Found problems: 321
2014 NIMO Problems, 3
Let $ABCD$ be a square with side length $2$. Let $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{CD}$ respectively, and let $X$ and $Y$ be the feet of the perpendiculars from $A$ to $\overline{MD}$ and $\overline{NB}$, also respectively. The square of the length of segment $\overline{XY}$ can be written in the form $\tfrac pq$ where $p$ and $q$ are positive relatively prime integers. What is $100p+q$?
[i]Proposed by David Altizio[/i]
2024 JHMT HS, 6
Let $N_5$ be the answer to problem 5.
Triangle $JHU$ satisfies $JH=N_5$ and $JU=6$. Point $X$ lies on $\overline{HU}$ such that $\overline{JX}$ is an altitude of $\triangle{JHU}$, point $Y$ is the midpoint of $\overline{JU}$, and $\overline{JX}$ and $\overline{HY}$ intersect at $Z$. Assume that $\triangle{HZX}$ is similar to $\triangle{JZY}$ (in this vertex order). Compute the area of $\triangle{JHU}$.
2013 Harvard-MIT Mathematics Tournament, 19
An isosceles trapezoid $ABCD$ with bases $AB$ and $CD$ has $AB=13$, $CD=17$, and height $3$. Let $E$ be the intersection of $AC$ and $BD$. Circles $\Omega$ and $\omega$ are circumscribed about triangles $ABE$ and $CDE$. Compute the sum of the radii of $\Omega$ and $\omega$.
2006 Mediterranean Mathematics Olympiad, 3
The side lengths $a,b,c$ of a triangle $ABC$ are integers with $\gcd(a,b,c)=1$. The bisector of angle $BAC$ meets $BC$ at $D$.
(a) show that if triangles $DBA$ and $ABC$ are similar then $c$ is a square.
(b) If $c=n^2$ is a square $(n\ge 2)$, find a triangle $ABC$ satisfying (a).
2019 AIME Problems, 7
Triangle $ABC$ has side lengths $AB=120$, $BC=220$, and $AC=180$. Lines $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$ are drawn parallel to $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$, respectively, such that the intersection of $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$ with the interior of $\triangle ABC$ are segments of length $55$, $45$, and $15$, respectively. Find the perimeter of the triangle whose sides lie on $\ell_{A}$, $\ell_{B}$, and $\ell_{C}$.
1982 IMO Longlists, 54
The right triangles $ABC$ and $AB_1C_1$ are similar and have opposite orientation. The right angles are at $C$ and $C_1$, and we also have $ \angle CAB = \angle C_1AB_1$. Let $M$ be the point of intersection of the lines $BC_1$ and $B_1C$. Prove that if the lines $AM$ and $CC_1$ exist, they are perpendicular.
2013 AIME Problems, 13
Triangle $AB_0C_0$ has side lengths $AB_0 = 12$, $B_0C_0 = 17$, and $C_0A = 25$. For each positive integer $n$, points $B_n$ and $C_n$ are located on $\overline{AB_{n-1}}$ and $\overline{AC_{n-1}}$, respectively, creating three similar triangles $\triangle AB_nC_n \sim \triangle B_{n-1}C_nC_{n-1} \sim \triangle AB_{n-1}C_{n-1}$. The area of the union of all triangles $B_{n-1}C_nB_n$ for $n\geq1$ can be expressed as $\tfrac pq$, where $p$ and $q$ are relatively prime positive integers. Find $q$.
2000 Austrian-Polish Competition, 7
Triangle $A_0B_0C_0$ is given in the plane. Consider all triangles $ABC$ such that:
(i) The lines $AB,BC,CA$ pass through $C_0,A_0,B_0$, respectvely,
(ii) The triangles $ABC$ and $A_0B_0C_0$ are similar.
Find the possible positions of the circumcenter of triangle $ABC$.
2016 Saudi Arabia Pre-TST, 1.2
Let $ABC$ be a non isosceles triangle inscribed in a circle $(O)$ and $BE, CF$ are two angle bisectors intersect at $I$ with $E$ belongs to segment $AC$ and $F$ belongs to segment $AB$. Suppose that $BE, CF$ intersect $(O)$ at $M,N$ respectively. The line $d_1$ passes through $M$ and perpendicular to $BM$ intersects $(O)$ at the second point $P,$ the line $d_2$ passes through $N$ and perpendicular to $CN$ intersect $(O)$ at the second point $Q$. Denote $H, K$ are two midpoints of $MP$ and $NQ$ respectively.
