Olympiad problems

1995 All-Russian Olympiad Regional Round, 9.6

Circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$ respectively intersect at $A$ and $B$. The circle passing through $O_1$, $O_2$, and $A$ intersects $S_1$, $S_2$ and line $AB$ again at $D$, $E$, and $C$, respectively. Show that $CD = CB = CE$.

1956 Moscow Mathematical Olympiad, 325

Tags: midpoint , locus , geometry , equal segments

On sides $AB$ and $CB$ of $\vartriangle ABC$ there are drawn equal segments, $AD$ and $CE$, respectively, of arbitrary length (but shorter than min($AB,BC$)). Find the locus of midpoints of all possible segments $DE$.

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 $\Gamma$.

Mathley 2014-15, 2

Tags: cyclic quadrilateral , parallel , equal segments

A quadrilateral $ABCD$ is inscribed in a circle and its two diagonals $AC,BD$ meet at $G$. Let $M$ be the center of $CD, E,F$ be the points on $BC, AD$ respectively such that $ME \parallel AC$ and $MF \parallel BD$. Point $H$ is the projection of $G$ onto $CD$. The circumcircle of $MEF$ meets $CD$ at $N$ distinct from $M$. Prove that $MN = MH$ Tran Quang Hung, Nguyen Le Phuoc, Thanh Xuan, Hanoi

2017 Flanders Math Olympiad, 1

Tags: parabola , equal segments , analytic geometry , conic , area of triangle

On the parabola $y = x^2$ lie three different points $P, Q$ and $R$. Their projections $P', Q'$ and $R'$ on the $x$-axis are equidistant and equal to $s$ , i.e. $| P'Q'| = | Q'R'| = s$. Determine the area of $\vartriangle PQR$ in terms of $s$

Cono Sur Shortlist - geometry, 1993.11

Tags: geometry , equal segments , semicircle

Let $\Gamma$ be a semicircle with center $O$ and diameter $AB$. $D$ is the midpoint of arc $AB$. On the ray $OD$, we take $E$ such that $OE = BD$. $BE$ intersects the semicircle at $F$ and $ P$ is the point on $AB$ such that $FP$ is perpendicular to $AB$. Prove that $BP=\frac13 AB$.

2020 Tournament Of Towns, 5

Tags: equal segments , geometry , right angle , trapezoid , trapezium

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

2017 QEDMO 15th, 10

Tags: equal segments , geometry , inradius

Let $\ell$ be a straight line and $P \notin \ell$ be a point in the plane. On $\ell$ are, in this arrangement, points $A_1, A_2,...$ such that the radii of the incircles of all triangles $P A_iA_{i + 1}$ are equal. Let $k \in N$. Show that the radius of the incircle of the triangle $P A_j A_{j + k}$ does not depend on the choice of $j \in N$ .

Kyiv City MO Seniors 2003+ geometry, 2006.11.3

Tags: equal segments , circumcircle , geometry , midpoint

Let $O$ be the center of the circle $\omega$ circumscribed around the acute-angled triangle $\vartriangle ABC$, and $W$ be the midpoint of the arc $BC$ of the circle $\omega$, which does not contain the point $A$, and $H$ be the point of intersection of the heights of the triangle $\vartriangle ABC$. Find the angle $\angle BAC$, if $WO = WH$. (O. Clurman)

May Olympiad L2 - geometry, 2010.2

Tags: equal segments , geometry , rectangle

Let $ABCD$ be a rectangle and the circle of center $D$ and radius $DA$, which cuts the extension of the side $AD$ at point $P$. Line $PC$ cuts the circle at point $Q$ and the extension of the side $AB$ at point $R$. Show that $QB = BR$.

2015 Caucasus Mathematical Olympiad, 2

Tags: equal segments , convex quadrilateral , equal angles , geometry

In the convex quadrilateral $ABCD$, point $K$ is the midpoint of $AB$, point $L$ is the midpoint of $BC$, point $M$ is the midpoint of CD, and point $N$ is the midpoint of $DA$. Let $S$ be a point lying inside the quadrilateral $ABCD$ such that $KS = LS$ and $NS = MS$ .Prove that $\angle KSN = \angle MSL$.

2015 Oral Moscow Geometry Olympiad, 3

Tags: geometry , construction , equal segments

In triangle $ABC$, points $D, E$, and $F$ are marked on sides $AC, BC$, and $AB$ respectively, so that $AD = AB$, $EC = DC$, $BF = BE$. After that, they erased everything except points $E, F$ and $D$. Reconstruct the triangle $ABC$ (no study required).

Champions Tournament Seniors - geometry, 2017.4

Tags: circles , equal segments , geometry , angle bisector

Let $AD$ be the bisector of triangle $ABC$. Circle $\omega$ passes through the vertex $A$ and touches the side $BC$ at point $D$. This circle intersects the sides $AC$ and $AB$ for the second time at points $M$ and $N$ respectively. Lines $BM$ and $CN$ intersect the circle for the second time $\omega$ at points $P$ and $Q$, respectively. Lines $AP$ and $AQ$ intersect side $BC$ at points $K$ and $L$, respectively. Prove that $KL=\frac12 BC$

2003 District Olympiad, 2

Tags: equal segments , geometry , perpendicular , parallel , right triangle

In the right triangle $ABC$ ( $\angle A = 90^o$), $D$ is the intersection of the bisector of the angle $A$ with the side $(BC)$, and $P$ and $Q$ are the projections of the point $D$ on the sides $(AB),(AC)$ respectively . If $BQ \cap DP=\{M\}$, $CP \cap DQ=\{N\}$, $BQ\cap CP=\{H\}$, show that: a) $PM = DN$ b) $MN \parallel BC$ c) $AH \perp BC$.

