Olympiad problems

2020 Australian Maths Olympiad, 3

Tags: tangent , circumcircle , parallel , right triangle , perpendicular bisector , midpoint , geometry

Let $ABC$ be a triangle with $\angle ACB=90^{\circ}$. Suppose that the tangent line at $C$ to the circle passing through $A,B,C$ intersects the line $AB$ at $D$. Let $E$ be the midpoint of $CD$ and let $F$ be a point on $EB$ such that $AF$ is parallel to $CD$. Prove that the lines $AB$ and $CF$ are perpendicular.

1983 IMO Longlists, 6

Tags: combinatorics , ramsey theory , extremal combinatorics , right triangle , geometry

Let $ABC$ be an equilateral triangle and $\mathcal{E}$ the set of all points contained in the three segments $AB$, $BC$, and $CA$ (including $A$, $B$, and $C$). Determine whether, for every partition of $\mathcal{E}$ into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle.

Denmark (Mohr) - geometry, 2010.1

Tags: circles , area , geometry , right triangle

Four right triangles, each with the sides $1$ and $2$, are assembled to a figure as shown. How large a fraction does the area of the small circle make up of that of the big one? [img]https://1.bp.blogspot.com/-XODK1XKCS0Q/XzXDtcA-xAI/AAAAAAAAMWA/zSLPpf3IcX0rgaRtOxm_F2begnVdUargACLcBGAsYHQ/s0/2010%2BMohr%2Bp1.png[/img]

2008 Hanoi Open Mathematics Competitions, 9

Tags: right triangle , geometry

Consider a right -angle triangle $ABC$ with $A=90^{o}$, $AB=c$ and $AC=b$. Let $P\in AC$ and $Q\in AB$ such that $\angle APQ=\angle ABC$ and $\angle AQP = \angle ACB$. Calculate $PQ+PE+QF$, where $E$ and $F$ are the projections of $B$ and $Q$ onto $BC$, respectively.

1999 Cono Sur Olympiad, 2

Tags: geometry , right triangle , rectangle , construction , area

Let $ABC$ be a triangle right in $A$. Construct a point $P$ on the hypotenuse $BC$ such that if $Q$ is the foot of the perpendicular drawn from $P$ to side $AC$, then the area of the square of side $PQ$ is equal to the area of the rectangle of sides $PB$ and $PC$. Show construction steps.

1966 Spain Mathematical Olympiad, 5

Tags: right triangle , geometry

The length of the hypotenuse $BC$ of a right triangle $ABC$ is $a$, and on it the points $M$ and $N$ are taken such that $BM = NC = k$, with $k < a/2$. Assuming that (only) the data $a$ and $k$ are known, calculate: a) The value of the sum of the squares of the lengths $AM$ and $AN$. b) The ratio of the areas of triangles $ABC$ and $AMN$. c) The area enclosed by the circle that passes through the points $A, M' , N'$ , where $M'$ is the orthogonal projection of $M$ onto $AC$ and $N'$ that of $N$ onto $AB$.

1997 Tuymaada Olympiad, 8

Tags: right triangle , congruent triangles , geometry

Find a right triangle that can be cut into $365$ equal triangles.

2003 Estonia National Olympiad, 3

Tags: geometry , rectangle , right triangle , isosceles

In the rectangle $ABCD$ with $|AB|<2 |AD|$, let $E$ be the midpoint of $AB$ and $F$ a point on the chord $CE$ such that $\angle CFD = 90^o$. Prove that $FAD$ is an isosceles triangle.

Denmark (Mohr) - geometry, 2003.1

Tags: geometry , right triangle

In a right-angled triangle, the sum $a + b$ of the sides enclosing the right angle equals $24$ while the length of the altitude $h_c$ on the hypotenuse $c$ is $7$. Determine the length of the hypotenuse.

2013 Oral Moscow Geometry Olympiad, 3

Tags: right triangle , angle bisector , midpoint , geometry

The bisectors $AA_1$ and $CC_1$ of the right triangle $ABC$ ($\angle B = 90^o$) intersect at point $I$. The line passing through the point $C_1$ and perpendicular on the line $AA_1$ intersects the line that passes through $A_1$ and is perpendicular on $CC_1$, at the point $K$. Prove that the midpoint of the segment $KI$ lies on segment $AC$.

May Olympiad L2 - geometry, 2011.3

Tags: right triangle , angle , geometry , perpendicular bisector , isosceles

In a right triangle rectangle $ABC$ such that $AB = AC$, $M$ is the midpoint of $BC$. Let $P$ be a point on the perpendicular bisector of $AC$, lying in the semi-plane determined by $BC$ that does not contain $A$. Lines $CP$ and $AM$ intersect at $Q$. Calculate the angles that form the lines $AP$ and $BQ$.

2018 Oral Moscow Geometry Olympiad, 1

Tags: right triangle , geometry , projection , inradius

In a right triangle $ABC$ with a right angle $C$, let $AK$ and $BN$ be the angle bisectors. Let $D,E$ be the projections of $C$ on $AK, BN$ respectively. Prove that the length of the segment $DE$ is equal to the radius of the inscribed circle.

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 $AK = KH$.

2021 Estonia Team Selection Test, 2

Tags: geometry , right triangle , isosceles triangle

Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$. Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.

