , convex , line , combinatorics

Given a convex pentagon $ABCDE$, prove that if an arbitrary point $M$ inside the pentagon is connected by lines with all the pentagon’s vertices, then either one or three or five of these lines cross the sides of the pentagon opposite the vertices they pass. Note: In reality, we need to exclude the points of the diagonals, because that in this case the drawn lines can pass not through the internal points of the sides, but through the vertices. But if the drawn diagonals are not considered or counted twice (because they are drawn from two vertices), then the statement remains true.

2016 Austria Beginners' Competition, 4

Tags: pentagon , geometry

Let $ABCDE$ be a convex pentagon with ﬁve equal sides and right angles at $C$ and $D$. Let $P$ denote the intersection point of the diagonals $AC$ and $BD$. Prove that the segments $PA$ and $PD$ have the same length. (Gottfried Perz)

2006 IMO Shortlist, 3

Tags: circumcircle , geometry , pentagon

Let $ ABCDE$ be a convex pentagon such that \[ \angle BAC \equal{} \angle CAD \equal{} \angle DAE\qquad \text{and}\qquad \angle ABC \equal{} \angle ACD \equal{} \angle ADE. \]The diagonals $BD$ and $CE$ meet at $P$. Prove that the line $AP$ bisects the side $CD$. [i]Proposed by Zuming Feng, USA[/i]

Brazil L2 Finals (OBM) - geometry, 2008.3

Tags: geometry , pentagon , tiling

Let $P$ be a convex pentagon with all sides equal. Prove that if two of the angles of $P$ add to $180^o$, then it is possible to cover the plane with $P$, without overlaps.

2023 HMNT, 9

Tags: geometry , pentagon

Pentagon $SPEAK$ is inscribed in triangle $NOW$ such that $S$ and $P$ lie on segment $NO$, $K$ and $A$ lie on segment $NW$, and $E$ lies on segment $OW$. Suppose that $NS = SP = PO$ and $NK = KA = AW$. Given that $EP = EK = 5$ and $EA = ES = 6$, compute $OW$.

Kyiv City MO Seniors Round2 2010+ geometry, 2019.10.3.1

Tags: equal segments , geometry , pentagon , regular pentagon

Let $ABCDE$ be a regular pentagon with center $M$. Point $P \ne M$ is selected on segment $MD$. The circumscribed circle of triangle $ABP$ intersects the line $AE$ for second time at point $Q$, and a line that is perpendicular to the $CD$ and passes through $P$, for second time at the point $R$. Prove that $AR = QR$.

1986 IMO Longlists, 38

Tags: algorithm , invariant , pentagon , combinatorics , game strategy

To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x,y,z$ respectively, and $y<0$, then the following operation is allowed: $x,y,z$ are replaced by $x+y,-y,z+y$ respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps.

VI Soros Olympiad 1999 - 2000 (Russia), 10.6

Tags: geometry , geometric inequality , pentagon , construction , area

Points $A$ and $B$ are given on a circle. With the help of a compass and a ruler, construct on this circle the points $C,$ $D$, $E$ that lie on one side of the straight line $AB$ and for which the pentagon with vertices $A$, $B$, $C$, $D$, $E$ has the largest possible area

2015 Abels Math Contest (Norwegian MO) Final, 3

Tags: pentagon , regular , distance , maximum , product , geometry

The five sides of a regular pentagon are extended to lines $\ell_1, \ell_2, \ell_3, \ell_4$, and $\ell_5$. Denote by $d_i$ the distance from a point $P$ to $\ell_i$. For which point(s) in the interior of the pentagon is the product $d_1d_2d_3d_4d_5$ maximal?

2024 Abelkonkurransen Finale, 4b

Tags: incenter , pentagon , geometry , cyclic

The pentagons $P_1P_2P_3P_4P_5$ and$I_1I_2I_3I_4I_5$ are cyclic, where $I_i$ is the incentre of the triangle $P_{i-1}P_iP_{i+1}$ (reckoned cyclically, that is $P_0=P_5$ and $P_6=P_1$). Show that the lines $P_1I_1, P_2I_2, P_3I_3, P_4I_4$ and $P_5I_5$ meet in a single point.

1962 All Russian Mathematical Olympiad, 020

Tags: minimum , geometry , maximum , pentagon

Given regular pentagon $ABCDE$. $M$ is an arbitrary point inside $ABCDE$ or on its side. Let the distances $|MA|, |MB|, ... , |ME|$ be renumerated and denoted with $$r_1\le r_2\le r_3\le r_4\le r_5.$$ Find all the positions of the $M$, giving $r_3$ the minimal possible value. Find all the positions of the $M$, giving $r_3$ the maximal possible value.

2018 Thailand TST, 1

Tags: geometry , pentagon , inscribed circle

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

II Soros Olympiad 1995 - 96 (Russia), 9.3

Tags: combinatorial geometry , geometry , pentagon

Is there a convex pentagon in which each diagonal is equal to some side?

2017 Pan-African Shortlist, G3

Tags: pentagon , circumcircle , geometry

Let $ABCDE$ be a regular pentagon, and $F$ some point on the arc $AB$ of the circumcircle of $ABCDE$. Show that \[ \frac{FD}{FE + FC} = \frac{FB + FA}{FD} = \frac{-1 + \sqrt{5}}{2}, \] and that $FD + FB + FA = FE + FC$.

1987 All Soviet Union Mathematical Olympiad, 450

Tags: pentagon , geometry , equal angles

Given a convex pentagon $ABCDE$ with $\angle ABC= \angle ADE$ and $\angle AEC= \angle ADB$ . Prove that $\angle BAC = \angle DAE$ .

2022/2023 Tournament of Towns, P3

Tags: pentagon , angle chasing , geometry

A pentagon $ABCDE$ is circumscribed about a circle. The angles at the vertices $A{}$, $C{}$ and $E{}$ of the pentagon are equal to $100^\circ$. Find the measure of the angle $\angle ACE$.

2018 Morocco TST., 4

Tags: pentagon , inscribed circle , geometry