2001 Junior Balkan Team Selection Tests - Moldova, 3

Tags: equal segments , geometry , equilateral

Let the convex quadrilateral $ABCD$ with $AD = BC$ ¸and $\angle A + \angle B = 120^o$. Take a point $P$ in the plane so that the line $CD$ separates the points $A$ and $P$, and the $DCP$ triangle is equilateral. Show that the triangle $ABP$ is equilateral. It is the true statement for a non-convex quadrilateral?

2019 Czech-Polish-Slovak Junior Match, 2

Tags: geometry , centroid , equal segments

Let $ABC$ be a triangle with centroid $T$. Denote by $M$ the midpoint of $BC$. Let $D$ be a point on the ray opposite to the ray $BA$ such that $AB = BD$. Similarly, let $E$ be a point on the ray opposite to the ray $CA$ such that $AC = CE$. The segments $T D$ and $T E$ intersect the side $BC$ in $P$ and $Q$, respectively. Show that the points $P, Q$ and $M$ split the segment $BC$ into four parts of equal length.

Cono Sur Shortlist - geometry, 2018.G3

Tags: pentagon , equal segments , equal angles , geometry

Consider the pentagon $ABCDE$ such that $AB = AE = x$, $AC = AD = y$, $\angle BAE = 90^o$ and $\angle ACB = \angle ADE = 135^o$. It is known that $C$ and $D$ are inside the triangle $BAE$. Determine the length of $CD$ in terms of $x$ and $y$.

Swiss NMO - geometry, 2019.7

Tags: equal segments , geometry , angle , equal angles

Let $ABC$ be a triangle with $\angle CAB = 2 \angle ABC$. Assume that a point $D$ is inside the triangle $ABC$ exists such that $AD = BD$ and $CD = AC$. Show that $\angle ACB = 3 \angle DCB$.

2017 Saudi Arabia IMO TST, 2

Tags: geometry , incircle , tangential quadrilateral , equal segments

Let $ABCD$ be the circumscribed quadrilateral with the incircle $(I)$. The circle $(I)$ touches $AB, BC, C D, DA$ at $M, N, P,Q$ respectively. Let $K$ and $L$ be the circumcenters of the triangles $AMN$ and $APQ$ respectively. The line $KL$ cuts the line $BD$ at $R$. The line $AI$ cuts the line $MQ$ at $J$. Prove that $RA = RJ$.

1955 Moscow Mathematical Olympiad, 313

Tags: equal segments , combinatorics , combinatorial geometry , arithmetic progression

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.

1993 Tournament Of Towns, (370) 2

Tags: equal segments , cyclic quadrilateral , geometry

Quadrilateral $ABCD$ is inscribed in a circle, $M$ is the intersection point of the lines $AB$ and $CD$ and $N$ is the intersection point of the lines $BC$ and $AD$. It is known that $BM = DN$. Prove that $CM = CN$. (F Nazarov)

2011 Sharygin Geometry Olympiad, 19

Tags: equal segments , geometry , median , altitude , bisector

Does there exist a nonisosceles triangle such that the altitude from one vertex, the bisectrix from the second one and the median from the third one are equal?

2018 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: symmetry , equal segments , circumcircle , geometry

Let $P$ be a point outside a circumference $\Gamma$, and let $PA$ be one of the tangents from $P$ to $\Gamma$. Line $l$ passes through $P$ and intersects $\Gamma$ at $B$ and $C$, with $B$ between $P$ and $C$. Let $D$ be the symmetric of $B$ with respect to $P$. Let $\omega_1$ and $\omega_2$ be the circles circumscribed to the triangles $DAC$ and $PAB$ respectively. $\omega_1$ and $\omega _2$ intersect at $E \neq A$. Line $EB$ cuts back to $\omega _1 $ in $F$. Prove that $CF = AB$.

2015 Czech-Polish-Slovak Junior Match, 3

Tags: equal segments , geometry , perpendicular , geometric inequality

Different points $A$ and $D$ are on the same side of the line $BC$, with $|AB| = | BC|= |CD|$ and lines $AD$ and $BC$ are perpendicular. Let $E$ be the intersection point of lines $AD$ and $BC$. Prove that $||BE| - |CE|| < |AD| \sqrt3$

Kyiv City MO Seniors Round2 2010+ geometry, 2019.10.3.1

Tags: equal segments , geometry , pentagon , regular pentagon

Let $ABCDE$ be a regular pentagon with center $M$. Point $P \ne M$ is selected on segment $MD$. The circumscribed circle of triangle $ABP$ intersects the line $AE$ for second time at point $Q$, and a line that is perpendicular to the $CD$ and passes through $P$, for second time at the point $R$. Prove that $AR = QR$.