Found problems: 361
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?
2007 Sharygin Geometry Olympiad, 15
In a triangle $ABC$, let $AA', BB'$ and $CC'$ be the bisectors. Suppose $A'B' \cap CC' =P$ and $A'C' \cap BB'= Q$. Prove that $\angle PAC = \angle QAB$.
Kyiv City MO Juniors Round2 2010+ geometry, 2015.8.41
On the sides $AB, \, \, BC, \, \, CA$ of the triangle $ABC$ the points ${{C} _ {1}}, \, \, {{A} _ { 1}},\, \, {{B} _ {1}}$ are selected respectively, that are different from the vertices. It turned out that $\Delta {{A} _ {1}} {{B} _ {1}} {{C} _ {1}}$ is equilateral, $\angle B{{C}_{1}}{{A}_{1}}=\angle {{C}_{1}}{{B}_{1}}A$ and $\angle B{{A}_{1}}{{C}_{1}}=\angle {{A}_{1}}{{B}_{1}}C$ . Is $ \Delta ABC$ equilateral?
2021 SAFEST Olympiad, 4
Let $ABC$ be a triangle with $AB > AC$. Let $D$ be a point on the side $AB$ such that $DB = DC$ and let $M$ be the midpoint of $AC$. The line parallel to $BC$ passing through $D$ intersects the line $BM$ in $K$. Show that $\angle KCD = \angle DAC.$
2016 Silk Road, 2
Around the acute-angled triangle $ABC$ ($AC>CB$) a circle is circumscribed, and the point $N$ is midpoint of the arc $ACB$ of this circle. Let the points $A_1$ and $B_1$ be the feet of perpendiculars on the straight line $NC$, drawn from points $A$ and $B$ respectively (segment $NC$ lies inside the segment $A_1B_1$). Altitude $A_1A_2$ of triangle $A_1AC$ and altitude $B_1B_2$ of triangle $B_1BC$ intersect at a point $K$ . Prove that $\angle A_1KN=\angle B_1KM$, where $M$ is midpoint of the segment $A_2B_2$ .
2004 Olympic Revenge, 1
$ABC$ is a triangle and $D$ is an internal point such that $\angle DAB=\angle DBC =\angle DCA$. $O_a$ is the circumcenter of $DBC$. $O_b$ is the circumcenter of $DAC$. $O_c$ is the circumcenter of $DAB$. Show that if the area of $ABC$ and $O_aO_bO_c$ are equal then $ABC$ is equilateral.
Indonesia Regional MO OSP SMA - geometry, 2015.3
Given the isosceles triangle $ABC$, where $AB = AC$. Let $D$ be a point in the segment $BC$ so that $BD = 2DC$. Suppose also that point $P$ lies on the segment $AD$ such that: $\angle BAC = \angle BP D$. Prove that $\angle BAC = 2\angle DP C$.
Swiss NMO - geometry, 2018.4
Let $D$ be a point inside an acute triangle $ABC$, such that $\angle BAD = \angle DBC$ and $\angle DAC = \angle BCD$. Let $P$ be a point on the circumcircle of the triangle $ADB$. Suppose $P$ are itself outside the triangle $ABC$. A line through $P$ intersects the ray $BA$ in $X$ and ray $CA$ in $Y$, so that $\angle XPB = \angle PDB$. Show that $BY$ and $CX$ intersect on $AD$.
1999 Poland - Second Round, 4
Let $P$ be a point inside a triangle $ABC$ such that $\angle PAB = \angle PCA$ and $\angle PAC =
\angle PBA$.
If $O \ne P$ is the circumcenter of $\triangle ABC$, prove that $\angle APO$ is right.
1903 Eotvos Mathematical Competition, 3
Let $A,B,C,D$ be the vertices of a rhombus, let $k_1$ be the circle through $B,C$ and $D$, let $k_2$ be the circle through $A,C$ and $D$, let $k_3$ be the circle through $A,B$ and $D$, let $k_4$ be the circle through $A,B$ and $C$. Prove that the tangents to $k_1$ and $k_3$ at $B$ form the same angle as the tangents to $k_2$ and $k_4$ at $A$.
2012 NZMOC Camp Selection Problems, 5
Let $ABCD$ be a quadrilateral in which every angle is smaller than $180^o$. If the bisectors of angles $\angle DAB$ and $\angle DCB$ are parallel, prove that $\angle ADC = \angle ABC$
2018 India PRMO, 17
Triangles $ABC$ and $DEF$ are such that $\angle A = \angle D, AB = DE = 17, BC = EF = 10$ and $AC - DF = 12$. What is $AC + DF$?
