Olympiad problems

2015 Sharygin Geometry Olympiad, P4

In a parallelogram $ABCD$ the trisectors of angles $A$ and $B$ are drawn. Let $O$ be the common points of the trisectors nearest to $AB$. Let $AO$ meet the second trisector of angle $B$ at point $A_1$, and let $BO$ meet the second trisector of angle $A$ at point $B_1$. Let $M$ be the midpoint of $A_1B_1$. Line $MO$ meets $AB$ at point $N$ Prove that triangle $A_1B_1N$ is equilateral.

1991 Tournament Of Towns, (313) 3

Tags: geometry , obtuse , equal ratio , angle , ratio

Point $D$ lies on side $AB$ of triangle $ABC$, and $$\frac{AD}{DC} = \frac{AB}{BC}.$$ Prove that angle $C$ is obtuse. (Sergey Berlov)

1969 IMO Longlists, 71

Tags: geometry , angle , rhombus

$(YUG 3)$ Let four points $A_i (i = 1, 2, 3, 4)$ in the plane determine four triangles. In each of these triangles we choose the smallest angle. The sum of these angles is denoted by $S.$ What is the exact placement of the points $A_i$ if $S = 180^{\circ}$?

2000 Estonia National Olympiad, 4

Tags: trigonometry , angle , sidelenghts

Prove that for any triangle the equation holds $a \cdot \cos (\beta + \gamma ) + b \cdot \cos (\gamma +\alpha) + c\cdot \cos (\alpha -\beta) = 0$, where $a, b, c$ are the sides of the triangle and $\alpha, \beta,\gamma$ according to their angles sizes of opposite angles.

Novosibirsk Oral Geo Oly VIII, 2017.7

Tags: angle , geometry

A car is driving along a straight highway at a speed of $60$ km per hour. Not far from the highway there is a parallel to him a $100$-meter fence. Every second, the passenger of the car measures the angle at which the fence is visible. Prove that the sum of all the angles he measured is less than $1100^o$

2015 Romania National Olympiad, 4

Tags: angle chasing , angle , geometry

Consider $\vartriangle ABC$ where $\angle ABC= 60 ^o$. Points $M$ and $D$ are on the sides $(AC)$, respectively $(AB)$, such that $\angle BCA = 2 \angle MBC$, and $BD = MC$. Determine $\angle DMB$.

2019 Philippine TST, 6

Tags: parallelogram , isogonal lines , angle , geometry

Let $D$ be an interior point of triangle $ABC$. Lines $BD$ and $CD$ intersect sides $AC$ and $AB$ at points $E$ and $F$, respectively. Points $X$ and $Y$ are on the plane such that $BFEX$ and $CEFY$ are parallelograms. Suppose lines $EY$ and $FX$ intersect at a point $T$ inside triangle $ABC$. Prove that points $B$, $C$, $E$, and $F$ are concyclic if and only if $\angle BAD = \angle CAT$.

2017 Oral Moscow Geometry Olympiad, 1

Tags: geometry , square , 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]

1996 IMO, 2

Tags: concurrency , incenter , angle , triangle , geometry

Let $ P$ be a point inside a triangle $ ABC$ such that \[ \angle APB \minus{} \angle ACB \equal{} \angle APC \minus{} \angle ABC. \] Let $ D$, $ E$ be the incenters of triangles $ APB$, $ APC$, respectively. Show that the lines $ AP$, $ BD$, $ CE$ meet at a point.

Estonia Open Senior - geometry, 1994.1.4

Tags: ratio , angle

Prove that if $\frac{AC}{BC}=\frac{AB + BC}{AC}$ in a triangle $ABC$ , then $\angle B = 2 \angle A$ .

2021 Novosibirsk Oral Olympiad in Geometry, 5

Tags: equal segments , angle , geometry

In an acute-angled triangle $ABC$ on the side $AC$, point $P$ is chosen in such a way that $2AP = BC$. Points $X$ and $Y$ are symmetric to $P$ with respect to vertices $A$ and $C$, respectively. It turned out that $BX = BY$. Find $\angle BCA$.

VI Soros Olympiad 1999 - 2000 (Russia), 10.2

Tags: angle , geometry

In the triangle $ABC$, the point $X$ is the projection of the touchpoint of the inscribed circle to the side $BC$ on the middle line parallel to $BC$. It is known that $\angle BAC \ge 60^o$. Prove that the angle $BXC$ is obtuse.

2017 Bundeswettbewerb Mathematik, 2

Tags: combinatorics , gon , angle , acute

What is the maximum number of acute interior angles a non-overlapping planar $2017$-gon can have?

2021 Saudi Arabia Training Tests, 4

Tags: geometry , geometric inequality , incircle , angle

Let $ABC$ be a triangle with incircle $(I)$, tangent to $BC$, $CA$, $AB$ at $D, E, F$ respectively. On the line $DF$, take points $M, P$ such that $CM \parallel AB$, $AP \parallel BC$. On the line $DE$, take points $N$, $Q$ such that $BN \parallel AC$, $AQ \parallel BC$. Denote $X$ as intersection of $PE$, $QF$ and $K$ as the midpoint of $BC$. Prove that if $AX = IK$ then $\angle BAC \le 60^o$.

2021 Yasinsky Geometry Olympiad, 2

Tags: geometry , angle

In the quadrilateral $ABCD$ it is known that $\angle A = 90^o$, $\angle C = 45^o$ . Diagonals $AC$ and $BD$ intersect at point $F$, and $BC = CF$, and the diagonal $AC$ is the bisector of angle $A$. Determine the other two angles of the quadrilateral $ABCD$. (Maria Rozhkova)

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?

2015 Bangladesh Mathematical Olympiad, 6

Tags: geometry , angle , trapezium

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 $m+n$?

1998 Belarus Team Selection Test, 1

Tags: geometry , angle , locus , circles

Two circles $S_1$ and $S_2$ intersect at different points $P,Q$. The arc of $S_1$ lying inside $S_2$ measures $2a$ and the arc of $S_2$ lying inside $S_1$ measures $2b$. Let $T$ be any point on $S_1$. Let $R,S$ be another points of intersection of $S_2$ with $TP$ and $TQ$ respectively. Let $a+2b<\pi$ . Find the locus of the intersection points of $PS$ and $RQ$. S.Shikh

1952 Moscow Mathematical Olympiad, 229

Tags: geometry , isosceles , angle , angle chasing

In an isosceles triangle $\vartriangle ABC, \angle ABC = 20^o$ and $BC = AB$. Points $P$ and $Q$ are chosen on sides $BC$ and $AB$, respectively, so that $\angle PAC = 50^o$ and $\angle QCA = 60^o$ . Prove that $\angle PQC = 30^o$ .

2011 Oral Moscow Geometry Olympiad, 2

Tags: geometry , equal segments , angle , isosceles , right angle

In an isosceles triangle $ABC$ ($AB=AC$) on the side $BC$, point $M$ is marked so that the segment $CM$ is equal to the altitude of the triangle drawn on this side, and on the side $AB$, point $K$ is marked so that the angle $\angle KMC$ is right. Find the angle $\angle ACK$.