Olympiad problems

Durer Math Competition CD Finals - geometry, 2019.D3

a) Does there exist a quadrilateral with (both of) the following properties: three of its edges are of the same length, but the fourth one is different, and three of its angles are equal, but the fourth one is different? b) Does there exist a pentagon with (both of) the following properties: four of its edges are of the same length, but the fifth one is different, and four of its angles are equal, but the fifth one is different?

2021 Science ON all problems, 2

Tags: geometry , angle

In triangle $ABC$, we have $\angle ABC=\angle ACB=44^o$. Point $M$ is in its interior such that $\angle MBC=16^o$ and $\angle MCB=30^o$. Prove that $\angle MAC=\angle MBC$. [i] (Andra Elena Mircea)[/i]

2020 German National Olympiad, 1

Tags: maximal , geometry , angle , circles

Let $k$ be a circle with center $M$ and let $B$ be another point in the interior of $k$. Determine those points $V$ on $k$ for which $\measuredangle BVM$ becomes maximal.

1968 IMO Shortlist, 20

Tags: 3d geometry , angle , permutation , point set , geometry

Given $n \ (n \geq 3)$ points in space such that every three of them form a triangle with one angle greater than or equal to $120^\circ$, prove that these points can be denoted by $A_1,A_2, \ldots,A_n$ in such a way that for each $i, j, k, 1 \leq i < j < k \leq n$, angle $A_iA_jA_k$ is greater than or equal to $120^\circ . $

2008 Tournament Of Towns, 4

Tags: geometry , angle

Each of Peter and Basil draws a convex quadrilateral with no parallel sides. The angles between a diagonal and the four sides of Peter's quadrilateral are $\alpha, \alpha, \beta$ and $\gamma$ in some order. The angles between a diagonal and the four sides of Basil's quadrilateral are also $\alpha, \alpha, \beta$ and $\gamma$ in some order. Prove that the acute angle between the diagonals of Peter's quadrilateral is equal to the acute angle between the diagonals of Basil's quadrilateral.

2015 Puerto Rico Team Selection Test, 2

Tags: geometry , angle , equal segments

In the triangle $ABC$, let $P$, $Q$, and $R$ lie on the sides $BC$, $AC$, and $AB$ respectively, such that $AQ = AR$, $BP = BR$ and $CP = CQ$. Let $\angle PQR=75^o$ and $\angle PRQ=35^o$. Calculate the measures of the angles of the triangle $ABC$.

2020 BMT Fall, Tie 1

Tags: angle , geometry

An [i]exterior [/i] angle is the supplementary angle to an interior angle in a polygon. What is the sum of the exterior angles of a triangle and dodecagon ($12$-gon), in degrees?

1997 Tournament Of Towns, (532) 4

Tags: hexagon , angle , area , geometry

$AC' BA'C B'$ is a convex hexagon such that $AB' = AC'$, $BC' = BA'$, $CA' = CB'$ and $\angle A +\angle B + \angle C = \angle A' + \angle B' + \angle C'$. Prove that the area of the triangle $ABC$ is half the area of the hexagon. (V Proizvolov)

May Olympiad L2 - geometry, 2022.3

Tags: angle , geometry

Let $ABCD$ be a square, $E$ a point on the side $CD$, and $F$ a point inside the square such that that triangle $BFE$ is isosceles and $\angle BFE = 90^o$ . If $DF=DE$, find the measure of angle $\angle FDE$.

2018 Peru Iberoamerican Team Selection Test, P2

Tags: geometry , angle chasing , equal segments , angle

Let $ABC$ be a triangle with $AB = AC$ and let $D$ be the foot of the height drawn from $A$ to $BC$. Let $P$ be a point inside the triangle $ADC$ such that $\angle APB> 90^o$ and $\angle PAD + \angle PBD = \angle PCD$. The $CP$ and $AD$ lines are cut at $Q$ and the $BP$ and $AD$ lines cut into $R$. Let $T$ be a point in segment $AB$ such that $\angle TRB = \angle DQC$ and let S be a point in the extension of the segment $AP$ (on the $P$ side) such that $\angle PSR = 2 \angle PAR$. Prove that $RS = RT$.

VI Soros Olympiad 1999 - 2000 (Russia), 11.10

Tags: angle , geometry

In triangle $ABC$, angle $A$ is equal to $a$ and angle $B$ is equal to $2a$. A circle with center at point $C$ of radius $CA$ intersects the line containing the bisector of the exterior angle at vertex $B$, at points $M$ and $N$. Find the angles of triangle $MAN$.

Ukrainian TYM Qualifying - geometry, I.5

Tags: 3d geometry , pyramid , angle , geometry

The heights of a triangular pyramid intersect at one point. Prove that all flat angles at any vertex of the surface are either acute, or right, or obtuse.

