Olympiad problems

2016 BMT Spring, 4

Let $ABC$ have side lengths $3$, $4$, and $5$. Let $P$ be a point inside $ABC$. What is the minimum sum of lengths of the altitudes from $P$ to the side lengths of $ABC$?

Triangle $ABC$ has side lengths $AB = 3$, $BC = 4$, and $CD = 5$. Draw line $\ell_A$ such that $\ell_A$ is parallel to $BC$ and splits the triangle into two polygons of equal area. Define lines $\ell_B$ and $\ell_C$ analogously. The intersection points of $\ell_A$, $\ell_B$, and $\ell_C$ form a triangle. Determine its area.

2009 Moldova National Olympiad, 8.4

Tags: geometry , right triangle , incenter

Prove that a right triangle has an angle equal to $30^o$ if and only if the center of the circle inscribed in this triangle is located on the perpendicular bisector of the median taken from the vertex of the right angle of the triangle.

2020-21 KVS IOQM India, 22

Tags: geometry , right triangle , inradius

Let $ABC$ be a triangle with $\angle BAC = 90^o$ and $D$ be the point on the side $BC$ such that $AD \perp BC$. Let$ r, r_1$, and $r_2$ be the inradii of triangles $ABC, ABD$, and $ACD$, respectively. If $r, r_1$, and $r_2$ are positive integers and one of them is $5$, find the largest possible value of $r+r_1+ r_2$.

2006 Junior Balkan Team Selection Tests - Romania, 1

Tags: right triangle , geometry , angle , equal ratio , ratio

Let $ABC$ be a triangle right in $C$ and the points $D, E$ on the sides $BC$ and $CA$ respectively, such that $\frac{BD}{AC} =\frac{AE}{CD} = k$. Lines $BE$ and $AD$ intersect at $O$. Show that the angle $\angle BOD = 60^o$ if and only if $k =\sqrt3$.

2004 Dutch Mathematical Olympiad, 5

Tags: number theory , right triangle

A right triangle with perpendicular sides $a$ and $b$ and hypotenuse $c$ has the following properties: $a = p^m$ and $b = q^n$ with $p$ and $q$ prime numbers and $m$ and $n$ positive integers, $c = 2k +1$ with $k$ a positive integer. Determine all possible values of $c$ and the associated values of $a$ and $b$.

2012 IMO Shortlist, G5

Tags: right triangle , geometry , circumcircle

Let $ABC$ be a triangle with $\angle BCA=90^{\circ}$, and let $D$ be the foot of the altitude from $C$. Let $X$ be a point in the interior of the segment $CD$. Let $K$ be the point on the segment $AX$ such that $BK=BC$. Similarly, let $L$ be the point on the segment $BX$ such that $AL=AC$. Let $M$ be the point of intersection of $AL$ and $BK$. Show that $MK=ML$. [i]Proposed by Josef Tkadlec, Czech Republic[/i]

Ukraine Correspondence MO - geometry, 2013.7

Tags: right triangle , collinear , geometry

An arbitrary point $D$ is marked on the hypotenuse $AB$ of a right triangle $ABC$. The circle circumscribed around the triangle $ACD$ intersects the line $BC$ at the point $E$ for the second time, and the circle circumscribed around the triangle $BCD$ intersects the line $AC$ for the second time at the point $F$. Prove that the line $EF$ passes through the point $D$.

1993 All-Russian Olympiad Regional Round, 9.6

Tags: geometry , right triangle

Three right-angled triangles have been placed in a halfplane determined by a line $\ell$, each with one leg lying on $\ell$. Assume that there is a line parallel to $\ell$ cutting the triangles in three congruent segments. Show that, if each of the triangles is rotated so that its other leg lies on $\ell$, then there still exists a line parallel to $\ell$ cutting them in three congruent segments.

Kyiv City MO Juniors Round2 2010+ geometry, 2018.9.1

Tags: geometry , right triangle , isosceles

Cut a right triangle with an angle of $30^o$ into three isosceles non-acute triangles, among which there are no congruent ones. (Maria Rozhkova)

1999 Tournament Of Towns, 2

Tags: right triangle , geometry , ratio , square

$ABC$ is a right-angled triangle. A square $ABDE$ is constructed on the opposite side of the hypothenuse $AB$ from $C$. The bisector of $\angle C$ cuts $DE$ at $F$. If $AC = 1$ and $BC = 3$, compute $\frac{DF}{EF}$. (A Blinkov)

Kyiv City MO Juniors Round2 2010+ geometry, 2016.8.1

Tags: right triangle , ratio , radius , geometry

In a right triangle, the point $O$ is the center of the circumcircle. Another circle of smaller radius centered at the point $O$ touches the larger leg and the altitude drawn from the top of the right angle. Find the acute angles of a right triangle and the ratio of the radii of the circumscribed and smaller circles.

