Olympiad problems

2012 Bundeswettbewerb Mathematik, 3

The incircle of the triangle $ABC$ touches the sides $BC, CA$ and $AB$ in points $A_1, B_1$ and $C_1$ respectively. $C_1D$ is a diameter of the incircle. Finally, let $E$ be the intersection of the lines $B_1C_1$ and $A_1D$. Prove that the segments $CE$ and $CB_1$ have equal length.

2004 Oral Moscow Geometry Olympiad, 4

Tags: equal segments , centroid , circumcenter , geometry

In triangle $ABC$, $M$ is the intersection point of the medians, $O$ is the center of the inscribed circle. Prove that if the line $OM$ is parallel to the side $BC$, then the point $O$ is equidistant from the sides $AB$ and $AC$.

2009 Junior Balkan Team Selection Tests - Romania, 2

Tags: equal segments , rhombus , geometry

Consider a rhombus $ABCD$. Point $M$ and $N$ are given on the line segments $AC$ and $BC$ respectively, such that $DM = MN$. Lines $AC$ and $DN$ meet at point $P$ and lines $AB$ and $DM$ meet at point $R$. Prove that $RP = PD$.

Kyiv City MO Juniors Round2 2010+ geometry, 2012.9.4

Tags: circumcircle , geometry , angle , parallel , orthocenter , equal segments

In an acute-angled triangle $ABC$, the point $O$ is the center of the circumcircle, and the point $H$ is the orthocenter. It is known that the lines $OH$ and $BC$ are parallel, and $BC = 4OH $. Find the value of the smallest angle of triangle $ ABC $. (Black Maxim)

2020 Regional Olympiad of Mexico Center Zone, 3

Tags: geometry , equal segments

In an acute triangle $ABC$, an arbitrary point $P$ is chosen on the altitude $AH$. The points $E$ and $F$ are the midpoints of $AC$ and $AB$, respectively. The perpendiculars from $E$ on $CP$ and from $F$ on $BP$ intersect at the point $K$. Show that $KB = KC$.

2022 Saudi Arabia BMO + EGMO TST, 1.3

Tags: geometry , equal segments

Given is triangle $ABC$ with $AB > AC$. Circles $O_B$, $O_C$ are inscribed in angle $BAC$ with $O_B$ tangent to $AB$ at $B$ and $O_C$ tangent to $AC$ at $C$. Tangent to $O_B$ from $C$ different than $AC$ intersects $AB$ at $K$ and tangent to $O_C$ from $B$ different than $AB$ intersects $AC$ at $L$. Line $KL$ and the angle bisector of $BAC$ intersect $BC$ at points $P$ and $M$, respectively. Prove that $BP = CM$.

2009 Thailand Mathematical Olympiad, 4

Tags: ratio , equal segments , geometry , area

In triangle $\vartriangle ABC$, $D$ is the midpoint of $BC$. Points $E$ and $F$ are chosen on side $AC$ so that $AF = F E = EC$. Let $AD$ intersect $BE$ and $BF$ and $G$ and $H$, respectively. Find the ratio of the areas of $\vartriangle BGH$ and $\vartriangle ABC$.

1975 Chisinau City MO, 105

Tags: geometry , concyclic , equal segments

Let $M$ be the point of intersection of the diagonals, and $K$ be the point of intersection of the bisectors of the angles $B$ and $C$ of the convex quadrilateral $ABCD$. Prove that points $A, B, M, K$ lie on the same circle if the following relation holds: $|AB|=|BC|=|CD|$

1993 All-Russian Olympiad Regional Round, 9.7

Tags: rhombus , equal segments , collinear , geometry

On the diagonal $AC$ of the rhombus $ABCD$, a point $E$ is taken, which is different from points $A$ and $C$, and on the lines $AB$ and $BC$ are points $N$ and $M$, respectively, with $AE = NE$ and $CE = ME$. Let $K$ be the intersection point of lines $AM$ and $CN$. Prove that points $K, E$ and $D$ are collinear.

Kyiv City MO Seniors Round2 2010+ geometry, 2017.11.2

Tags: geometry , concyclic , cyclic quadrilateral , equal segments

The median $CM$ is drawn in the triangle $ABC$ intersecting bisector angle $BL$ at point $O$. Ray $AO$ intersects side $BC$ at point $K$, beyond point $K$ draw the segment $KT = KC$. On the ray $BC$ beyond point $C$ draw a segment $CN = BK$. Prove that is a quadrilateral $ABTN$ is cyclic if and only if $AB = AK$. (Vladislav Yurashev)

1996 Singapore Team Selection Test, 1

Tags: equal segments , square , perpendicular , geometry

Let $P$ be a point on the side $AB$ of a square $ABCD$ and $Q$ a point on the side $BC$. Let $H$ be the foot of the perpendicular from $B$ to $PC$. Suppose that $BP = BQ$. Prove that $QH$ is perpendicular to $HD$.

2021 Novosibirsk Oral Olympiad in Geometry, 5

Tags: equal segments , angle , geometry

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 $\angle BCA$.

