Found problems: 721
2008 Oral Moscow Geometry Olympiad, 3
In the regular hexagon $ABCDEF$ on the line $AF$, the point $X$ is taken so that the angle $XCD$ is $45^o$. Find the angle $\angle FXE$.
(Kiev Olympiad)
2022 IFYM, Sozopol, 7
Points $M$, $N$, $P$ and $Q$ are midpoints of the sides $AB$, $BC$, $CD$ and $DA$ of the inscribed quadrilateral $ABCD$ with intersection point $O$ of its diagonals. Let $K$ be the second intersection point of the circumscribed circles of $MOQ$ and $NOP$. Prove that $OK\perp AC$.
1985 IMO Shortlist, 2
A polyhedron has $12$ faces and is such that:
[b][i](i)[/i][/b] all faces are isosceles triangles,
[b][i](ii)[/i][/b] all edges have length either $x$ or $y$,
[b][i](iii)[/i][/b] at each vertex either $3$ or $6$ edges meet, and
[b][i](iv)[/i][/b] all dihedral angles are equal.
Find the ratio $x/y.$
Indonesia MO Shortlist - geometry, g1
Given a cyclic quadrilateral $ABCD$ so that $AB = AD$ and $AB + BC <CD$. Prove that the angle $ABC$ is more than $120$ degrees.
Ukraine Correspondence MO - geometry, 2003.8
In the triangle $ABC$, $D$ is the midpoint of $AB$, and $E$ is the point on the side $BC$, for which $CE = \frac13 BC$. It is known that $\angle ADC =\angle BAE$. Find $\angle BAC$.
2011 District Olympiad, 3
Let $ABCA'B'C'$ a right triangular prism with the bases equilateral triangles. A plane $\alpha$ containing point $A$ intersects the rays $BB'$ and $CC'$ at points E and $F$, so that $S_ {ABE} + S_{ACF} = S_{AEF}$. Determine the measure of the angle formed by the plane $(AEF)$ with the plane $(BCC')$.
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 $BD$.
2015 Iran Geometry Olympiad, 2
Let $ABC$ be a triangle with $\angle A = 60^o$. The points $M,N,K$ lie on $BC,AC,AB$ respectively such that $BK = KM = MN = NC$. If $AN = 2AK$, find the values of $\angle B$ and $\angle C$.
by Mahdi Etesami Fard
2023 Novosibirsk Oral Olympiad in Geometry, 6
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 $\angle CFA$.
2009 Cuba MO, 8
Let $ABC$ be an isosceles triangle with base $BC$ and $\angle BAC = 20^o$. Let $D$ a point on side $AB$ such that $AD = BC$. Determine $\angle DCA$.
1988 IMO Longlists, 47
In the convex pentagon $ ABCDE,$ the sides $ BC, CD, DE$ are equal. Moreover each diagonal of the pentagon is parallel to a side ($ AC$ is parallel to $ DE$, $ BD$ is parallel to $ AE$ etc.). Prove that $ ABCDE$ is a regular pentagon.
1997 Tournament Of Towns, (532) 4
$AC' BA'C B'$ 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)
2022 May Olympiad, 3
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 $\angle FDE$.
2019 Oral Moscow Geometry Olympiad, 4
Given a right triangle $ABC$ ($\angle C=90^o$). The $C$-excircle touches the hypotenuse $AB$ at point $C_1, A_1$ is the touchpoint of $B$-excircle with line $BC, B_1$ is the touchpoint of $A$-excircle with line $AC$. Find the angle $\angle A_1C_1B_1$.
Brazil L2 Finals (OBM) - geometry, 2007.1
Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be the intersection of straight lines $BO$ and $AC$ and $\omega$ be the circumcircle of triangle $AOP$. Suppose that $BO = AP$ and that the measure of the arc $OP$ in $\omega$, that does not contain $A$, is $40^o$. Determine the measure of the angle $\angle OBC$.
[img]https://3.bp.blogspot.com/-h3UVt-yrJ6A/XqBItXzT70I/AAAAAAAAL2Q/7LVv0gmQWbo1_3rn906fTn6wosY1-nIfwCK4BGAYYCw/s1600/2007%2Bomb%2Bl2.png[/img]
2015 Gulf Math Olympiad, 2
a) Let $UVW$ , $U'V'W'$ be two triangles such that $ VW = V'W' , UV = U'V' , \angle WUV = \angle W'U'V'.$
Prove that the angles $\angle VWU , \angle V'W'U'$ are equal or supplementary.
b) $ABC$ is a triangle where $\angle A$ is [b]obtuse[/b]. take a point $P$ inside the triangle , and extend $AP,BP,CP$ to meet the sides $BC,CA,AB$ in $K,L,M$ respectively. Suppose that $PL = PM .$
1) If $AP$ bisects $\angle A$ , then prove that $AB = AC$ .