2024 Mozambique National Olympiad, P6

Tags: right triangle , geometry , area , parallelogram

Let $ABC$ be an isosceles right triangle with $\angle BCA=90^{\circ}, BC=AC=10$. Let $P$ be a point on $AB$ that is a distance $x$ from $A$, $Q$ be a point on $AC$ such that $PQ$ is parallel to $BC$. Let $R$ and $S$ be points on $BC$ such that $QR$ is parallel to $AB$ and $PS$ is parallel to $AC$. The union of the quadrilaterals $PBRQ$ and $PSCQ$ determine a shaded area $f(x)$. Evaluate $f(2)$

2017 BMT Spring, 9

Tags: right triangle , geometry

Let $AB = 10$ be a diameter of circle $P$. Pick point $C$ on the circle such that $AC = 8$. Let the circle with center $O$ be the incircle of $\vartriangle ABC$. Extend line $AO$ to intersect circle $P$ again at $D$. Find the length of $BD$.

2022 New Zealand MO, 4

Tags: right triangle , geometry , incircle

Triangle $ABC$ is right-angled at $B$ and has incentre $I$. Points $D$, $E$ and $F$ are the points where the incircle of the triangle touches the sides $BC$, $AC$ and AB respectively. Lines $CI$ and $EF$ intersect at point $P$. Lines $DP$ and $AB$ intersect at point $Q$. Prove that $AQ = BF$.

Kyiv City MO Juniors 2003+ geometry, 2004.7.3

Tags: right triangle , right angle , geometry

Given a right triangle $ABC$ ($\angle A <45^o$,$ \angle C = 90^o$), on the sides $AC$ and $AB$ which are selected points $D,E$ respectively, such that $BD = AD$ and $CB = CE$. Let the segments $BD$ and $CE$ intersect at the point $O$. Prove that $\angle DOE = 90^o$.

Durer Math Competition CD 1st Round - geometry, 2019.D4

Tags: right triangle , angle , geometry

Let $ABC$ be an isosceles right-angled triangle, having the right angle at vertex $C$. Let us consider the line through $C$ which is parallel to $AB$ and let $D$ be a point on this line such that $AB = BD$ and $D$ is closer to $B$ than to $A$. Find the angle $\angle CBD$.

2019 Thailand Mathematical Olympiad, 1

Tags: right triangle , circumcircle , geometry , pentagon , concyclic

Let $ABCDE$ be a convex pentagon with $\angle AEB=\angle BDC=90^o$ and line $AC$ bisects $\angle BAE$ and $\angle DCB$ internally. The circumcircle of $ABE$ intersects line $AC$ again at $P$. (a) Show that $P$ is the circumcenter of $BDE$. (b) Show that $A, C, D, E$ are concyclic.

May Olympiad L2 - geometry, 2002.3

Tags: right triangle , square , isosceles , geometry

In a triangle $ABC$, right in $A$ and isosceles, let $D$ be a point on the side $AC$ ($A \ne D \ne C$) and $E$ be the point on the extension of $BA$ such that the triangle $ADE$ is isosceles. Let $P$ be the midpoint of segment $BD$, $R$ be the midpoint of the segment $CE$ and $Q$ the intersection point of $ED$ and $BC$. Prove that the quadrilateral $ARQP$ is a square.

1992 Denmark MO - Mohr Contest, 2

Tags: geometry , right triangle , tangent circle

In a right-angled triangle, $a$ and $b$ denote the lengths of the two catheti. A circle with radius $r$ has the center on the hypotenuse and touches both catheti. Show that $\frac{1}{a}+\frac{1}{b}=\frac{1}{r}$.

2006 Abels Math Contest (Norwegian MO), 4

Tags: tangent circle , right triangle , circumcircle , mixtilinear circle , inradius , incircle

Let $\gamma$ be the circumscribed circle about a right-angled triangle $ABC$ with right angle $C$. Let $\delta$ be the circle tangent to the sides $AC$ and $BC$ and tangent to the circle $\gamma$ internally. (a) Find the radius $i$ of $\delta$ in terms of $a$ when $AC$ and $BC$ both have length $a$. (b) Show that the radius $i$ is twice the radius of the inscribed circle of $ABC$.

2018 Lusophon Mathematical Olympiad, 2

Tags: geometry , right triangle , square , isosceles

In a triangle $ABC$, right in $A$ and isosceles, let $D$ be a point on the side $AC$ ($A \ne D \ne C$) and $E$ be the point on the extension of $BA$ such that the triangle $ADE$ is isosceles. Let $P$ be the midpoint of segment $BD$, $R$ be the midpoint of the segment $CE$ and $Q$ the intersection point of $ED$ and $BC$. Prove that the quadrilateral $ARQP$ is a square.

2017 Costa Rica - Final Round, G2

Tags: right triangle , geometry , incenter

Consider the right triangle $\vartriangle ABC$ right at $A$ and let $D$ be a point on the hypotenuse $BC$. Consider the line that passes through the incenters of $\vartriangle ABD$ and $\vartriangle ACD$, and let $K$ and $ L$ the intersections of said line with $AB$ and $AC$ respectively. Show that if $AK = AL$ then $D$ is the foot of the altitude on the hypotenuse.