Olympiad problems

Estonia Open Junior - geometry, 2013.2.3

In an isosceles right triangle $ABC$ the right angle is at vertex $C$. On the side $AC$ points $K, L$ and on the side $BC$ points $M, N$ are chosen so that they divide the corresponding side into three equal segments. Prove that there is exactly one point $P$ inside the triangle $ABC$ such that $\angle KPL = \angle MPN = 45^o$.

2008 Singapore Junior Math Olympiad, 1

Tags: geometry , right triangle

In $\vartriangle ABC, \angle ACB = 90^o, D$ is the foot of the altitude from $C$ to $AB$ and $E$ is the point on the side $BC$ such that $CE = BD/2$. Prove that $AD + CE = AE$.

2020 Thailand TST, 1

Tags: right angle , right triangle , geometry , circumcircle , tangent circle

Let $ABC$ be a triangle with circumcircle $\Gamma$. Let $\omega_0$ be a circle tangent to chord $AB$ and arc $ACB$. For each $i = 1, 2$, let $\omega_i$ be a circle tangent to $AB$ at $T_i$ , to $\omega_0$ at $S_i$ , and to arc $ACB$. Suppose $\omega_1 \ne \omega_2$. Prove that there is a circle passing through $S_1, S_2, T_1$, and $T_2$, and tangent to $\Gamma$ if and only if $\angle ACB = 90^o$. .

Ukrainian TYM Qualifying - geometry, 2017.4

Tags: integer , right triangle , geometry

Specify at least one right triangle $ABC$ with integer sides, inside which you can specify a point $M$ such that the lengths of the segments $MA, MB, MC$ are integers. Are there many such triangles, none of which are are similar?

1992 Tournament Of Towns, (325) 2

Tags: right triangle , geometry , ratio

Consider a right triangle $ABC$, where $A$ is the right angle, and $AC > AB$. Points $E$ on $AC$ and $D$ on $BC$ are chosen so that$ AB = AE = BD$. Prove that the triangle $ADE$ is right if and only if the ratio $AB : AC : BC$ of sides of the triangle $ABC$ is $3 : 4 : 5$. (A. Parovan)

2012 Sharygin Geometry Olympiad, 5

Tags: angle chasing , geometry , right triangle , isosceles

Let $ABC$ be an isosceles right-angled triangle. Point $D$ is chosen on the prolongation of the hypothenuse $AB$ beyond point $A$ so that $AB = 2AD$. Points $M$ and $N$ on side $AC$ satisfy the relation $AM = NC$. Point $K$ is chosen on the prolongation of $CB$ beyond point $B$ so that $CN = BK$. Determine the angle between lines $NK$ and $DM$. (M.Kungozhin)

2017 Saudi Arabia BMO TST, 1

Tags: product , prime , right triangle , integer sidelengths , number theory , geometry

Let $n = p_1p_2... p_{2017}$ be the positive integer where $p_1, p_2, ..., p_{2017}$ are $2017$ distinct odd primes. A triangle is called [i]nice [/i] if it is a right triangle with integer side lengths and the inradius is $n$. Find the number of nice triangles (two triangles are consider different if their tuples of length of sides are different)

1997 Argentina National Olympiad, 2

Tags: midpoint , angle , right triangle , isosceles , geometry

Let $ABC$ be a triangle and $M$ be the midpoint of $AB$. If it is known that $\angle CAM + \angle MCB = 90^o$, show that triangle $ABC$ is isosceles or right.

2022 Sharygin Geometry Olympiad, 3

Tags: geometry , right triangle , equal segments

Let $CD$ be an altitude of right-angled triangle $ABC$ with $\angle C = 90$. Regular triangles$ AED$ and $CFD$ are such that $E$ lies on the same side from $AB$ as $C$, and $F$ lies on the same side from $CD$ as $B$. The line $EF$ meets $AC$ at $L$. Prove that $FL = CL + LD$

2021 Germany Team Selection Test, 2

Tags: isosceles triangle , right triangle , geometry

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 $\angle ACB = 90^{\circ}$.

1999 Tournament Of Towns, 1

Tags: folding , area , right triangle , geometry , ratio

A right-angled triangle made of paper is folded along a straight line so that the vertex at the right angle coincides with one of the other vertices of the triangle and a quadrilateral is obtained . (a) What is the ratio into which the diagonals of this quadrilateral divide each other? (b) This quadrilateral is cut along its longest diagonal. Find the area of the smallest piece of paper thus obtained if the area of the original triangle is $1$ . (A Shapovalov)

2022 Yasinsky Geometry Olympiad, 5

Tags: geometric inequality , right triangle , geometry

Let $ABC$ be a right triangle with leg $CB = 2$ and hypotenuse $AB= 4$. Point $K$ is chosen on the hypotenuse $AB$, and point $L$ is chosen on the leg $AC$. a) Describe and justify how to construct such points $K$ and $ L$ so that the sum of the distances $CK+KL$ is the smallest possible. b) Find the smallest possible value of $CK+KL$. (Olexii Panasenko)