Durer Math Competition CD Finals - geometry, 2010.C5

Tags: equal segments , geometry , incircle

Let $D$ the touchpoint of the inscribed circle of triangle $ABC$ be with side $AB$ . From $A$ the perpendicular lines on the angle bisectors of vertices $B$ and $C$ intersect them at points $A_1$ and $A_2$ respectively . Prove that $A_1A_2 = AD$.

2021 Saudi Arabia Training Tests, 5

Tags: geometry , rectangle , equal segments

Let $ABCD$ be a rectangle with $P$ lies on the segment $AC$. Denote $Q$ as a point on minor arc $PB$ of $(PAB)$ such that $QB = QC$. Denote $R$ as a point on minor arc $PD$ of $(PAD)$ such that $RC = RD$. The lines $CB$, $CD$ meet $(CQR)$ again at $M, N$ respectively. Prove that $BM = DN$. by Tran Quang Hung

2021 Yasinsky Geometry Olympiad, 4

Tags: equal segments , geometry , incenter , concurrency

Given an acute triangle $ABC$, in which $\angle BAC = 60^o$. On the sides $AC$ and $AB$ take the points $T$ and $Q$, respectively, such that $CT = TQ = QB$. Prove that the center of the inscribed circle of triangle $ATQ$ lies on the side $BC$. (Dmitry Shvetsov)

2016 Oral Moscow Geometry Olympiad, 1

Tags: geometry , equal segments , angle , diagonal , hexagon

Angles are equal in a hexagon, three main diagonals are equal and the other six diagonals are also equal. Is it true that it has equal sides?

Ukrainian From Tasks to Tasks - geometry, 2011.3

Tags: geometry , equal segments , circumcenter

Let $O$ be the center of the circumcircle, and $AD$ be the angle bisector of the acute triangle $ABC$. The perpendicular drawn from point $D$ on the line $AO$ intersects the line $AC$ at the point $P$. Prove that $AP = AB$.

2013 Regional Competition For Advanced Students, 4

Tags: symmetry , equal segments , geometry , pentagon , equal angles

We call a pentagon [i]distinguished [/i] if either all side lengths or all angles are equal. We call it [i]very distinguished[/i] if in addition two of the other parts are equal. i.e. $5$ sides and $2$ angles or $2$ sides and $5$ angles.Show that every very distinguished pentagon has an axis of symmetry.

2011 Oral Moscow Geometry Olympiad, 2

Tags: geometry , equal segments , angle , isosceles , right angle

In an isosceles triangle $ABC$ ($AB=AC$) on the side $BC$, point $M$ is marked so that the segment $CM$ is equal to the altitude of the triangle drawn on this side, and on the side $AB$, point $K$ is marked so that the angle $\angle KMC$ is right. Find the angle $\angle ACK$.

1979 All Soviet Union Mathematical Olympiad, 279

Tags: relatively prime , equal segments , combinatorics , number theory

Natural $p$ and $q$ are relatively prime. The $[0,1]$ is divided onto $(p+q)$ equal segments. Prove that every segment except two marginal contain exactly one from the $(p+q-2)$ numbers $$\{1/p, 2/p, ... , (p-1)/p, 1/q, 2/q, ... , (q-1)/q\}$$

2012 Dutch BxMO/EGMO TST, 4

Tags: equal segments , area , geometry

Let $ABCD$ a convex quadrilateral (this means that all interior angles are smaller than $180^o$), such that there exist a point $M$ on line segment $AB$ and a point $N$ on line segment $BC$ having the property that $AN$ cuts the quadrilateral in two parts of equal area, and such that the same property holds for $CM$. Prove that $MN$ cuts the diagonal $BD$ in two segments of equal length.

2016 Iranian Geometry Olympiad, 2

Tags: circumcircle , equal segments , angle chasing , geometry

Let $\omega$ be the circumcircle of triangle $ABC$ with $AC > AB$. Let $X$ be a point on $AC$ and $Y$ be a point on the circle $\omega$, such that $CX = CY = AB$. (The points $A$ and $Y$ lie on different sides of the line $BC$). The line $XY$ intersects $\omega$ for the second time in point $P$. Show that $PB = PC$. by Iman Maghsoudi

2019 Saudi Arabia Pre-TST + Training Tests, 2.3

Tags: equal segments , geometry , equilateral , collinear

Consider equilateral triangle $ABC$ and suppose that there exist three distinct points $X, Y,Z$ lie inside triangle $ABC$ such that i) $AX = BY = CZ$ ii) The triplets of points $(A,X,Z), (B,Y,X), (C,Z,Y )$ are collinear in that order. Prove that $XY Z$ is an equilateral triangle.

2004 Argentina National Olympiad, 5

Tags: geometry , equal segments , pentagon , area

The pentagon $ABCDE$ has $AB = BC, CD = DE, \angle ABC = 120^o, \angle CDE = 60^o$ and $BD = 2$. Calculate the area of the pentagon.

2008 Tournament Of Towns, 3

Tags: geometry , right triangle , circumcircle , equal segments

In triangle $ABC, \angle A = 90^o$. $M$ is the midpoint of $BC$ and $H$ is the foot of the altitude from $A$ to $BC$. The line passing through $M$ and perpendicular to $AC$ meets the circumcircle of triangle $AMC$ again at $P$. If $BP$ intersects $AH$ at $K$, prove that $AK = KH$.