Olympiad problems

2008 AMC 12/AHSME, 18

Tags: geometry , 3d geometry , tetrahedron , pyramid , calculus , analytic geometry , area of a triangle

Triangle $ ABC$, with sides of length $ 5$, $ 6$, and $ 7$, has one vertex on the positive $ x$-axis, one on the positive $ y$-axis, and one on the positive $ z$-axis. Let $ O$ be the origin. What is the volume of tetrahedron $ OABC$? $ \textbf{(A)}\ \sqrt{85} \qquad \textbf{(B)}\ \sqrt{90} \qquad \textbf{(C)}\ \sqrt{95} \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ \sqrt{105}$

2008 Indonesia TST, 1

Tags: geometry , combinatorial geometry , combinatorics , area of a triangle

Let $ABCD$ be a square with side $20$ and $T_1, T_2, ..., T_{2000}$ are points in $ABCD$ such that no $3$ points in the set $S = \{A, B, C, D, T_1, T_2, ..., T_{2000}\}$ are collinear. Prove that there exists a triangle with vertices in $S$, such that the area is less than $1/10$.

1993 Mexico National Olympiad, 3

Tags: combinatorial geometry , geometry , pentagon , area of a triangle

Given a pentagon of area $1993$ and $995$ points inside the pentagon, let $S$ be the set containing the vertices of the pentagon and the $995$ points. Show that we can find three points of $S$ which form a triangle of area $\le 1$.

2013 AIME Problems, 13

Tags: geometry , ratio , pythagorean theorem , area of a triangle , heron's formula , algebra , system of equations

In $\triangle ABC$, $AC = BC$, and point $D$ is on $\overline{BC}$ so that $CD = 3 \cdot BD$. Let $E$ be the midpoint of $\overline{AD}$. Given that $CE = \sqrt{7}$ and $BE = 3$, the area of $\triangle ABC$ can be expressed in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n$.

2014 BMT Spring, 8

Tags: geometry , area of a triangle , area

Line segment $AB$ has length $4$ and midpoint $M$. Let circle $C_1$ have diameter $AB$, and let circle $C_2$ have diameter $AM$. Suppose a tangent of circle $C_2$ goes through point $ B$ to intersect circle $C_1$ at $N$. Determine the area of triangle $AMN$.

2019 Tournament Of Towns, 2

Tags: geometry , geometric inequality , area of a triangle , triangle inequality , inequalities

Two acute triangles $ABC$ and $A_1B_1C_1$ are such that $B_1$ and $C_1$ lie on $BC$, and $A_1$ lies inside the triangle $ABC$. Let $S$ and $S_1$ be the areas of those triangles respectively. Prove that $\frac{S}{AB + AC}> \frac{S_1}{A_1B_1 + A_1C_1}$ (Nairi Sedrakyan, Ilya Bogdanov)

1998 Tuymaada Olympiad, 3

Tags: geometric inequality , inequalities , inradius , area of a triangle , area , geometry

The segment of length $\ell$ with the ends on the border of a triangle divides the area of that triangle in half. Prove that $\ell >r\sqrt2$, where $r$ is the radius of the inscribed circle of the triangle.

2013 AIME Problems, 12

Tags: geometry , trigonometry , number theory , relatively prime , area of a triangle

Let $\triangle PQR$ be a triangle with $\angle P = 75^\circ$ and $\angle Q = 60^\circ$. A regular hexagon $ABCDEF$ with side length 1 is drawn inside $\triangle PQR$ so that side $\overline{AB}$ lies on $\overline{PQ}$, side $\overline{CD}$ lies on $\overline{QR}$, and one of the remaining vertices lies on $\overline{RP}$. There are positive integers $a$, $b$, $c$, and $d$ such that the area of $\triangle PQR$ can be expressed in the form $\tfrac{a+b\sqrt c}d$, where $a$ and $d$ are relatively prime and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.

2020 Malaysia IMONST 1, 13

Tags: right triangle , area of a triangle

Given a right-angled triangle with perimeter $18$. The sum of the squares of the three side lengths is $128$. What is the area of the triangle?