Olympiad problems

2021 Saudi Arabia Training Tests, 15

Tags: geometry , angle

Let $ABC$ be convex quadrilateral and $X$ lying inside it such that $XA \cdot XC^2 = XB \cdot XD^2$ and $\angle AXD + \angle BXC = \angle CXD$. Prove that $\angle XAD + \angle XCD = \angle XBC + \angle XDC$.

2010 Sharygin Geometry Olympiad, 1

Tags: geometry , angle bisector , angle chasing , altitude , angle

For each vertex of triangle $ABC$, the angle between the altitude and the bisectrix from this vertex was found. It occurred that these angle in vertices $A$ and $B$ were equal. Furthermore the angle in vertex $C$ is greater than two remaining angles. Find angle $C$ of the triangle.

Denmark (Mohr) - geometry, 2009.1

Tags: geometry , rotation , angle

In the figure, triangle $ADE$ is produced from triangle $ABC$ by a rotation by $90^o$ about the point $A$. If angle $D$ is $60^o$ and angle $E$ is $40^o$, how large is then angle $u$? [img]https://1.bp.blogspot.com/-6Fq2WUcP-IA/Xzb9G7-H8jI/AAAAAAAAMWY/hfMEAQIsfTYVTdpd1Hfx15QPxHmfDLEkgCLcBGAsYHQ/s0/2009%2BMohr%2Bp1.png[/img]

2002 Estonia Team Selection Test, 4

Tags: equal angles , cyclic , geometry , angle

Let $ABCD$ be a cyclic quadrilateral such that $\angle ACB = 2\angle CAD$ and $\angle ACD = 2\angle BAC$. Prove that $|CA| = |CB| + |CD|$.

Kyiv City MO 1984-93 - geometry, 1993.9.2

Tags: geometry , area , angle

Let $a, b, c$ be the lengths of the sides of a triangle, and let $S$ be its area. We know that $S = \frac14 (c^2 - a^2 - b^2)$. Prove that $\angle C = 135^o$.

Estonia Open Junior - geometry, 2010.2.3

Tags: geometry , fixed , equilateral , angle

On the side $BC$ of the equilateral triangle $ABC$, choose any point $D$, and on the line $AD$, take the point $E$ such that $| B A | = | BE |$. Prove that the size of the angle $AEC$ is of does not depend on the choice of point $D$, and find its size.

2022 Mexican Girls' Contest, 2

Tags: geometry , isosceles , angle

Consider $\triangle ABC$ an isosceles triangle such that $AB = BC$. Let $P$ be a point satisfying $$\angle ABP = 80^\circ, \angle CBP = 20^\circ, \textrm{and} \hspace{0.17cm} AC = BP$$ Find all possible values of $\angle BCP$.

2015 Costa Rica - Final Round, 1

Tags: geometry , circumcircle , angle

Let $\vartriangle ABC$ be such that $\angle BAC$ is acute. The line perpendicular on side $AB$ from $C$ and the line perpendicular on $AC$ from $B$ intersect the circumscribed circle of $\vartriangle ABC$ at $D$ and $E$ respectively. If $DE = BC$ , calculate $\angle BAC$.

2014 Flanders Math Olympiad, 3

Tags: geometry , equal segments , angle chasing , angle

Let $PQRS$ be a quadrilateral with $| P Q | = | QR | = | RS |$, $\angle Q= 110^o$ and $\angle R = 130^o$ . Determine $\angle P$ and $\angle S$ .

2021 Yasinsky Geometry Olympiad, 1

Tags: geometry , equal segments , angle

The quadrilateral $ABCD$ is known to have $BC = CD = AC$, and the angle $\angle ABC= 70^o$. Calculate the degree measure of the angle $\angle ADB$. (Alexey Panasenko)

2015 Bangladesh Mathematical Olympiad, 6

Tags: geometry , trapezium , angle

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 $m+n$?

2020 Ukrainian Geometry Olympiad - April, 5

Tags: geometry , geometric inequality , angle

Inside the convex quadrilateral $ABCD$ there is a point $M$ such that $\angle AMB = \angle ADM + \angle BCM$ and $\angle AMD = \angle ABM + \angle DCM$. Prove that $AM \cdot CM + BM \cdot DM \ge \sqrt{AB \cdot BC\cdot CD \cdot DA}$

2016 Argentina National Olympiad Level 2, 2

Tags: angle , equal segments , geometry

Point $D$ on the side $BC$ of the acute triangle $ABC$ is chosen so that $AD = AC$. Let $P$ and $Q$ be the feet of the perpendiculars from $C$ and $D$ on the side $AB$, respectively. Suppose that $AP^2 + 3BP^2 = AQ^2 + 3BQ^2$. Determine the measure of angle $\angle ABC$.