1. Prove that triangles $IEF$ and $OKH$ are similar.
2. Suppose that S is the intersection of two lines $d_1$ and $d_2$. Prove that $SO$ is perpendicular to $EF$.
1997 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 6
Let $ ABCD$ be a trapezoid with $ AB\parallel{}CD$. Let $ a \equal{} AB$ and $ b \equal{} CD$. For $ MN\parallel{}AB$ such that $ M$ lies on $ AD,$ $ N$ lies on $ BC$, and the trapezoids $ ABNM$ and $ MNCD$ have the same area, the length of $ MN$ equals
[img]http://i250.photobucket.com/albums/gg265/geometry101/NielsHenrikAbel1997Number6.jpg[/img]
A. $ \sqrt{ab}$
B. $ \frac{a\plus{}b}{2}$
C. $ \frac{a^2 \plus{} b^2}{a\plus{}b}$
D. $ \sqrt{\frac{a^2 \plus{} b^2}{2}}$
E. $ \frac{a^2 \plus{} (2 \sqrt{2} \minus{} 2)ab \plus{} b^2}{\sqrt{2} (a\plus{}b)}$
2001 AMC 12/AHSME, 15
An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common endpoint.)
$ \displaystyle \textbf{(A)} \ \frac {1}{2} \sqrt {3} \qquad \textbf{(B)} \ 1 \qquad \textbf{(C)} \ \sqrt {2} \qquad \textbf{(D)} \ \frac {3}{2} \qquad \textbf{(E)} \ 2$
2000 AMC 12/AHSME, 21
Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is $ m$ times the area of the square. The ratio of the area of the other small right triangle to the area of the square is
$ \textbf{(A)}\ \frac {1}{2m \plus{} 1} \qquad \textbf{(B)}\ m \qquad \textbf{(C)}\ 1 \minus{} m \qquad \textbf{(D)}\ \frac {1}{4m} \qquad \textbf{(E)}\ \frac {1}{8m^2}$
2024 Vietnam Team Selection Test, 3
Let $ABC$ be an acute scalene triangle. Incircle of $ABC$ touches $BC,CA,AB$ at $D,E,F$ respectively. Let $X,Y,Z$ be feet the altitudes of from $A,B,C$ to the sides $BC,CA,AB$ respectively. Let $A',B',C'$ be the reflections of $X,Y,Z$ in $EF,FD,DE$ respectively. Prove that triangles $ABC$ and $A'B'C'$ are similar.
1984 AMC 12/AHSME, 17
A right triangle $ABC$ with hypotenuse $AB$ has side $AC = 15$. Altitude $CH$ divides $AB$ into segments $AH$ And $HB$, with $HB = 16$. The area of $\triangle ABC$ is:
[asy]
size(200);
defaultpen(linewidth(0.8)+fontsize(11pt));
pair A = origin, H = (5,0), B = (13,0), C = (5,6.5);
draw(C--A--B--C--H^^rightanglemark(C,H,B,16));
label("$A$",A,W);
label("$B$",B,E);
label("$C$",C,N);
label("$H$",H,S);
label("$15$",C/2,NW);
label("$16$",(H+B)/2,S);
[/asy]
$\textbf{(A) }120\qquad
\textbf{(B) }144\qquad
\textbf{(C) }150\qquad
\textbf{(D) }216\qquad
\textbf{(E) }144\sqrt5$
1998 Poland - First Round, 3
In the isosceles triangle $ ABC$ the angle $ BAC$ is a right angle. Point $ D$ lies on the side $ BC$ and satisfies $ BD \equal{} 2 \cdot CD$. Point $ E$ is the foot of the perpendicular of the point $ B$ on the line $ AD$. Find the angle $ CED$.
2021 USA TSTST, 8
Let $ABC$ be a scalene triangle. Points $A_1,B_1$ and $C_1$ are chosen on segments $BC,CA$ and $AB$, respectively, such that $\triangle A_1B_1C_1$ and $\triangle ABC$ are similar. Let $A_2$ be the unique point on line $B_1C_1$ such that $AA_2=A_1A_2$. Points $B_2$ and $C_2$ are defined similarly. Prove that $\triangle A_2B_2C_2$ and $\triangle ABC$ are similar.
