Olympiad problems

2018 Yasinsky Geometry Olympiad, 4

In the quadrilateral $ABCD$, the length of the sides $AB$ and $BC$ is equal to $1, \angle B= 100^o , \angle D= 130^o$ . Find the length of $BD$.

2014 Contests, 3

Tags: geometry , equal segments , right triangle , semicircle , angle

(i) $ABC$ is a triangle with a right angle at $A$, and $P$ is a point on the hypotenuse $BC$. The line $AP$ produced beyond $P$ meets the line through $B$ which is perpendicular to $BC$ at $U$. Prove that $BU = BA$ if, and only if, $CP = CA$. (ii) $A$ is a point on the semicircle $CB$, and points $X$ and $Y$ are on the line segment $BC$. The line $AX$, produced beyond $X$, meets the line through $B$ which is perpendicular to $BC$ at $U$. Also the line $AY$, produced beyond $Y$, meets the line through $C$ which is perpendicular to $BC$ at $V$. Given that $BY = BA$ and $CX = CA$, determine the angle $\angle VAU$.

1995 Singapore Team Selection Test, 2

Tags: equal angles , angle , geometry

$ABC$ is a triangle with $\angle A > 90^o$ . On the side $BC$, two distinct points $P$ and $Q$ are chosen such that $\angle BAP = \angle PAQ$ and $BP \cdot CQ = BC \cdot PQ$. Calculate the size of $\angle PAC$.

2010 District Olympiad, 3

Tags: geometry , isosceles , equal segments , angle

Consider triangle $ABC$ with $AB = AC$ and $\angle A = 40 ^o$. The points $S$ and $T$ are on the sides $AB$ and $BC$, respectively, so that $\angle BAT = \angle BCS= 10 ^o$. The lines $AT$ and $CS$ intersect at point $P$. Prove that $BT = 2PT$.

2015 Sharygin Geometry Olympiad, P5

Tags: circles , angle , geometry

Let a triangle $ABC$ be given. Two circles passing through $A$ touch $BC$ at points $B$ and $C$ respectively. Let $D$ be the second common point of these circles ($A$ is closer to $BC$ than $D$). It is known that $BC = 2BD$. Prove that $\angle DAB = 2\angle ADB.$

2020 Romanian Master of Mathematics Shortlist, G1

Tags: angle , geometry

The incircle of a scalene triangle $ABC$ touches the sides $BC, CA$, and $AB$ at points $D, E$, and $F$, respectively. Triangles $APE$ and $AQF$ are constructed outside the triangle so that \[AP =PE, AQ=QF, \angle APE=\angle ACB,\text{ and }\angle AQF =\angle ABC.\]Let $M$ be the midpoint of $BC$. Find $\angle QMP$ in terms of the angles of the triangle $ABC$. [i]Iran, Shayan Talaei[/i]

2007 Thailand Mathematical Olympiad, 6

Tags: ratio , geometry , max , angle

Let $M$ be the midpoint of a given segment $BC$. Point $A$ is chosen to maximize $\angle ABC$ while subject to the condition that $\angle MAC = 20^o$ . What is the ratio $BC/BA$ ?

2017 Oral Moscow Geometry Olympiad, 1

Tags: square , geometry , angle , circles

One square is inscribed in a circle, and another square is circumscribed around the same circle so that its vertices lie on the extensions of the sides of the first (see figure). Find the angle between the sides of these squares. [img]https://3.bp.blogspot.com/-8eLBgJF9CoA/XTodHmW87BI/AAAAAAAAKY0/xsHTx71XneIZ8JTn0iDMHupCanx-7u4vgCK4BGAYYCw/s400/sharygin%2Boral%2B2017%2B10-11%2Bp1.png[/img]

1991 IMO, 2

Tags: geometry , geometric inequality , triangle , brocard , angle

Let $ \,ABC\,$ be a triangle and $ \,P\,$ an interior point of $ \,ABC\,$. Show that at least one of the angles $ \,\angle PAB,\;\angle PBC,\;\angle PCA\,$ is less than or equal to $ 30^{\circ }$.

2020 Ukrainian Geometry Olympiad - December, 5

Tags: geometry , circumcircle , altitude , angle

In an acute triangle $ABC$ with an angle $\angle ACB =75^o$, altitudes $AA_3,BB_3$ intersect the circumscribed circle at points $A_1,B_1$ respectively. On the lines $BC$ and $CA$ select points $A_2$ and $B_2$ respectively suchthat the line $B_1B_2$ is parallel to the line $BC$ and the line $A_1A_2$ is parallel to the line $AC$ . Let $M$ be the midpoint of the segment $A_2B_2$. Find in degrees the measure of the angle $\angle B_3MA_3$.

1996 Singapore MO Open, 2

Tags: geometry , angle , square

In the following figure, $ABCD$ is a square of unit length and $P, Q$ are points on $AD$ and $AB$ respectively. Find $\angle PCQ$ if $|AP| + |AQ| + |PQ| = 2$. [img]https://cdn.artofproblemsolving.com/attachments/2/c/2f40db978c1d3fcbc0161f874b5cbec926058e.png[/img]

2019 Saudi Arabia JBMO TST, 3

Tags: equal angles , angle , geometry

Consider a triangle $ABC$ and let $M$ be the midpoint of the side $BC$. Suppose $\angle MAC = \angle ABC$ and $\angle BAM = 105^o$. Find the measure of $\angle ABC$.

