Olympiad problems

2015 Sharygin Geometry Olympiad, 6

Lines $b$ and $c$ passing through vertices $B$ and $C$ of triangle $ABC$ are perpendicular to sideline $BC$. The perpendicular bisectors to $AC$ and $AB$ meet $b$ and $c$ at points $P$ and $Q$ respectively. Prove that line $PQ$ is perpendicular to median $AM$ of triangle $ABC$. (D. Prokopenko)

1952 Moscow Mathematical Olympiad, 225

Tags: perpendicular , tangent , geometry

From a point $C$, tangents $CA$ and $CB$ are drawn to a circle $O$. From an arbitrary point $N$ on the circle, perpendiculars $ND, NE, NF$ are drawn on $AB, CA$ and $CB$, respectively. Prove that the length of $ND$ is the mean proportional of the lengths of $NE$ and $NF$.

2018 Hanoi Open Mathematics Competitions, 9

Tags: geometry , perpendicular , circumcenter , similar triangles

Let $ABC$ be acute, non-isosceles triangle, inscribed in the circle $(O)$. Let $D$ be perpendicular projection of $A$ onto $BC$, and $E, F$ be perpendicular projections of $D$ onto $CA,AB$ respectively. (a) Prove that $AO \perp EF$. (b) The line $AO$ intersects $DE,DF$ at $I,J$ respectively. Prove that $\vartriangle DIJ$ and $\vartriangle ABC$ are similar. (c) Prove that circumcenter of $\vartriangle DIJ$ is equidistant from $B$ and $C$

2012 Tournament of Towns, 6

Tags: combinatorial geometry , geometry , centroid , regular polygon , perpendicular

(a) A point $A$ is marked inside a circle. Two perpendicular lines drawn through $A$ intersect the circle at four points. Prove that the centre of mass of these four points does not depend on the choice of the lines. (b) A regular $2n$-gon ($n \ge 2$) with centre $A$ is drawn inside a circle (A does not necessarily coincide with the centre of the circle). The rays going from $A$ to the vertices of the $2n$-gon mark $2n$ points on the circle. Then the $2n$-gon is rotated about $A$. The rays going from $A$ to the new locations of vertices mark new $2n$ points on the circle. Let $O$ and $N$ be the centres of gravity of old and new points respectively. Prove that $O = N$.

1977 IMO Shortlist, 8

Tags: convex quadrilateral , perpendicular , geometry

Let $S$ be a convex quadrilateral $ABCD$ and $O$ a point inside it. The feet of the perpendiculars from $O$ to $AB, BC, CD, DA$ are $A_1, B_1, C_1, D_1$ respectively. The feet of the perpendiculars from $O$ to the sides of $S_i$, the quadrilateral $A_iB_iC_iD_i$, are $A_{i+1}B_{i+1}C_{i+1}D_{i+1}$, where $i = 1, 2, 3.$ Prove that $S_4$ is similar to S.

2004 Federal Competition For Advanced Students, P2, 3

Tags: trapezoid , geometry , circles , diagonal , perpendicular , area

A trapezoid $ABCD$ with perpendicular diagonals $AC$ and $BD$ is inscribed in a circle $k$. Let $k_a$ and $k_c$ respectively be the circles with diameters $AB$ and $CD$. Compute the area of the region which is inside the circle $k$, but outside the circles $k_a$ and $k_c$.

Croatia MO (HMO) - geometry, 2022.7

Tags: geometry , perpendicular

In the triangle $ABC$ holds $|AB| = |AC|$ and the inscribed circle touches the sides $\overline{BC}$, $\overline{AC}$ and $\overline{AB}$ at the points $D$, $E$ and $F$ respectively . The perpendicular from the point $D$ to the line $EF$ intersects the side $\overline{AB}$ at the point $G$, and the circles circumscribed around the triangles $AEF$ and $ABC$ intersect at the points $A $and $T$. Prove that the lines $T G$ and $T F$ are perpendicular.