2) Find the angles of the triangle $ABC$ if you know that $AK,BL,CM$ are angle bisectors of the triangle $ABC$ and that $2AK = BL$.
1986 Greece Junior Math Olympiad, 3
Inside a triangle ABC, consider points $D, E$ such that $\angle ABD =\angle DBE=\angle EBC$ and $\angle ACD=\angle DC E=\angle ECB$. Calculate angles $\angle BDE$, $\angle B EC$, $\angle D E C$ in terms of the angle of the triangle $ABC$.
2017 Dutch Mathematical Olympiad, 2
A parallelogram $ABCD$ with $|AD| =|BD|$ has been given. A point $E$ lies on line segment $|BD|$ in such a way that $|AE| = |DE|$. The (extended) line $AE$ intersects line segment $BC$ in $F$. Line $DF$ is the angle bisector of angle $CDE$. Determine the size of angle $ABD$.
[asy]
unitsize (3 cm);
pair A, B, C, D, E, F;
D = (0,0);
A = dir(250);
B = dir(290);
C = B + D - A;
E = extension((A + D)/2, (A + D)/2 + rotate(90)*(A - D), B, D);
F = extension(A, E, B, C);
draw(A--B--C--D--cycle);
draw(A--F--D--B);
dot("$A$", A, SW);
dot("$B$", B, SE);
dot("$C$", C, NE);
dot("$D$", D, NW);
dot("$E$", E, S);
dot("$F$", F, SE);
[/asy]
1991 Greece National Olympiad, 2
Let $\widehat{xOy}$ be an acute angle , $A$ a point on ray $Oy$ and $B$ a point on ray $Ox$ such that $AB \perp OX$ .Prove that there are two points on $Ox$, each of the equidistant from $A$ and $Ox$.
Novosibirsk Oral Geo Oly IX, 2021.5
The pentagon $ABCDE$ is inscribed in the circle. Line segments $AC$ and $BD$ intersect at point $K$. Line segment $CE$ touches the circumcircle of triangle $ABK$ at point $N$. Find the angle $CNK$ if $\angle ECD = 40^o.$
2016 Japan MO Preliminary, 3
A hexagon $ABCDEF$ is inscribed in a circle. Let $P, Q, R, S$ be intersections of $AB$ and $DC$, $BC$ and $ED$, $CD$ and $FE$, $DE$ and $AF$, then $\angle BPC=50^{\circ}$, $\angle CQD=45^{\circ}$, $\angle DRE=40^{\circ}$, $\angle ESF=35^{\circ}$.
Let $T$ be an intersection of $BE$ and $CF$. Find $\angle BTC$.
2024 Nepal TST, P5
Let $ABC$ be an acute triangle so that $2BC = AB + AC,$ with incenter $I{}.$ Let $AI{}$ meet $BC{}$ at point $A'.{}$ The perpendicular bisector of $AA'{}$ meets $BI{}$ and $CI{}$ at points $B'{}$ and $C'{}$ respectively. Let $AB'{}$ intersect $(ABC)$ at $X{}$ and let $XI{}$ intersect $AC'{}$ at $X'{}.$ Prove that $2\angle XX'A'=\angle ABC.{}$
[i](Proposed by Kang Taeyoung, South Korea)[/i]
2016 Peru MO (ONEM), 1
Let $ABCD$ be a trapezoid of parallel bases $ BC$ and $AD$. If $\angle CAD = 2\angle CAB, BC = CD$ and $AC = AD$, determine all the possible values of the measure of the angle $\angle CAB$.
2012 District Olympiad, 2
The pyramid $VABCD$ has base the rectangle ABCD, and the side edges are congruent. Prove that the plane $(VCD)$ forms congruent angles with the planes $(VAC)$ and $(BAC)$ if and only if $\angle VAC = \angle BAC $.
OIFMAT II 2012, 3
In the interior of an equilateral triangle $ ABC $ a point $ P $ is chosen such that $ PA ^2 = PB ^2 + PC ^2 $. Find the measure of $ \angle BPC $.