1999 Harvard-MIT Mathematics Tournament, 2

Tags: right triangle , geometry

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)?

2003 Nordic, 3

Tags: geometry , right triangle , equilateral triangle

The point ${D}$ inside the equilateral triangle ${\triangle ABC}$ satisfies ${\angle ADC = 150^o}$. Prove that a triangle with side lengths ${|AD|, |BD|, |CD|}$ is necessarily a right-angled triangle.

2018 Thailand TSTST, 6

Tags: right triangle , perpendicular bisector , geometry

In a right-angled triangle $ABC$ ($\angle A = 90^o$), the perpendicular bisector of $BC$ intersects the line $AC$ in $K$ and the perpendicular bisector of $BK$ intersects the line $AB$ in $L$. If the line $CL$ be the internal bisector of angle $C$, find all possible values for angles $B$ and $C$. by Mahdi Etesami Fard

1992 Austrian-Polish Competition, 2

Tags: right triangle , square , combinatorics , coloring

Each point on the boundary of a square has to be colored in one color. Consider all right triangles with the vertices on the boundary of the square. Determine the least number of colors for which there is a coloring such that no such triangle has all its vertices of the same color.

2007 Oral Moscow Geometry Olympiad, 1

Tags: geometry , right triangle , similar triangles

The triangle was divided into five triangles similar to it. Is it true that the original triangle is right-angled? (S. Markelov)

2019 Costa Rica - Final Round, 6

Tags: right triangle , geometry , isosceles , collinear

Consider the right isosceles $\vartriangle ABC$ at $ A$. Let $L$ be the intersection of the bisector of $\angle ACB$ with $AB$ and $K$ the intersection point of $CL$ with the bisector of $BC$. Let $X$ be the point on line $AK$ such that $\angle KCX = 90^o$ and let $Y$ be the point of intersection of $CX$ with the circumcircle of $\vartriangle ABC$. Let $Y'$ the reflection of point $Y$ wrt $BC$. Prove that $B - K -Y'$. Notation: $A-B-C$ means than points $A,B,C$ are collinear in that order i.e. $ B$ lies between $ A$ and $C$.

1996 Portugal MO, 1

Tags: right triangle , square , geometry

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.

2010 Saudi Arabia BMO TST, 2

Tags: right triangle , geometry , geometric inequality

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

Durer Math Competition CD Finals - geometry, 2011.D2

Tags: right triangle , isosceles , geometry

In an right isosceles triangle $ABC$, there are two points on the hypotenuse $AB, K$ and $M$, respectively, such that $KCM$ angle is $45^o$ (point $K$ lies between $A$ and $M$). Prove that $AK^2 + MB^2 = KM^2$ [img]https://cdn.artofproblemsolving.com/attachments/2/c/e7c57e0651e5a4c492cc4ae4b115bf68a7a833.png[/img]

2021 Brazil Team Selection Test, 2

Tags: right triangle , isosceles 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 $\angle ACB = 90^{\circ}$.

1949-56 Chisinau City MO, 59

Tags: right triangle , trigonometry , geometry

Show that triangle $ABC$ is right-angled if its angles satisfy the ratio $\cos^2A + \cos ^2B +\ cos ^2C=1$.

2020 Yasinsky Geometry Olympiad, 1

Tags: right triangle , angle chasing , circumcircle , angle , geometry

Given a right triangle $ABC$, the point $M$ is the midpoint of the hypotenuse $AB$. A circle is circumscribed around the triangle $BCM$, which intersects the segment $AC$ at a point $Q$ other than $C$. It turned out that the segment $QA$ is twice as large as the side $BC$. Find the acute angles of triangle $ABC$. (Mykola Moroz)

1949-56 Chisinau City MO, 32

Tags: right triangle , midpoint , locus , geometry

Determine the locus of points that are the midpoints of segments of equal length, the ends of which lie on the sides of a given right angle.