2010 Austria Beginners' Competition, 4

Tags: angle , geometry , right triangle

In the right-angled triangle $ABC$ with a right angle at $C$, the side $BC$ is longer than the side $AC$. The perpendicular bisector of $AB$ intersects the line $BC$ at point $D$ and the line $AC$ at point $E$. The segments $DE$ has the same length as the side $AB$. Find the measures of the angles of the triangle $ABC$. (R. Henner, Vienna)

Novosibirsk Oral Geo Oly VIII, 2020.3

Tags: geometry , right triangle

Maria Ivanovna drew on the blackboard a right triangle $ABC$ with a right angle $B$. Three students looked at her and said: $\bullet$ Yura said: "The hypotenuse of this triangle is $10$ cm." $\bullet$ Roma said: "The altitude drawn from the vertex $B$ on the side $AC$ is $6$ cm." $\bullet$ Seva said: "The area of the triangle $ABC$ is $25$ cm$^2$." Determine which of the students was mistaken if it is known that there is exactly one such person.

2021 South Africa National Olympiad, 2

Tags: ratio , right triangle , geometry

Let $PAB$ and $PBC$ be two similar right-angled triangles (in the same plane) with $\angle PAB = \angle PBC = 90^\circ$ such that $A$ and $C$ lie on opposite sides of the line $PB$. If $PC = AC$, calculate the ratio $\frac{PA}{AB}$.

2017 Thailand Mathematical Olympiad, 8

Tags: algebra , sides of a triangle , minimum , right triangle , inequalities

Let $a, b, c$ be side lengths of a right triangle. Determine the minimum possible value of $\frac{a^3 + b^3 + c^3}{abc}$.

Brazil L2 Finals (OBM) - geometry, 2002.1

Tags: symmetric , area , right triangle , area of a triangle , geometry

Let $XYZ$ be a right triangle of area $1$ m$^2$ . Consider the triangle $X'Y'Z'$ such that $X'$ is the symmetric of X wrt side $YZ$, $Y'$ is the symmetric of $Y$ wrt side $XZ$ and $Z' $ is the symmetric of $Z$ wrt side $XY$. Calculate the area of the triangle $X'Y'Z'$.

2014 Romania National Olympiad, 2

Tags: rhombus , right triangle , geometry , perpendicular , square , isosceles

Outside the square $ABCD$, the rhombus $BCMN$ is constructed with angle $BCM$ obtuse . Let $P$ be the intersection point of the lines $BM$ and $AN$ . Prove that $DM \perp CP$ and the triangle $DPM$ is right isosceles .

1994 Denmark MO - Mohr Contest, 4

Tags: integer , right triangle , geometry

In a right-angled triangle in which all side lengths are integers, one has a cathetus length $1994$. Determine the length of the hypotenuse.

2009 Brazil Team Selection Test, 1

Tags: right triangle , inradius , geometry

Let $r$ be a positive real number. Prove that the number of right triangles with prime positive integer sides that have an inradius equal to $r$ are zero or a power of $2$. [hide=original wording]Seja r um numero real positivo. Prove que o numero de triangulos retangulos com lados inteiros positivos primos entre si que possuem inraio igual a r e zero ou uma potencia de 2.[/hide]

Champions Tournament Seniors - geometry, 2008.2

Tags: right triangle , incenter , geometry

Given a right triangle $ABC$ with $ \angle C=90^o$. On its hypotenuse $AB$ is arbitrary mark the point$ P$. The point $Q$ is symmetric to the point $P$ wrt $AC$. Let the lines $PQ$ and $BQ$ intersect $AC$ at points $O$ and $R$ respectively. Denote by $S$ the foot of the perpendicular from the point $R$ on the line $AB$ ($S \ne P$), and let $T$ be the intersection point of lines $OS$ and $BR$. Prove that $R$ is the center of the circle inscribed in the triangle $CST$.

2000 Belarus Team Selection Test, 1.2

Tags: angle bisector , right triangle , geometry

Let $P$ be a point inside a triangle $ABC$ with $\angle C = 90^o$ such that $AP = AC$, and let $M$ be the midpoint of $AB$ and $CH$ be the altitude. Prove that $PM$ bisects $\angle BPH$ if and only if $\angle A = 60^o$.

1999 Junior Balkan Team Selection Tests - Moldova, 2

Tags: angle , right triangle , geometry , isosceles

Let $ABC$ be an isosceles right triangle with $\angle A=90^o$. Point $D$ is the midpoint of the side $[AC]$, and point $E \in [AC]$ is so that $EC = 2AE$. Calculate $\angle AEB + \angle ADB$ .

2016 Oral Moscow Geometry Olympiad, 3

Tags: right triangle , geometry , distance

A circle with center $O$ passes through the ends of the hypotenuse of a right-angled triangle and intersects its legs at points $M$ and $K$. Prove that the distance from point $O$ to line $MK$ is half the hypotenuse.

2017 Grand Duchy of Lithuania, 3

Tags: right triangle , circumcircle , parallel , geometry

Let $ABC$ be a triangle with $\angle A = 90^o$ and let $D$ be an orthogonal projection of $A$ onto $BC$. The midpoints of $AD$ and $AC$ are called $E$ and $F$, respectively. Let $M$ be the circumcentre of $\vartriangle BEF$. Prove that $AC\parallel BM$.