2000 All-Russian Olympiad Regional Round, 8.6

Tags: algebra , geometry , angle

The electric train traveled from platform A to platform B in $X$ minutes ($0< X<60$). Find $X$ if it is known that as at the moment departure from A, and at the time of arrival at B, the angle between hourly and the minute hand was equal to $X$ degrees.

1989 IMO Shortlist, 32

Tags: circumcircle , trigonometry , parallelogram , rhombus , angle , geometry

The vertex $ A$ of the acute triangle $ ABC$ is equidistant from the circumcenter $ O$ and the orthocenter $ H.$ Determine all possible values for the measure of angle $ A.$

1992 Tournament Of Towns, (342) 4

Tags: geometry , geometric inequality , angle

(a) In triangle $ABC$, angle $A$ is greater than angle $B$. Prove that the length of side $BC$ is greater than half the length of side $AB$. (b) In the convex quadrilateral $ABCD$, the angle at $A$ is greater than the angle at $C$ and the angle at $D$ is greater than the angle at $B$. Prove that the length of side $BC$ is greater than half of the length of side $AD$. (F Nazarov)

Estonia Open Senior - geometry, 2001.1.1

Tags: angle , concurrency , geometry

Points $A, B, C, D, E$ and F are given on a circle in such a way that the three chords $AB, CD$ and $EF$ intersect in one point. Express angle $\angle EFA$ in terms of angles $\angle ABC$ and $\angle CDE$ (find all possibilities).

Kyiv City MO Juniors Round2 2010+ geometry, 2013.7.3

Tags: geometry , square , angle chasing , equal angles , angle

In the square $ABCD$ on the sides $AD$ and $DC$, the points $M$ and $N$ are selected so that $\angle BMA = \angle NMD = 60 { } ^ \circ $. Find the value of the angle $MBN$.

Kyiv City MO 1984-93 - geometry, 1988.7.1

Tags: trapezoid , equilateral , angle , geometry

An isosceles trapezoid is divided by each diagonal into two isosceles triangles. Determine the angles of the trapezoid.

1985 Tournament Of Towns, (090) T1

Tags: geometry , angle

In quadrilateral ABCD it is given that $AB = BC = 1, \angle ABC = 100^o$ , and $\angle CDA = 130^o$ . Find the length of $BD$.

2018 Istmo Centroamericano MO, 5

Tags: isosceles , geometry , angle

Let $ABC$ be an isosceles triangle with $CA = CB$. Let $D$ be the foot of the alttiude from $C$, and $\ell$ be the external angle bisector at $C$. Take a point $N$ on $\ell$ so that $AN> AC$ , on the same side as $A$ wrt $CD$. The bisector of the angle $NAC$ cuts $\ell$'at $F$. Show that $\angle NCD + \angle BAF> 180^o.$

2011 German National Olympiad, 4

Tags: point , angle , maximal , sum , set , geometry

There are two points $A$ and $B$ in the plane. a) Determine the set $M$ of all points $C$ in the plane for which $|AC|^2 +|BC|^2 = 2\cdot|AB|^2.$ b) Decide whether there is a point $C\in M$ such that $\angle ACB$ is maximal and if so, determine this angle.

2016 ASMT, 2

Tags: geometry , angle

Points $D$ and $E$ are chosen on the exterior of $\vartriangle ABC$ such that $\angle ADC = \angle BEC = 90^o$. If $\angle ACB = 40^o$, $AD = 7$, $CD = 24$, $CE = 15$, and $BE = 20$, what is the measure of $\angle ABC $ in,degrees?

Estonia Open Senior - geometry, 2020.2.5

Tags: geometry , ratio , angle

The bisector of the interior angle at the vertex $B$ of the triangle $ABC$ and the perpendicular line on side $BC$ passing through the vertex $C$ intersects at $D$. Let $M$ and $N$ be the midpoints of the segments $BC$ and $BD$, respectively, with $N$ on the side $AC$. Find all possibilities of the angles of the triangles $ABC$, if it is known that $\frac{| AM |}{| BC |}=\frac{|CD|}{|BD|}$. .

2003 Estonia National Olympiad, 1

Tags: regular polygon , geometry , angle

Let $A_1, A_2, ..., A_m$ and $B_2 , B_3,..., B_n$ be the points on a circle such that $A_1A_2... A_n$ is a regular $m$-gon and $A_1B_2...B_n$ is a regular $n$-gon whereby $n > m$ and the point $B_2$ lies between $A_1$ and $A_2$. Find $\angle B_2A_1A_2$.