Olympiad problems

IV Soros Olympiad 1997 - 98 (Russia), 11.5

Tags: perpendicular , geometry

Let $M$ be the point of intersection of the diagonals of the inscribed quadrilateral $ABCD$, and let the angle $\angle AMB$ be an acute angle. On the side $BC$, as a base, an isosceles triangle $BCK$ is constructed in the direction external to the quadrilateral such that $\angle KBC+\angle AMB= 90^o$. Prove that line $KM$ is perpendicular to $AD$.

2022 Costa Rica - Final Round, 1

Tags: geometry , rhombus , perpendicular , perpendicularity

Let $\Gamma$ be a circle with center $O$. Consider the points $A$, $B$, $C$, $D$, $E$ and $F$ in $\Gamma$, with $D$ and $E$ in the (minor) arc $BC$ and $C$ in the (minor) arc $EF$, such that $DEFO$ is a rhombus and $\vartriangle ABC$ It is equilateral. Show that $\overleftrightarrow{BD}$ and $\overleftrightarrow{CE}$ are perpendicular.

2016 Romania National Olympiad, 2

Tags: cube , perpendicular , ratio , geometry , 3d geometry

In a cube $ABCDA'B'C'D' $two points are considered, $M \in (CD')$ and $N \in (DA')$. Show that the $MN$ is common perpendicular to the lines $CD'$ and $DA'$ if and only if $$\frac{D'M}{D'C}=\frac{DN}{DA'} =\frac{1}{3}.$$

Croatia MO (HMO) - geometry, 2020.3

Tags: geometry , right angle , perpendicular

Given a triangle $ABC$ such that $AB<AC$ . On sides $AB$ and $BC$, points $P$ and $Q$ are marked respectively such that the lines $AQ$ and $CP$ are perpendicular, and the circle inscribed in the triangle $ABC$ touches the length $PQ$. The line $CP$ intersects the circle circumscribed around the triangle $ABC$ at the points $C$ and $T$. If the lines $CA,PQ$ and $BT$ intersect at one point, prove that the angle $\angle CAB$ is right.

2008 Postal Coaching, 2

Tags: geometry , perpendicular , distance , geometric inequalities

Let $ABC$ be a triangle, $AD$ be the altitude from $A$ on to $BC$. Draw perpendiculars $DD_1$ and $DD_2$ from $D$ on to $AB$ and $AC$ respectively and let $p(A)$ be the length of the segment $D_1D_2$. Similarly define $p(B)$ and $p(C)$. Prove that $\frac{p(A)p(B)p(C)}{s^3}\le \frac18$ , where s is the semi-perimeter of the triangle $ABC$.

1995 Abels Math Contest (Norwegian MO), 2a

Tags: perpendicular , tangent circle , circles , geometry

Two circles $k_1,k_2$ touch each other at $P$, and touch a line $\ell$ at $A$ and $B$ respectively. Line $AP$ meets $k_2$ at $C$. Prove that $BC$ is perpendicular to $\ell$.

2020 Ukrainian Geometry Olympiad - April, 3

Tags: geometry , circumcircle , parallel , perpendicular , equal segments

Triangle $ABC$. Let $B_1$ and $C_1$ be such points, that $AB= BB_1, AC=CC_1$ and $B_1, C_1$ lie on the circumscribed circle $\Gamma$ of $\vartriangle ABC$. Perpendiculars drawn from from points $B_1$ and $C_1$ on the lines $AB$ and $AC$ intersect $\Gamma$ at points $B_2$ and $C_2$ respectively, these points lie on smaller arcs $AB$ and $AC$ of circle $\Gamma$ respectively, Prove that $BB_2 \parallel CC_2$.

1982 Polish MO Finals, 2

Tags: perpendicular , cyclic , parallel

In a cyclic quadrilateral $ABCD$ the line passing through the midpoint of $AB$ and the intersection point of the diagonals is perpendicular to $CD$. Prove that either the sides $AB$ and $CD$ are parallel or the diagonals are perpendicular.

2016 Czech-Polish-Slovak Junior Match, 1

Tags: perpendicular , geometry , area , max

Let $AB$ be a given segment and $M$ be its midpoint. We consider the set of right-angled triangles $ABC$ with hypotenuses $AB$. Denote by $D$ the foot of the altitude from $C$. Let $K$ and $L$ be feet of perpendiculars from $D$ to the legs $BC$ and $AC$, respectively. Determine the largest possible area of the quadrilateral $MKCL$. Czech Republic

2019 Romania National Olympiad, 2

Tags: equal segments , geometry , square , perpendicular

Let $ABCD$ be a square and $E$ a point on the side $(CD)$. Squares $ENMA$ and $EBQP$ are constructed outside the triangle $ABE$. Prove that: a) $ND = PC$ b) $ND\perp PC$.

1957 Moscow Mathematical Olympiad, 358

Tags: geometry , combinatorial geometry , broken line , 3d geometry , perpendicular

The segments of a closed broken line in space are of equal length, and each three consecutive segments are mutually perpendicular. Prove that the number of segments is divisible by $6$.

2005 Sharygin Geometry Olympiad, 9

Tags: geometry , equilateral , equilateral triangle , perpendicular , midpoint , centroid

Let $O$ be the center of a regular triangle $ABC$. From an arbitrary point $P$ of the plane, the perpendiculars were drawn on the sides of the triangle. Let $M$ denote the intersection point of the medians of the triangle , having vertices the feet of the perpendiculars. Prove that $M$ is the midpoint of the segment $PO$.