Olympiad problems

2018 India PRMO, 5

Let $ABCD$ be a trapezium in which $AB //CD$ and $AD \perp AB$. Suppose $ABCD$ has an incircle which touches $AB$ at $Q$ and $CD$ at $P$. Given that $PC = 36$ and $QB = 49$, find $PQ$.

2018 Polish Junior MO Finals, 2

Let $ABCD$ be a trapezium with bases $AB$ and $CD$ in which $AB + CD = AD$. Diagonals $AC$ and $BD$ intersect in point $E$. Line passing through point $E$ and parallel to bases of trapezium cuts $AD$ in point $F$. Prove that $\sphericalangle BFC = 90 ^{\circ}$.

geometry trapezium plane geometry

2021 Bangladeshi National Mathematical Olympiad, 4

$ABCD$ is an isosceles trapezium such that $AD=BC$, $AB=5$ and $CD=10$. A point $E$ on the plane is such that $AE\perp{EC}$ and $BC=EC$. The length of $AE$ can be expressed as $a\sqrt{b}$, where $a$ and $b$ are integers and $b$ is not divisible by any square number other than $1$. Find the value of $(a+b)$.

perpendicular egmo book geometry trapezium number theory

2015 Bangladesh Mathematical Olympiad, 6

Trapezoid $ABCD$ has sides $AB=92,BC=50,CD=19,AD=70$ $AB$ is parallel to $CD$ A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$.Given that $AP=\dfrac mn$ (Where $m,n$ are relatively prime).What is $m+n$?

geometry trapezium angle number theory

2008 Singapore Senior Math Olympiad, 1

Let $ABCD$ be a trapezium with $AD // BC$. Suppose $K$ and $L$ are, respectively, points on the sides $AB$ and $CD$ such that $\angle BAL = \angle CDK$. Prove that $\angle BLA = \angle CKD$.

equal angles geometry trapezium trapezoid

2020 Tournament Of Towns, 5

Let $ABCD$ be an inscribed trapezoid. The base $AB$ is $3$ times longer than $CD$. Tangents to the circumscribed circle at the points $A$ and $C$ intersect at the point $K$. Prove that the angle $KDA$ is a right angle. Alexandr Yuran

right angle geometry trapezium equal segments trapezoid symmedians

2019 Adygea Teachers' Geometry Olympiad, 4

From which two statements about the trapezoid follows the third: 1) the trapezoid is tangential, 2) the trapezoid is right, 3) its area is equal to the product of the bases?

right angle geometry trapezium trapezoid tangential

2013 India PRMO, 8

Let $AD$ and $BC$ be the parallel sides of a trapezium $ABCD$. Let $P$ and $Q$ be the midpoints of the diagonals $AC$ and $BD$. If $AD = 16$ and $BC = 20$, what is the length of $PQ$?

geometry trapezium trapezoid similar triangles

2021 Cyprus JBMO TST, 3

Let $AB\varGamma\varDelta$ be a rhombus. (a) Prove that you can construct a circle $(c)$ which is inscribed in the rhombus and is tangent to its sides. (b) The points $\varTheta,H,K,I$ are on the sides $\varDelta\varGamma,B\varGamma,AB,A\varDelta$ of the rhombus respectively, such that the line segments $KH$ and $I\varTheta$ are tangent on the circle $(c)$. Prove that the quadrilateral defined from the points $\varTheta,H,K,I$ is a trapezium.

geometry trapezium rhombus