[i]Fedir Yudin [/i]
2010 Bundeswettbewerb Mathematik, 3
On the sides of a triangle $XYZ$ to the outside construct similar triangles $YDZ, EXZ ,YXF$ with circumcenters $K, L ,M$ respectively. Here are $\angle ZDY = \angle ZXE = \angle FXY$ and $\angle YZD = \angle EZX = \angle YFX$. Show that the triangle $KLM$ is similar to the triangles .
[img]https://cdn.artofproblemsolving.com/attachments/e/f/fe0d0d941015d32007b6c00b76b253e9b45ca5.png[/img]
2018 Singapore Senior Math Olympiad, 2
In a convex quadrilateral $ABCD, \angle A < 90^o, \angle B < 90^o$ and $AB > CD$. Points $P$ and $Q$ are on the segments $BC$ and $AD$ respectively. Suppose the triangles $APD$ and $BQC$ are similar. Prove that $AB$ is parallel to $CD$.
1957 AMC 12/AHSME, 35
Side $ AC$ of right triangle $ ABC$ is divide into $ 8$ equal parts. Seven line segments parallel to $ BC$ are drawn to $ AB$ from the points of division. If $ BC \equal{} 10$, then the sum of the lengths of the seven line segments:
$ \textbf{(A)}\ \text{cannot be found from the given information} \qquad
\textbf{(B)}\ \text{is }{33}\qquad
\textbf{(C)}\ \text{is }{34}\qquad
\textbf{(D)}\ \text{is }{35}\qquad
\textbf{(E)}\ \text{is }{45}$
2012 Math Prize For Girls Problems, 5
The figure below shows a semicircle inscribed in a right triangle.
[asy]
draw((0, 0) -- (15, 0) -- (0, 8) -- cycle);
real r = 120 / 23;
real theta = -aTan(8/15);
draw(arc((r, r), r, theta + 180, theta + 360));
[/asy]
The triangle has legs of length 8 and 15. The semicircle is tangent to the two legs, and its diameter is on the hypotenuse. What is the radius of the semicircle?
2010 Romania Team Selection Test, 1
Each point of the plane is coloured in one of two colours. Given an odd integer number $n \geq 3$, prove that there exist (at least) two similar triangles whose similitude ratio is $n$, each of which has a monochromatic vertex-set.
[i]Vasile Pop[/i]
2010 China Team Selection Test, 1
Let $\triangle ABC$ be an acute triangle, and let $D$ be the projection of $A$ on $BC$. Let $M,N$ be the midpoints of $AB$ and $AC$ respectively. Let $\Gamma_1$ and $\Gamma_2$ be the circumcircles of $\triangle BDM$ and $\triangle CDN$ respectively, and let $K$ be the other intersection point of $\Gamma_1$ and $\Gamma_2$. Let $P$ be an arbitrary point on $BC$ and $E,F$ are on $AC$ and $AB$ respectively such that $PEAF$ is a parallelogram. Prove that if $MN$ is a common tangent line of $\Gamma_1$ and $\Gamma_2$, then $K,E,A,F$ are concyclic.
2007 Stanford Mathematics Tournament, 13
A rope of length 10 [i]m[/i] is tied tautly from the top of a flagpole to the ground 6 [i]m[/i] away from the base of the pole. An ant crawls up the rope and its shadow moves at a rate of 30 [i]cm/min[/i]. How many meters above the ground is the ant after 5 minutes? (This takes place on the summer solstice on the Tropic of Cancer so that the sun is directly overhead.)
2010 Romania Team Selection Test, 4
Let $n$ be an integer number greater than or equal to $2$, and let $K$ be a closed convex set of area greater than or equal to $n$, contained in the open square $(0, n) \times (0, n)$. Prove that $K$ contains some point of the integral lattice $\mathbb{Z} \times \mathbb{Z}$.
[i]Marius Cavachi[/i]
2011 Purple Comet Problems, 15
A pyramid has a base which is an equilateral triangle with side length $300$ centimeters. The vertex of the pyramid is $100$ centimeters above the center of the triangular base. A mouse starts at a corner of the base of the pyramid and walks up the edge of the pyramid toward the vertex at the top. When the mouse has walked a distance of $134$ centimeters, how many centimeters above the base of the pyramid is the mouse?