Found problems: 509
2022 Federal Competition For Advanced Students, P2, 2
Let $ ABC$ be an acute-angled, non-isosceles triangle with orthocenter $H$, $M$ midpoint of side $AB$ and $w$ bisector of angle $\angle ACB$. Let $S$ be the point of intersection of the perpendicular bisector of side $AB$ with $w$ and $F$ the foot of the perpendicular from $H$ on $w$. Prove that the segments $MS$ and $MF$ are equal.
[i](Karl Czakler)[/i]
Novosibirsk Oral Geo Oly VII, 2023.6
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 $\angle CFA$.
Kyiv City MO Juniors Round2 2010+ geometry, 2020.8.2
Given a convex quadrilateral $ABCD$, in which $\angle CBD = 90^o$, $\angle BCD =\angle CAD$ and $AD= 2BC$. Prove that $CA =CD$.
(Anton Trygub)
1955 Moscow Mathematical Olympiad, 313
On the numerical line, arrange a system of closed segments of length $1$ without common points (endpoints included) so that any infinite arithmetic progression with any non zero difference and any first term has a common point with a segment of the system.
2018 Ecuador NMO (OMEC), 5
Let $ABC$ be an acute triangle and let $M$, $N$, and $ P$ be on $CB$, $AC$, and $AB$, respectively, such that $AB = AN + PB$, $BC = BP + MC$, $CA = CM + AN$. Let $\ell$ be a line in a different half plane than $C$ with respect to to the line $AB$ such that if $X, Y$ are the projections of $A, B$ on $\ell$ respectively, $AX = AP$ and $BY = BP$. Prove that $NXYM$ is a cyclic quadrilateral.
2000 Tournament Of Towns, 3
In a triangle $ABC, AB = c, BC = a, CA = b$, and $a < b < c$. Points $B'$ and $A'$ are chosen on the rays $BC$ and $AC$ respectively so that $BB'= AA'= c$. Points $C''$ and $B''$ are chosen on the rays $CA$ and $BA$ so that $CC'' = BB'' = a$. Find the ratio of the segment $A'B'$ to the segment $C'' B''$.
(R Zhenodarov)
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 $\angle KCD = \angle DAC.$
2012 Denmark MO - Mohr Contest, 5
In the hexagon $ABCDEF$, all angles are equally large. The side lengths satisfy $AB = CD = EF = 3$ and $BC = DE = F A = 2$. The diagonals $AD$ and $CF$ intersect each other in the point $G$. The point $H$ lies on the side $CD$ so that $DH = 1$. Prove that triangle $EGH$ is equilateral.
2020 Tournament Of Towns, 5
Let $ABCD$ be an inscribed trapezoid. The base $AB$ is $3$ times longer than $CD$. Tangents to the circumscribed circle at the points $A$ and $C$ intersect at the point $K$. Prove that the angle $KDA$ is a right angle.
Alexandr Yuran
2010 Oral Moscow Geometry Olympiad, 5
Points $K$ and $M$ are taken on the sides $AB$ and $CD$ of square $ABCD$, respectively, and on the diagonal $AC$ - point $L$ such that $ML = KL$. Let $P$ be the intersection point of the segments $MK$ and $BD$. Find the angle $\angle KPL$.
1998 Tournament Of Towns, 2
$ABCD$ is a parallelogram. A point $M$ is found on the side $AB$ or its extension such that $\angle MAD = \angle AMO$ where $O$ is the intersection point of the diagonals of the parallelogram. Prove that $MD = MG$.
(M Smurov)
2018 Dutch IMO TST, 3
Let $ABC$ be an acute triangle, and let $D$ be the foot of the altitude through $A$. On $AD$, there are distinct points $E$ and $F$ such that $|AE| = |BE|$ and $|AF| =|CF|$. A point$ T \ne D$ satis es $\angle BTE = \angle CTF = 90^o$. Show that $|TA|^2 =|TB| \cdot |TC|$.
Denmark (Mohr) - geometry, 2009.4
Let $E$ be an arbitrary point different from $A$ and $B$ on the side $AB$ of a square $ABCD$, and let $F$ and $G$ be points on the segment $CE$ so that $BF$ and $DG$ are perpendicular to $CE$. Prove that $DF = AG$.