XMO (China) 2-15 - geometry, 4.1

Tags: geometry , perpendicular , perpendicularity

As shown in the figure, it is known that $BC= AC$ in $\vartriangle ABC$, $M$ is the midpoint of $AB$, points $D$, $E$ lie on $AB$ such that $\angle DCE= \angle MCB$, the circumscribed circle of $\vartriangle BDC$ and the circumscribed circle of $\vartriangle AEC$ intersect at point $F $(different from point $C$), point $H$ lies on $AB$ such that the straight line $CM$ bisects the line segment $HF$. Let the circumcenters of $\vartriangle HFE$ and $\vartriangle BFM$ be $O_1$, $O_2$ respectively. Prove that $O_1O_2 \perp CF$. [img]https://cdn.artofproblemsolving.com/attachments/8/c/62d4ecbc18458fb4f2bf88258d5024cddbc3b0.jpg[/img]

2011 Sharygin Geometry Olympiad, 3

Tags: concurrency , geometry , circumcircle , perpendicular

The line passing through vertex $A$ of triangle $ABC$ and parallel to $BC$ meets the circumcircle of $ABC$ for the second time at point $A_1$. Points $B_1$ and $C_1$ are defined similarly. Prove that the perpendiculars from $A_1, B_1, C_1$ to $BC, CA, AB$ respectively concur.

1987 Tournament Of Towns, (148) 5

Tags: locus , perpendicular , equilateral , geometry

Perpendiculars are drawn from an interior point $M$ of the equilateral triangle $ABC$ to its sides , intersecting them at points $D, E$ and $F$ . Find the locus of all points $M$ such that $DEF$ is a right triangle . (J . Tabov , Sofia)

XMO (China) 2-15 - geometry, 7.1

Tags: perpendicular , geometry

As shown in the figure, it is known that $BC = AC$ in $ABC$, $M$ is the midpoint of $AB$, points $D$ and $E$ lie on $AB$ satisfying $\angle DCE = \angle MCB$, the circumscribed circle of $\vartriangle BDC$ and the circumscribed circle of $\vartriangle AEC$ intersect at point $F$ (different from point $C$), point $H$ lies on $AB$ such that the straight line $CM$ bisects the line segment $HF$. Let the circumcenters of $\vartriangle HFE$ and $\vartriangle BFM$ be $O_1$ and $O_2$ respectively. Prove that $O_1O_2\perp CF$. [img]https://cdn.artofproblemsolving.com/attachments/e/4/e8fc62735b8cfbd382e490617f26d335c46823.png[/img]

Novosibirsk Oral Geo Oly VIII, 2020.4

Tags: equal segments , geometry , perpendicular

Point $P$ is chosen inside triangle $ABC$ so that $\angle APC+\angle ABC=180^o$ and $BC=AP.$ On the side $AB$, a point $K$ is chosen such that $AK = KB + PC$. Prove that $CK \perp AB$.

2016 Federal Competition For Advanced Students, P2, 2

Tags: incircle , midpoint , perpendicular , geometry

Let $ABC$ be a triangle. Its incircle meets the sides $BC, CA$ and $AB$ in the points $D, E$ and $F$, respectively. Let $P$ denote the intersection point of $ED$ and the line perpendicular to $EF$ and passing through $F$, and similarly let $Q$ denote the intersection point of $EF$ and the line perpendicular to $ED$ and passing through $D$. Prove that $B$ is the mid-point of the segment $PQ$. Proposed by Karl Czakler

2000 Czech and Slovak Match, 5

Tags: trapezoid , incircle , geometry , perpendicular , isosceles

Let $ABCD$ be an isosceles trapezoid with bases $AB$ and $CD$. The incircle of the triangle $BCD$ touches $CD$ at $E$. Point $F$ is chosen on the bisector of the angle $DAC$ such that the lines $EF$ and $CD$ are perpendicular. The circumcircle of the triangle $ACF$ intersects the line $CD$ again at $G$. Prove that the triangle $AFG$ is isosceles.

VMEO III 2006, 12.1

Tags: perpendicular , geometry

Given a circle $(O)$ and a point $P$ outside that circle. $M$ is a point running on the circle $(O)$. The circle with center $I$ and diameter $PM$ intersects circle $(O)$ again at $N$. The tangent of $(I)$ at $P$ intersects $MN$ at $Q$. The line through $Q$ perpendicular to $PO$ intersects $PM$ at $ A$. $AN$ intersects $(O)$ further at $ B$. $BM$ intersects $PO$ at $C$. Prove that $AC$ is perpendicular to $OQ$.

