Olympiad problems

2013 Argentina National Olympiad Level 2, 2

Let $ABC$ be a right triangle. It is known that there are points $D$ on the side $AC$ and $E$ on the side $BC$ such that $AB = AD = BE$ and $BD$ is perpendicular to $DE$. Calculate the ratios $\frac{AB}{BC}$ and $\frac{BC}{CA}$.

1999 Harvard-MIT Mathematics Tournament, 2

Tags: geometry , right triangle

A ladder is leaning against a house with its lower end $15$ feet from the house. When the lower end is pulled $9$ feet farther from the house, the upper end slides $13$ feet down. How long is the ladder (in feet)?

2022 Sharygin Geometry Olympiad, 3

Tags: geometry , equal segments , right triangle

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 $FL = CL + LD$

2020 Thailand TST, 1

Tags: geometry , circumcircle , right angle , right triangle , 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 $\angle ACB = 90^o$. .

2018 Lusophon Mathematical Olympiad, 2

Tags: right triangle , isosceles , square , 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.

2006 Sharygin Geometry Olympiad, 10.5

Tags: geometry , tetrahedron , 3d geometry , right triangle

Can a tetrahedron scan turn out to be a triangle with sides $3, 4$ and $5$ (a tetrahedron can be cut only along the edges)?

2013 Saudi Arabia BMO TST, 1

Tags: geometry , cyclic quadrilateral , right triangle , isosceles

In triangle $ABC$, $AB = AC = 3$ and $\angle A = 90^o$. Let $M$ be the midpoint of side $BC$. Points $D$ and $E$ lie on sides $AC$ and $AB$ respectively such that $AD > AE$ and $ADME$ is a cyclic quadrilateral. Given that triangle $EMD$ has area $2$, find the length of segment $CD$.

2016 Singapore Junior Math Olympiad, 3

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

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.

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 $AD + CE = AE$.

2009 Ukraine Team Selection Test, 5

Tags: geometry , circumcircle , circumcenter , right triangle , orthocenter

Let $A,B,C,D,E$ be consecutive points on a circle with center $O$ such that $AC=BD=CE=DO$. Let $H_1,H_2,H_3$ be the orthocenters triangles $ACD,BCD,BCE$ respectively. Prove that the triangle $H_1H_2H_3$ is right.

2011 Saudi Arabia Pre-TST, 1

Tags: geometry , right triangle , geometric inequality

Let $ABC$ be a triangle with $\angle A = 90^o$ and let $P$ be a point on the hypotenuse $BC$. Prove that $$\frac{AB^2}{PC}+\frac{AC^2}{PB} \ge \frac{BC^3}{PA^2 + PB \cdot PC}$$

2004 Paraguay Mathematical Olympiad, 2

Tags: geometry , right triangle , pythagorean theorem

Determine for what values of $x$ the expressions $2x + 2$,$x + 4$, $x + 2$ can represent the sidelengths of a right triangle.

Denmark (Mohr) - geometry, 1991.3

Tags: geometry , perimeter , right triangle

A right-angled triangle has perimeter $60$ and the altitude of the hypotenuse has a length $12$. Determine the lengths of the sides.

2023 European Mathematical Cup, 2

Tags: geometry , right triangle , circumcircle

Let $ABC$ be a triangle such that $\angle BAC = 90^{\circ}$. The incircle of triangle $ABC$ is tangent to the sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D,E,F$ respectively. Let $M$ be the midpoint of $\overline{EF}$. Let $P$ be the projection of $A$ onto $BC$ and let $K$ be the intersection of $MP$ and $AD$. Prove that the circumcircles of triangles $AFE$ and $PDK$ have equal radius. [i]Kyprianos-Iason Prodromidis[/i]

1996 Portugal MO, 1

Tags: geometry , square , right triangle

Consider a square on the hypotenuse of a right triangle $[ABC]$ (right at $B$). Prove that the line segment that joins vertex $B$ with the center of the square makes $45^o$ angles with legs of the triangle.

Novosibirsk Oral Geo Oly VIII, 2021.5

Tags: geometry , right triangle , isosceles , equal segments

On the legs $AC$ and $BC$ of an isosceles right-angled triangle with a right angle $C$, points $D$ and $E$ are taken, respectively, so that $CD = CE$. Perpendiculars on line $AE$ from points $C$ and $D$ intersect segment $AB$ at points $P$ and $Q$, respectively. Prove that $BP = PQ$.

1993 Rioplatense Mathematical Olympiad, Level 3, 6

Tags: right triangle , geometry , pentagon , equal segments , angle

Let $ABCDE$ be pentagon such that $AE = ED$ and $BC = CD$. It is known that $\angle BAE + \angle EDC + \angle CB A = 360^o$ and that $P$ is the midpoint of $AB$. Show that the triangle $ECP$ is right.

2020 Australian Maths Olympiad, 3

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

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.

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 $\frac{PA}{AB}$.

2010 Silk Road, 1

Tags: convex quadrilateral , right triangle , angle , geometry

In a convex quadrilateral it is known $ABCD$ that $\angle ADB + \angle ACB = \angle CAB + \angle DBA = 30^{\circ}$ and $AD = BC$. Prove that from the lengths $DB$, $CA$ and $DC$, you can make a right triangle.

2010 Sharygin Geometry Olympiad, 5

Tags: geometry , equal segments , right triangle , altitude , circumcircle

Let $BH$ be an altitude of a right-angled triangle $ABC$ ($\angle B = 90^o$). The incircle of triangle $ABH$ touches $AB,AH$ in points $H_1, B_1$, the incircle of triangle $CBH$ touches $CB,CH$ in points $H_2, B_2$, point $O$ is the circumcenter of triangle $H_1BH_2$. Prove that $OB_1 = OB_2$.

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 $\angle A = 60^o$.

Denmark (Mohr) - geometry, 1996.1

Tags: geometry , right triangle , area

In triangle $ABC$, angle $C$ is right and the two catheti are both length $1$. For one given the choice of the point $P$ on the cathetus $BC$, the point $Q$ on the hypotenuse and the point $R$ are plotted on the second cathetus so that $PQ$ is parallel to $AC$ and $QR$ is parallel to $BC$. Thereby the triangle is divided into three parts. Determine the locations of point $P$ for which the rectangular part has a larger area than each of the other two parts.

2009 Junior Balkan Team Selection Tests - Romania, 1

Tags: geometry , right triangle , geometric inequality , inequalities

Show that in any triangle $ABC$ with $A = 90^0$ the following inequality holds: $$(AB -AC)^2(BC^2 + 4AB \cdot AC)^2 \le 2BC^6$$

2015 Hanoi Open Mathematics Competitions, 10

Tags: square , geometry , concyclic , right triangle

A right-angled triangle has property that, when a square is drawn externally on each side of the triangle, the six vertices of the squares that are not vertices of the triangle are concyclic. Assume that the area of the triangle is $9$ cm$^2$. Determine the length of sides of the triangle.