2020 Ukrainian Geometry Olympiad - April, 3
Triangle $ABC$. Let $B_1$ and $C_1$ be such points, that $AB= BB_1, AC=CC_1$ and $B_1, C_1$ lie on the circumscribed circle $\Gamma$ of $\vartriangle ABC$. Perpendiculars drawn from from points $B_1$ and $C_1$ on the lines $AB$ and $AC$ intersect $\Gamma$ at points $B_2$ and $C_2$ respectively, these points lie on smaller arcs $AB$ and $AC$ of circle $\Gamma$ respectively, Prove that $BB_2 \parallel CC_2$.
2020 Ukrainian Geometry Olympiad - April, 5
Given a convex pentagon $ABCDE$, with $\angle BAC = \angle ABE = \angle DEA - 90^o$, $\angle BCA = \angle ADE$ and also $BC = ED$. Prove that $BCDE$ is parallelogram.
1997 Belarusian National Olympiad, 1
Different points $A_1,A_2,A_3,A_4,A_5$ lie on a circle so that $A_1A_2 = A_2A_3 = A_3A_4 =A_4A_5$. Let $A_6$ be the diametrically opposite point to $A_2$, and $A_7$ be the intersection of $A_1A_5$ and $A_3A_6$. Prove that the lines $A_1A_6$ and $A_4A_7$ are perpendicular
Indonesia MO Shortlist - geometry, g4
Given an isosceles triangle $ABC$ with $AB = AC$, suppose $D$ is the midpoint of the $AC$. The circumcircle of the $DBC$ triangle intersects the altitude from $A$ at point $E$ inside the triangle $ABC$, and the circumcircle of the triangle $AEB$ cuts the side $BD$ at point $F$. If $CF$ cuts $AE$ at point $G$, prove that $AE = EG$.
Novosibirsk Oral Geo Oly VII, 2021.5
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 $\angle BCA$.
2021 Yasinsky Geometry Olympiad, 1
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)
2014 Contests, 2 juniors
Let $ABCD$ be a parallelogram with an acute angle at $A$. Let $G$ be a point on the line $AB$, distinct from $B$, such that $|CG| = |CB|$. Let $H$ be a point on the line $BC$, distinct from $B$, such that $|AB| =|AH|$. Prove that triangle $DGH$ is isosceles.
[asy]
unitsize(1.5 cm);
pair A, B, C, D, G, H;
A = (0,0);
B = (2,0);
D = (0.5,1.5);
C = B + D - A;
G = reflect(A,B)*(C) + C - B;
H = reflect(B,C)*(H) + A - B;
draw(H--A--D--C--G);
draw(interp(A,G,-0.1)--interp(A,G,1.1));
draw(interp(C,H,-0.1)--interp(C,H,1.1));
draw(D--G--H--cycle, dashed);
dot("$A$", A, SW);
dot("$B$", B, SE);
dot("$C$", C, E);
dot("$D$", D, NW);
dot("$G$", G, NE);
dot("$H$", H, SE);
[/asy]
2019 Saudi Arabia Pre-TST + Training Tests, 1.3
Let $ABCDEF$ be a convex hexagon satisfying $AC = DF, CE = FB$ and $EA = BD$. Prove that the lines connecting the midpoints of opposite sides of the hexagon $ABCDEF$ intersect in one point.
VMEO IV 2015, 12.3
Triangle $ABC$ is inscribed in circle $(O)$. $ P$ is a point on arc $BC$ that does not contain $ A$ such that $AP$ is the symmedian of triangle $ABC$. $E ,F$ are symmetric of $P$ wrt $CA, AB$ respectively . $K$ is symmetric of $A$ wrt $EF$. $L$ is the projection of $K$ on the line passing through $A$ and parallel to $BC$. Prove that $PA=PL$.
2017 Thailand TSTST, 3
In $\vartriangle ABC$ with $AB > AC$, the tangent to the circumcircle at $A$ intersects line $BC$ at $P$. Let $Q$ be the point on $AB$ such that $AQ = AC$, and $A$ lies between $B$ and $Q$. Let $R$ be the point on ray $AP$ such that $AR = CP$. Let $X, Y$ be the midpoints of $AP, CQ$ respectively. Prove that $CR = 2XY$ .
Novosibirsk Oral Geo Oly IX, 2023.3
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 $\angle CFA$.
Ukraine Correspondence MO - geometry, 2010.7
An arbitrary point $D$ was marked on the median $BM$ of the triangle $ABC$. It is known that the point $DE\parallel AB$ and $CE \parallel BM$. Prove that $BE = AD$