Olympiad problems

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$ satises $\angle BTE = \angle CTF = 90^o$. Show that $|TA|^2 =|TB| \cdot |TC|$.

2005 Mexico National Olympiad, 6

Tags: geometry , equal segments

Let $ABC$ be a triangle and $AD$ be the angle bisector of $<BAC$, with $D$ on $BC$. Let $E$ be a point on segment $BC$ such that $BD = EC$. Through $E$ draw $l$ a parallel line to $AD$ and let $P$ be a point in $l$ inside the triangle. Let $G$ be the point where $BP$ intersects $AC$ and $F$ be the point where $CP$ intersects $AB$. Show $BF = CG$.

2021 Poland - Second Round, 2

Tags: equal segments , parallelogram , equal angles , geometry

The point P lies on the side $CD$ of the parallelogram $ABCD$ with $\angle DBA = \angle CBP$. Point $O$ is the center of the circle passing through the points $D$ and $P$ and tangent to the straight line $AD$ at point $D$. Prove that $AO = OC$.

Kharkiv City MO Seniors - geometry, 2015.10.3

Tags: geometry , circumcircle , circles , equal segments

On side $AB$ of triangle $ABC$, point $M$ is selected. A straight line passing through $M$ intersects the segment $AC$ at point $N$ and the ray $CB$ at point $K$. The circumscribed circle of the triangle $AMN$ intersects $\omega$, the circumscribed circle of the triangle $ABC$, at points $A$ and $S$. Straight lines $SM$ and $SK$ intersect with $\omega$ for the second time at points $P$ and $Q$, respectively. Prove that $AC = PQ$.

2019 Denmark MO - Mohr Contest, 5

Tags: geometry , equilateral , right triangle , equal segments

In the figure below the triangles $BCD, CAE$ and $ABF$ are equilateral, and the triangle $ABC$ is right-angled with $\angle A = 90^o$. Prove that $|AD| = |EF|$. [img]https://1.bp.blogspot.com/-QMMhRdej1x8/XzP18QbsXOI/AAAAAAAAMUI/n53OsE8rwZcjB_zpKUXWXq6bg3o8GUfSwCLcBGAsYHQ/s0/2019%2Bmohr%2Bp5.png[/img]

1956 Moscow Mathematical Olympiad, 325

Tags: geometry , equal segments , midpoint , locus

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 $DE$.

2022 Novosibirsk Oral Olympiad in Geometry, 7

Tags: geometry , orthocenter , equal segments

Altitudes $AA_1$ and $CC_1$ of an acute-angled triangle $ABC$ intersect at point $H$. A straight line passing through point $H$ parallel to line $A_1C_1$ intersects the circumscribed circles of triangles $AHC_1$ and $CHA_1$ at points $X$ and $Y$, respectively. Prove that points $X$ and $Y$ are equidistant from the midpoint of segment $BH$.

2009 All-Russian Olympiad Regional Round, 11.6

Tags: geometry , equal segments

Point $D$ on side $BC$ of acute triangle ABC is such that $AB=AD$. The circumcircle of triangle $ABD$ intersects side $AC$ at points $A$ and $K$. Line $DK$ intersects the perpendicular drawn from $B$ on $AC$, at the point $L$. Prove that $CL= BC$

2003 Singapore Team Selection Test, 2

Tags: geometry , equal segments , semicircle

Let $M$ be a point on the diameter $AB$ of a semicircle $\Gamma$. The perpendicular at $M$ meets the semicircle $\Gamma$ at $P$. A circle inside $\Gamma$. touches $\Gamma$. and is tangent to $PM$ at $Q$ and $AM$ at $R$. Prove that $P B = RB$.

2019 Switzerland - Final Round, 7

Tags: geometry , equal angles , equal segments , angle

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 $\angle ACB = 3 \angle DCB$.

1994 Abels Math Contest (Norwegian MO), 1b

Tags: geometry , angle bisector , equal segments , tangent

Let $C$ be a point on the extension of the diameter $AB$ of a circle. A line through $C$ is tangent to the circle at point $N$. The bisector of $\angle ACN$ meets the lines $AN$ and $BN$ at $P$ and $Q$ respectively. Prove that $PN = QN$.

2020 BMT Fall, 20

Tags: equal segments , equal angles , geometry

Non-degenerate quadrilateral $ABCD$ with $AB = AD$ and $BC = CD$ has integer side lengths, and $\angle ABC = \angle BCD = \angle CDA$. If $AB = 3$ and $B \ne D$, how many possible lengths are there for $BC$?

