Olympiad problems

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 $\angle BCA$.

2020 Regional Olympiad of Mexico West, 6

Tags: geometry , equal segments

Let $ M $ be the midpoint of side $ BC $ of a scalene triangle $ ABC $. The circle passing through $ A $, $ B $ and $ M $ intersects side $ AC $ again at $ D $. The circle passing through $ A $, $ C $ and $ M $ cuts side $ AB $ again at $ E $. Let $ O $ be the circumcenter of triangle $ ADE $. Prove that $ OB=OC $.

Swiss NMO - geometry, 2006.7

Tags: geometry , equal segments , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral with $\angle ABC = 60^o$ and $| BC | = | CD |$. Prove that $|CD| + |DA| = |AB|$

1949-56 Chisinau City MO, 45

Tags: tangent , locus , circles , equal segments

Determine the locus of points, from which the tangent segments to two given circles are equal.

2000 Tournament Of Towns, 3

Tags: ratio , equal segments , geometry

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)

Kyiv City MO Juniors 2003+ geometry, 2019.8.3

Tags: equal angles , equal segments , angle , geometry

In the triangle $ABC$ it is known that $2AC=AB$ and $\angle A = 2\angle B$. In this triangle draw the angle bisector $AL$, and mark point $M$, the midpoint of the side $AB$. It turned out that $CL = ML$. Prove that $\angle B= 30^o$. (Hilko Danilo)

1995 Tournament Of Towns, (458) 3

Tags: geometry , trapezoid , equal segments

The non-parallel sides of a trapezium serve as the diameters of two circles. Prove that all four tangents to the circles drawn from the point of intersection of the diagonals are equal (if this point lies outside the circles). (S Markelov)

Kyiv City MO Juniors Round2 2010+ geometry, 2017.7.41

Tags: geometry , isosceles , equal segments

Let $AC$ be the largest side of the triangle $ABC$. The point M is selected on the ray $AC$ ray, and point $N$ on ray $CA$ such that $CN = CB$ and$ AM = AB$ . a) Prove that $\vartriangle ABC$ is isosceles if we know that $BM = BN$. b) Will the statement remain true if $AC$ is not necessarily the largest side of triangle $ABC$?

2013 District Olympiad, 4

Tags: square , equal segments , geometry

Consider the square $ABCD$ and the point $E$ inside the angle $CAB$, such that $\angle BAE =15^o$, and the lines $BE$ and $BD$ are perpendicular. Prove that $AE = BD$.

2019 Saudi Arabia Pre-TST + Training Tests, 1.3

Tags: geometry , hexagon , concurrency , equal segments

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.

2017 NZMOC Camp Selection Problems, 2

Tags: geometry , isosceles , parallelogram , equal segments

Let $ABCD$ be a parallelogram with an acute angle at $A$. Let $G$ be the point on the line $AB$, distinct from $B$, such that $CG = CB$. Let $H$ be the point on the line $BC$, distinct from $B$, such that $AB = AH$. Prove that triangle $DGH$ is isosceles.

2015 India Regional MathematicaI Olympiad, 5

Tags: circles , equal segments , geometry

Two circles $\Gamma$ and $\Sigma$ intersect at two distinct points $A$ and $B$. A line through $B$ intersects $\Gamma$ and $\Sigma$ again at $C$ and $D$, respectively. Suppose that $CA=CD$. Show that the centre of $\Sigma$ lies on $\Gamma$.

2022 Chile National Olympiad, 2

Tags: equal segments , geometry , incenter

Let $ABC$ be a triangle such that $\angle CAB = 60^o$. Consider $D, E$ points on sides $AC$ and $AB$ respectively such that $BD$ bisects angle $\angle ABC$ , $CE$ bisects angle $\angle BCA$ and let $I$ be the intersection of them. Prove that $|ID| =|IE|$.