1952 Moscow Mathematical Olympiad, 217

Tags: geometry , skew , perpendicular , 3d geometry

Given three skew lines. Prove that they are pair-wise perpendicular to their pair-wise perpendiculars.

2010 Estonia Team Selection Test, 4

Tags: geometry , angle bisector , tangent , circumcircle , perpendicular

In an acute triangle $ABC$ the angle $C$ is greater than the angle $A$. Let $AE$ be a diameter of the circumcircle of the triangle. Let the intersection point of the ray $AC$ and the tangent of the circumcircle through the vertex $B$ be $K$. The perpendicular to $AE$ through $K$ intersects the circumcircle of the triangle $BCK$ for the second time at point $D$. Prove that $CE$ bisects the angle $BCD$.

2014 India PRMO, 3

Tags: geometry , perpendicular , diagonal

Let $ABCD$ be a convex quadrilateral with perpendicular diagonals. If $AB = 20, BC = 70$ and $CD = 90$, then what is the value of $DA$?

2011 Greece Junior Math Olympiad, 1

Tags: geometry , perpendicular , circumcircle

Let $ABC$ be a triangle with $\angle BAC=120^o$, which the median $AD$ is perpendicular to side $AB$ and intersects the circumscribed circle of triangle $ABC$ at point $E$. Lines $BA$ and $EC$ intersect at $Z$. Prove that a) $ZD \perp BE$ b) $ZD=BC$

2015 Estonia Team Selection Test, 9

Tags: perpendicular , orthocenter , angle bisector , geometry

The orthocenter of an acute triangle $ABC$ is $H$. Let $K$ and $P$ be the midpoints of lines $BC$ and $AH$, respectively. The angle bisector drawn from the vertex $A$ of the triangle $ABC$ intersects with line $KP$ at $D$. Prove that $HD\perp AD$.

2006 Portugal MO, 2

Tags: perpendicular , geometry , equilateral

In the equilateral triangle $[ABC], D$ is the midpoint of $[AC], E$ and the orthogonal projection of $D$ over $[CB]$ and $F$ is the midpoint of $[DE]$. Prove that $[FB]$ and $[AE]$ are perpendicular. [img]https://1.bp.blogspot.com/-TjSyQotGIOM/X4XMolaXHvI/AAAAAAAAMng/cVsHfl-lrXAFE5LMdosE6vqK1Tf-8WOQgCLcBGAsYHQ/s0/2006%2Bportugal%2Bp2.png[/img]

2014 Thailand TSTST, 2

Tags: perpendicular , incircle , geometry

In a triangle $ABC$, the incircle with incenter $I$ is tangent to $BC$ at $A_1, CA$ at $B_1$, and $AB$ at $C_1$. Denote the intersection of $AA_1$ and $BB_1$ by $G$, the intersection of $AC$ and $A_1C_1$ by $X$, and the intersection of $BC$ and $B_1C_1$ by $Y$ . Prove that $IG \perp XY$ .

2016 Singapore Junior Math Olympiad, 3

Tags: angle bisector , right triangle , altitude , perpendicular , geometry , collinear

In the triangle $ABC$, $\angle A=90^\circ$, the bisector of $\angle B$ meets the altitude $AD$ at the point $E$, and the bisector of $\angle CAD$ meets the side $CD$ at $F$. The line through $F$ perpendicular to $BC$ intersects $AC$ at $G$. Prove that $B,E,G$ are collinear.

2016 Singapore Senior Math Olympiad, 4

Tags: geometry , integer , integer area , integer sidelengths , perpendicular

Let $P$ be a $2016$ sided polygon with all its adjacent sides perpendicular to each other, i.e., all its internal angles are either $90^o$ or $270^o$. If the lengths of its sides are odd integers, prove that its area is an even integer.

1998 Bosnia and Herzegovina Team Selection Test, 4

Tags: tangent , geometry , perpendicular , circles

Circle $k$ with radius $r$ touches the line $p$ in point $A$. Let $AB$ be a dimeter of circle and $C$ an arbitrary point of circle distinct from points $A$ and $B$. Let $D$ be a foot of perpendicular from point $C$ to line $AB$. Let $E$ be a point on extension of line $CD$, over point $D$, such that $ED=BC$. Let tangents on circle from point $E$ intersect line $p$ in points $K$ and $N$. Prove that length of $KN$ does not depend from $C$