Estonia Open Senior - geometry, 2019.1.5
Polygon $A_0A_1...A_{n-1}$ satisfies the following:
$\bullet$ $A_0A_1 \le A_1A_2 \le ...\le A_{n-1}A_0$ and
$\bullet$ $\angle A_0A_1A_2 = \angle A_1A_2A_3 = ... = \angle A_{n-2}A_{n-1}A_0$ (all angles are internal angles).
Prove that this polygon is regular.
1987 All Soviet Union Mathematical Olympiad, 450
Given a convex pentagon $ABCDE$ with $\angle ABC= \angle ADE$ and $\angle AEC= \angle ADB$ . Prove that $\angle BAC = \angle DAE$ .
2007 Postal Coaching, 5
Let $P$ be an interior point of triangle $ABC$ such that $\angle BPC = \angle CPA =\angle APB = 120^o$. Prove that the Euler lines of triangles $APB,BPC,CPA$ are concurrent.
Durer Math Competition CD 1st Round - geometry, 2015.C2
Given a rectangle $ABCD$, side $AB$ is longer than side $BC$. Find all the points $P$ of the side line $AB$ from which the sides $AD$ and $DC$ are seen from the point $P$ at an equal angle (i.e. $\angle APD = \angle DPC$)
2010 Mathcenter Contest, 3
$ABCD$ is a convex quadrilateral, and the point $K$ is a point on side $AB$, where $\angle KDA=\angle BCD$, let $L$ be a point on the diagonal $AC$, where $KL$ is parallel to $BC$. Prove that $$\angle KDB=\angle LDC.$$
[i](tatari/nightmare)[/i]
2014 Junior Balkan Team Selection Tests - Romania, 3
Let $ABC$ be an acute triangle and let $O$ be its circumcentre. Now, let the diameter $PQ$ of circle $ABC$ intersects sides $AB$ and $AC$ in their interior at$ D$ and $E$, respectively. Now, let $F$ and $G$ be the midpoints of $CD$ and $BE$. Prove that $\angle FOG=\angle BAC$
1994 Austrian-Polish Competition, 9
On the plane are given four distinct points $A,B,C,D$ on a line $g$ in this order, at the mutual distances $AB = a, BC = b, CD = c$.
(a) Construct (if possible) a point $P$ outside line $g$ such that $\angle APB =\angle BPC =\angle CPD$.
(b) Prove that such a point $P$ exists if and only if $ (a+b)(b+c) < 4ac$
1992 Spain Mathematical Olympiad, 5
Given a triangle $ABC$, show how to construct the point $P$ such that $\angle PAB= \angle PBC= \angle PCA$.
Express this angle in terms of $\angle A,\angle B,\angle C$ using trigonometric functions.
Geometry Mathley 2011-12, 11.3
Let $ABC$ be a triangle such that $AB = AC$ and let $M$ be a point interior to the triangle. If $BM$ meets $AC$ at $D$. show that $\frac{DM}{DA}=\frac{AM}{AB}$ if and only if $\angle AMB = 2\angle ABC$.
Michel Bataille
1997 Tournament Of Towns, (554) 4
Two circles intersect at points $A$ and $B$. A common tangent touches the first circle at point $C$ and the second at point $D$. Let $\angle CBD > \angle CAD$. Let the line $CB$ intersect the second circle again at point $E$. Prove that $AD$ bisects the angle $\angle CAE$.
(P Kozhevnikov)
Kyiv City MO Juniors 2003+ geometry, 2011.8.41
The medians $AL, BM$, and $CN$ are drawn in the triangle $ABC$. Prove that $\angle ANC = \angle ALB$ if and only if $\angle ABM =\angle LAC$.
(Veklich Bogdan)
2013 Argentina National Olympiad, 2
In a convex quadrilateral $ABCD$ the angles $\angle A$ and $\angle C$ are equal and the bisector of $\angle B$ passes through the midpoint of the side $CD$. If it is known that $CD = 3AD$, calculate $\frac{AB}{BC}$.
2018 Regional Olympiad of Mexico West, 3
A scalene acute triangle $ABC$ is drawn on the plane, in which $BC$ is the longest side. Points $P$ and $D$ are constructed, the first inside $ABC$ and the second outside, so that $\angle ABC = \angle CBD$, $\angle ACP = \angle BCD$ and that the area of triangle $ABC$ is equal to the area of quadrilateral $BPCD$. Prove that triangles $BCD$ and $ACP$ are similar.