2023 Assara - South Russian Girl's MO, 2

Tags: angle , geometry

In the convex quadrilateral $ABCD$, point $X$ is selected on side $AD$, and the diagonals intersect at point $E$. It is known that $AC = BD$, $\angle ABX = \angle AX B = 50^o$, $\angle CAD = 51^o$, $\angle AED = 80^o$. Find the value of angle $\angle AXC$.

1995 Czech And Slovak Olympiad IIIA, 5

Tags: geometry , angle , tangent circle

Let $A,B$ be points on a circle $k$ with center $S$ such that $\angle ASB = 90^o$ . Circles $k_1$ and $k_2$ are tangent to each other at $Z$ and touch $k$ at $A$ and $B$ respectively. Circle $k_3$ inside $\angle ASB$ is internally tangent to $k$ at $C$ and externally tangent to $k_1$ and $k_2$ at $X$ and $Y$, respectively. Prove that $\angle XCY = 45^o$

2016 Germany National Olympiad (4th Round), 5

Tags: geometry , cyclic quadrilateral , trigonometry , angle

Let $A,B,C,D$ be points on a circle with radius $r$ in this order such that $|AB|=|BC|=|CD|=s$ and $|AD|=s+r$. Find all possible values of the interior angles of the quadrilateral $ABCD$.

Kyiv City MO Juniors 2003+ geometry, 2018.8.3

Tags: geometry , rectangle , isosceles , angle

In the isosceles triangle $ABC$ with the vertex at the point $B$, the altitudes $BH$ and $CL$ are drawn. The point $D$ is such that $BDCH$ is a rectangle. Find the value of the angle $DLH$. (Bogdan Rublev)

2015 Thailand TSTST, 1

Tags: geometry , ratio , angle

Let $D$ be a point inside an acute triangle $ABC$ such that $\angle ADC = \angle A +\angle B$, $\angle BDA = \angle B + \angle C$ and $\angle CDB = \angle C + \angle A$. Prove that $\frac{AB \cdot CD}{AD} = \frac{AC \cdot CB} {AB}$.

2005 Sharygin Geometry Olympiad, 11.4

Tags: geometry , angle , incircle , chord , diameter

In the triangle $ABC , \angle A = \alpha, BC = a$. The inscribed circle touches the lines $AB$ and $AC$ at points $M$ and $P$. Find the length of the chord cut by the line $MP$ in a circle with diameter $BC$.

1959 Kurschak Competition, 2

Tags: geometry , angle

The angles subtended by a tower at distances $100$, $200$ and $300$ from its foot sum to $90^o$. What is its height?

1983 IMO, 2

Tags: geometry , circles , midpoint , angle

Let $A$ be one of the two distinct points of intersection of two unequal coplanar circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ respectively. One of the common tangents to the circles touches $C_1$ at $P_1$ and $C_2$ at $P_2$, while the other touches $C_1$ at $Q_1$ and $C_2$ at $Q_2$. Let $M_1$ be the midpoint of $P_1Q_1$ and $M_2$ the midpoint of $P_2Q_2$. Prove that $\angle O_1AO_2=\angle M_1AM_2$.

Novosibirsk Oral Geo Oly IX, 2023.3

Tags: geometry , equal segments , isosceles , angle

An isosceles triangle $ABC$ with base $AC$ is given. On the rays $CA$, $AB$ and $BC$, the points $D, E$ and $F$ were marked, respectively, in such a way that $AD = AC$, $BE = BA$ and $CF = CB$. Find the sum of the angles $\angle ADB$, $\angle BEC$ and $\angle CFA$.

2014 May Olympiad, 4

Tags: geometry , right triangle , isosceles triangle , equilateral triangle , angle

Let $ABC$ be a right triangle and isosceles, with $\angle C = 90^o$. Let $M$ be the midpoint of $AB$ and $N$ the midpoint of $AC$. Let $ P$ be such that $MNP$ is an equilateral triangle with $ P$ inside the quadrilateral $MBCN$. Calculate the measure of $\angle CAP$

Novosibirsk Oral Geo Oly VIII, 2021.2

Tags: geometry , angle

The extensions of two opposite sides of the convex quadrilateral intersect and form an angle of $20^o$ , the extensions of the other two sides also intersect and form an angle of $20^o$. It is known that exactly one angle of the quadrilateral is $80^o$. Find all of its other angles.

1954 Putnam, A4

Tags: angle , physics

A uniform rod of length $2k$ and weight $w$ rests with the end $A$ against a vertical wall, while the lower end $B$ is fastened by a string $BC$ of length $2b$ coming from a point $C$ in the wall above $A.$ If the system is in equilibrium, determine the angle $ABC.$

2022 Brazil EGMO TST, 2

Tags: geometry , angle

Let $\vartriangle ABC$ be a triangle in which $\angle ACB = 40^o$ and $\angle BAC = 60^o$ . Let $D$ be a point inside the segment $BC$ such that $CD =\frac{AB}{2}$ and let $M$ be the midpoint of the segment $AC$. How much is the angle $\angle CMD$ in degrees?