2016 Ecuador NMO (OMEC), 3

Tags: equal segments , equilateral , angle , geometry

Let $A, B, C, D$ be four different points on a line $\ell$, such that $AB = BC = CD$. In one of the semiplanes determined by the line $\ell$, the points $P$ and $Q$ are chosen in such a way that the triangle $CPQ$ is equilateral with its vertices named clockwise. Let $M$ and $N$ be two points on the plane such that the triangles $MAP$ and $NQD$ are equilateral (the vertices are also named clockwise). Find the measure of the angle $\angle MBN$.

2003 Oral Moscow Geometry Olympiad, 2

Tags: ratio , geometry , right angle , equal segments , equal angles

In a convex quadrilateral $ABCD$, $\angle ABC = 90^o$ , $\angle BAC = \angle CAD$, $AC = AD, DH$ is the alltitude of the triangle $ACD$. In what ratio does the line $BH$ divide the segment $CD$?

2017 Thailand TSTST, 3

Tags: circumcircle , equal segments , geometry

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 $CR = 2XY$ .

2021 Sharygin Geometry Olympiad, 8.6

Tags: geometry , circumcenter , equal segments , concyclic

Let $ABC$ be an acute-angled triangle. Point $P$ is such that $AP = AB$ and $PB\parallel AC$. Point $Q$ is such that $AQ = AC$ and $CQ\parallel AB$. Segments $CP$ and $BQ$ meet at point $X$. Prove that the circumcenter of triangle $ABC$ lies on the circle $(PXQ)$.

Durer Math Competition CD Finals - geometry, 2019.D3

Tags: geometry , equal segments , equal angles , angle

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?

Champions Tournament Seniors - geometry, 2017.4

Tags: equal segments , angle bisector , circles , geometry

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 $KL=\frac12 BC$

2015 Puerto Rico Team Selection Test, 2

Tags: geometry , angle , equal segments

In the triangle $ABC$, let $P$, $Q$, and $R$ lie on the sides $BC$, $AC$, and $AB$ respectively, such that $AQ = AR$, $BP = BR$ and $CP = CQ$. Let $\angle PQR=75^o$ and $\angle PRQ=35^o$. Calculate the measures of the angles of the triangle $ABC$.

2021 SAFEST Olympiad, 4

Tags: equal angles , equal segments , geometry

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 $\angle KCD = \angle DAC.$

2018 Peru Iberoamerican Team Selection Test, P2

Tags: geometry , angle chasing , equal segments , angle

Let $ABC$ be a triangle with $AB = AC$ and let $D$ be the foot of the height drawn from $A$ to $BC$. Let $P$ be a point inside the triangle $ADC$ such that $\angle APB> 90^o$ and $\angle PAD + \angle PBD = \angle PCD$. The $CP$ and $AD$ lines are cut at $Q$ and the $BP$ and $AD$ lines cut into $R$. Let $T$ be a point in segment $AB$ such that $\angle TRB = \angle DQC$ and let S be a point in the extension of the segment $AP$ (on the $P$ side) such that $\angle PSR = 2 \angle PAR$. Prove that $RS = RT$.

Kyiv City MO Seniors 2003+ geometry, 2015.10.5

Tags: circles , equal segments , geometry

Circles ${{w} _ {1}}$ and ${{w} _ {2}}$ with centers at points ${{O} _ {1}}$ and ${{ O} _ {2}}$ intersect at points $A$ and $B$, respectively. Around the triangle ${{O} _ {1}} {{O} _ {2}} B$ circumscribe a circle $w$ centered at the point $O$, which intersects the circles ${{w } _ {1}}$ and ${{w} _ {2}}$ for the second time at points $K$ and $L$, respectively. The line $OA$ intersects the circles ${{w} _ {1}}$ and ${{w} _ {2}}$ at the points $M$ and $N$, respectively. The lines $MK$ and $NL$ intersect at the point $P$. Prove that the point $P$ lies on the circle $w$ and $PM = PN$. (Vadym Mitrofanov)

1994 Tournament Of Towns, (404) 2

Tags: equal segments , geometry

Two circles intersect at the points $A$ and $B$. Tangent lines drawn to both of the circles at the point $A$ intersect the circles at the points $M$ and $N$. The lines $BM$ and $BN$ intersect the circles once more at the points $P$ and $Q$ respectively. Prove that the segments $MP$ and $NQ$ are equal. (I Nagel)

2021 Austrian MO Beginners' Competition, 2

Tags: equal segments , circumcenter , geometry

A triangle $ABC$ with circumcenter $U$ is given, so that $\angle CBA = 60^o$ and $\angle CBU = 45^o$ apply. The straight lines $BU$ and $AC$ intersect at point $D$. Prove that $AD = DU$. (Karl Czakler)

Kyiv City MO Juniors 2003+ geometry, 2014.851

Tags: geometry , ratio , median , equal segments

On the side $AB$ of the triangle $ABC$ mark the point $K$. The segment $CK$ intersects the median $AM$ at the point $F$. It is known that $AK = AF$. Find the ratio $MF: BK$.