Swiss NMO - geometry, 2016.8

Tags: geometry , orthocenter , equal segments

Let $ABC$ be an acute-angled triangle with height intersection $H$. Let $G$ be the intersection of parallel of $AB$ through $H$ with the parallel of $AH$ through $B$. Let $I$ be the point on the line $GH$, so that $AC$ bisects segment $HI$. Let $J$ be the second intersection of $AC$ and the circumcircle of the triangle $CGI$. Show that $IJ = AH$

Indonesia MO Shortlist - geometry, g10

Tags: equal segments , isosceles , geometry

Given a triangle $ABC$ with $AB = AC$, angle $\angle A = 100^o$ and $BD$ bisector of angle $\angle B$. Prove that $$BC = BD + DA.$$

2011 Korea Junior Math Olympiad, 2

Tags: geometry , equal segments , collinear , collinearity , circles , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral inscirbed in circle $O$. Let the tangent to $O$ at $A$ meet $BC$ at $S$, and the tangent to $O$ at $B$ meet $CD$ at $T$. Circle with $S$ as its center and passing $A$ meets $BC$ at $E$, and $AE$ meets $O$ again at $F(\ne A)$. The circle with $T$ as its center and passing $B$ meets $CD$ at $K$. Let $P = BK \cap AC$. Prove that $P,F,D$ are collinear if and only if $AB = AP$.

2020 Federal Competition For Advanced Students, P1, 2

Tags: equal segments , right triangle , geometry

Let $ABC$ be a right triangle with a right angle in $C$ and a circumcenter $U$. On the sides $AC$ and $BC$, the points $D$ and $E$ lie in such a way that $\angle EUD = 90 ^o$. Let $F$ and $G$ be the projection of $D$ and $E$ on $AB$, respectively. Prove that $FG$ is half as long as $AB$. (Walther Janous)

2013 Tournament of Towns, 5

Tags: geometry , equal segments , right angle , angle

In a quadrilateral $ABCD$, angle $B$ is equal to $150^o$, angle $C$ is right, and sides $AB$ and $CD$ are equal. Determine the angle between $BC$ and the line connecting the midpoints of sides $BC$ and $AD$.

2011 Oral Moscow Geometry Olympiad, 1

Tags: geometry , equal segments , perpendicular , rectangle

The bisector of angle $B$ and the bisector of external angle $D$ of rectangle $ABCD$ intersect side $AD$ and line $AB$ at points $M$ and $K$, respectively. Prove that the segment $MK$ is equal and perpendicular to the diagonal of the rectangle.

2009 Denmark MO - Mohr Contest, 4

Tags: geometry , square , equal segments

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 $DF = AG$.

2011 Sharygin Geometry Olympiad, 7

Tags: geometry , geometric inequality , equal segments

Points $P$ and $Q$ on sides $AB$ and $AC$ of triangle $ABC$ are such that $PB = QC$. Prove that $PQ < BC$.

1998 North Macedonia National Olympiad, 1

Tags: angle chasing , pentagon , equal segments , geometry

Let $ABCDE$ be a convex pentagon with $AB = BC =CA$ and $CD = DE = EC$. Let $T$ be the centroid of $\vartriangle ABC$, and $N$ be the midpoint of $AE$. Compute $\angle NT D$

Kharkiv City MO Seniors - geometry, 2016.11.5

Tags: geometry , circumcenter , circles , equal segments

The circle $\omega$ passes through the vertices $B$ and $C$ of triangle $ABC$ and intersects its sides $AC,AB$ at points $A,E$, respectively. On the ray $BD$, a point $K$ such that $BK = AC$ is chosen , and on the ray $CE$, a point $L$ such that $CL = AB$ is chosen. Prove that the center $O$ of the circumscribed circle of the triangle $AKL$ lies on the circle $\omega$.

2010 Dutch IMO TST, 4

Tags: geometry , square , circles , equal segments

Let $ABCD$ be a square with circumcircle $\Gamma_1$. Let $P$ be a point on the arc $AC$ that also contains $B$. A circle $\Gamma_2$ touches $\Gamma_1$ in $P$ and also touches the diagonal $AC$ in $Q$. Let $R$ be a point on $\Gamma_2$ such that the line $DR$ touches $\Gamma_2$. Proof that $|DR| = |DA|$.

Croatia MO (HMO) - geometry, 2012.3

Tags: geometry , cyclic quadrilateral , equal segments

Let $ABCD$ be a cyclic quadrilateral such that $|AD| =|BD|$ and let $M$ be the intersection of its diagonals. Furthermore, let $N$ be the second intersection of the diagonal $AC$ with the circle passing through points $B, M$ and the center of the circle inscribed in triangle $BCM$. Prove that $AN \cdot NC = CD \cdot BN$