Found problems: 107
2000 Tournament Of Towns, 2
Two parallel sides of a quadrilateral have integer lengths. Prove that this quadrilateral can be cut into congruent triangles.
(A Shapovalov)
2010 IFYM, Sozopol, 3
Let $ ABC$ is a triangle, let $ H$ is orthocenter of $ \triangle ABC$, let $ M$ is midpoint of $ BC$. Let $ (d)$ is a line perpendicular with $ HM$ at point $ H$. Let $ (d)$ meet $ AB, AC$ at $ E, F$ respectively. Prove that $ HE \equal{}HF$.
2019 AIME Problems, 1
Points $C\neq D$ lie on the same side of line $AB$ so that $\triangle ABC$ and $\triangle BAD$ are congruent with $AB = 9$, $BC=AD=10$, and $CA=DB=17$. The intersection of these two triangular regions has area $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2023 Israel National Olympiad, P3
A triangle $ABC$ is given together with an arbitrary circle $\omega$. Let $\alpha$ be the reflection of $\omega$ with respect to $A$, $\beta$ the reflection of $\omega$ with respect to $B$, and $\gamma$ the reflection of $\omega$ with respect to $C$. It is known that the circles $\alpha, \beta, \gamma$ don't intersect each other.
Let $P$ be the meeting point of the two internal common tangents to $\beta, \gamma$ (see picture). Similarly, $Q$ is the meeting point of the internal common tangents of $\alpha, \gamma$, and $R$ is the meeting point of the internal common tangents of $\alpha, \beta$.
Prove that the triangles $PQR, ABC$ are congruent.
1998 AMC 8, 13
What is the ratio of the area of the shaded square to the area of the large square? (The figure is drawn to scale)
[asy]
draw((0,0)--(0,4)--(4,4)--(4,0)--cycle);
draw((0,0)--(4,4));
draw((0,4)--(3,1)--(3,3));
draw((1,1)--(2,0)--(4,2));
fill((1,1)--(2,0)--(3,1)--(2,2)--cycle,black);[/asy]
$ \text{(A)}\ \frac{1}{6}\qquad\text{(B)}\ \frac{1}{7}\qquad\text{(C)}\ \frac{1}{8}\qquad\text{(D)}\ \frac{1}{12}\qquad\text{(E)}\ \frac{1}{16} $
1984 All Soviet Union Mathematical Olympiad, 381
Given triangle $ABC$ . From the $P$ point three lines $(PA),(PB),(PC)$ are drawn. They cross the circumscribed circle at $A_1, B_1,C_1$ points respectively. It comes out that the $A_1B_1C_1$ triangle equals to the initial one. Prove that there are not more than eight such a points $P$ in a plane.
2009 AMC 8, 20
How many non-congruent triangles have vertices at three of the eight points in the array shown below?
[asy]dot((0,0));
dot((0,.5));
dot((.5,0));
dot((.5,.5));
dot((1,0));
dot((1,.5));
dot((1.5,0));
dot((1.5,.5));[/asy]
$ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$