1997 Tournament Of Towns, (529) 2

Tags: angle , geometry

One side of a triangle is equal to one third of the sum of the other two. Prove that the angle opposite the first side is the smallest angle of the triangle. (AK Tolpygo)

2023 Novosibirsk Oral Olympiad in Geometry, 6

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 Junior Balkan Team Selection Tests - Romania, 4

Tags: geometry , ratio , angle

Let $ABCD$ be a quadrilateral with $\angle A + \angle C = 60^o$. If $AB \cdot CD = BC \cdot AD$, prove that $AB \cdot CD = AC \cdot BD$. Leonard Giugiuc

2000 Chile National Olympiad, 4

Tags: geometry , angle

Let $ AD $ be the bisector of a triangle $ ABC $ $ (D \in BC) $ such that $ AB + AD = CD $ and $ AC + AD = BC $. Determine the measure of the angles of $ \vartriangle ABC $

2020 Yasinsky Geometry Olympiad, 4

Tags: altitude , geometry , angle chasing , angle

Let $BB_1$ and $CC_1$ be the altitudes of the acute-angled triangle $ABC$. From the point $B_1$ the perpendiculars $B_1E$ and $B_1F$ are drawn on the sides $AB$ and $BC$ of the triangle, respectively, and from the point $C_1$ the perpendiculars $C_1 K$ and $C_1L$ on the sides $AC$ and $BC$, respectively. It turned out that the lines $EF$ and $KL$ are perpendicular. Find the measure of the angle $A$ of the triangle $ABC$. (Alexander Dunyak)

1995 Tournament Of Towns, (443) 3

Tags: square , geometry , angle

Suppose $L$ is the circle inscribed in the square $T_1$, and $T_2$ is the square inscribed in $L$, so that vertices of $T_1$ lie on the straight lines containing the sides of $T_2$. Find the angles of the convex octagon whose vertices are at the tangency points of $L$ with the sides of $T_1$ and at the vertices of $T_2$. (S Markelov)

Kyiv City MO Juniors 2003+ geometry, 2018.9.51

Tags: geometry , square , angle

Given a circle $\Gamma$ with center at point $O$ and diameter $AB$. $OBDE$ is square, $F$ is the second intersection point of the line $AD$ and the circle $\Gamma$, $C$ is the midpoint of the segment $AF$. Find the value of the angle $OCB$.

2019 Argentina National Olympiad, 3

Tags: geometry , equal segments , angle

In triangle $ABC$ it is known that $\angle ACB = 2\angle ABC$. Furthermore $P$ is an interior point of the triangle $ABC$ such that $AP = AC$ and $PB = PC$. Prove that $\angle BAC = 3 \angle BAP$.

1999 IMO Shortlist, 4

Tags: geometry , trigonometry , triangle , similarity , angle

For a triangle $T = ABC$ we take the point $X$ on the side $(AB)$ such that $AX/AB=4/5$, the point $Y$ on the segment $(CX)$ such that $CY = 2YX$ and, if possible, the point $Z$ on the ray ($CA$ such that $\widehat{CXZ} = 180 - \widehat{ABC}$. We denote by $\Sigma$ the set of all triangles $T$ for which $\widehat{XYZ} = 45$. Prove that all triangles from $\Sigma$ are similar and find the measure of their smallest angle.

1979 IMO Shortlist, 17

Tags: triangle , geometry , angle

Inside an equilateral triangle $ABC$ one constructs points $P, Q$ and $R$ such that \[\angle QAB = \angle PBA = 15^\circ,\\ \angle RBC = \angle QCB = 20^\circ,\\ \angle PCA = \angle RAC = 25^\circ.\] Determine the angles of triangle $PQR.$

2016 Latvia Baltic Way TST, 13

Tags: geometric inequality , geometry , angle

Suppose that $A, B, C$, and $X$ are any four distinct points in the plane with $$\max \,(BX,CX) \le AX \le BC.$$ Prove that $\angle BAC \le 150^o$.

2020 Estonia Team Selection Test, 2

Tags: geometry , combinatorial geometry , min , angle , combinatorics

Let $n$ be an integer, $n \ge 3$. Select $n$ points on the plane, none of which are three on the same line. Consider all triangles with vertices at selected points, denote the smallest of all the interior angles of these triangles by the variable $\alpha$. Find the largest possible value of $\alpha$ and identify all the selected $n$ point placements for which the max occurs.

1998 Swedish Mathematical Competition, 4

Tags: geometry , area , angle

$ABCD$ is a quadrilateral with $\angle A = 90o$, $AD = a$, $BC = b$, $AB = h$, and area $\frac{(a+b)h}{2}$. What can we say about $\angle B$?