Olympiad problems

2000 AMC 12/AHSME, 19

In triangle $ ABC$, $ AB \equal{} 13$, $ BC \equal{} 14$, and $ AC \equal{} 15$. Let $ D$ denote the midpoint of $ \overline{BC}$ and let $ E$ denote the intersection of $ \overline{BC}$ with the bisector of angle $ BAC$. Which of the following is closest to the area of the triangle $ ADE$? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 2.5 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 3.5 \qquad \textbf{(E)}\ 4$

2011 QEDMO 8th, 5

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

$9$ points are given in the interior of the unit square. Prove there exists a triangle of area $\le \frac18$ whose vertices are three of the points.

2013 National Olympiad First Round, 17

Tags: geometry , area of a triangle

Let $ABC$ be an equilateral triangle with side length $10$ and $P$ be a point inside the triangle such that $|PA|^2+ |PB|^2 + |PC|^2 = 128$. What is the area of a triangle with side lengths $|PA|,|PB|,|PC|$? $ \textbf{(A)}\ 6\sqrt 3 \qquad\textbf{(B)}\ 7 \sqrt 3 \qquad\textbf{(C)}\ 8 \sqrt 3 \qquad\textbf{(D)}\ 9 \sqrt 3 \qquad\textbf{(E)}\ 10 \sqrt 3 $

2019 BMT Spring, 16

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

Let $ABC$ be a triangle with $AB = 26$, $BC = 51$, and $CA = 73$, and let $O$ be an arbitrary point in the interior of $\vartriangle ABC$. Lines $\ell_1$, $\ell_2$, and $\ell_3$ pass through $O$ and are parallel to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. The intersections of $\ell_1$, $\ell_2$, and $\ell_3$ and the sides of $\vartriangle ABC$ form a hexagon whose area is $A$. Compute the minimum value of $A$.

1986 AIME Problems, 15

Tags: area of a triangle , geometry , analytic geometry , graphing lines , slope , trigonometry , pythagorean theorem

Let triangle $ABC$ be a right triangle in the xy-plane with a right angle at $C$. Given that the length of the hypotenuse $AB$ is 60, and that the medians through $A$ and $B$ lie along the lines $y=x+3$ and $y=2x+4$ respectively, find the area of triangle $ABC$.

1979 IMO Longlists, 35

Tags: geometry , inequalities , induction , area of a triangle , heron's formula

Given a sequence $(a_n)$, with $a_1 = 4$ and $a_{n+1} = a_n^2-2 (\forall n \in\mathbb{N})$, prove that there is a triangle with side lengths $a_{n-1}, a_n, a_{n+1},$ and that its area is equal to an integer.

2014 Harvard-MIT Mathematics Tournament, 3

Tags: angle bisector , geometry , area of a triangle , heron's formula

$ABC$ is a triangle such that $BC = 10$, $CA = 12$. Let $M$ be the midpoint of side $AC$. Given that $BM$ is parallel to the external bisector of $\angle A$, find area of triangle $ABC$. (Lines $AB$ and $AC$ form two angles, one of which is $\angle BAC$. The external angle bisector of $\angle A$ is the line that bisects the other angle.

2003 Paraguay Mathematical Olympiad, 4

Tags: area of a triangle , cevian , area , geometry

Triangle $ABC$ is divided into six smaller triangles by lines that pass through the vertices and through a common point inside of the triangle. The areas of four of these triangles are indicated. Calculate the area of triangle $ABC$. [img]https://cdn.artofproblemsolving.com/attachments/9/2/2013de890e438f5bf88af446692b495917b1ff.png[/img]

2019 Yasinsky Geometry Olympiad, p1

Tags: circles , analytic geometry , area of a triangle

The circle $x^2 + y^2 = 25$ intersects the abscissa in points $A$ and $B$. Let $P$ be a point that lies on the line $x = 11$, $C$ is the intersection point of this line with the $Ox$ axis, and the point $Q$ is the intersection point of $AP$ with the given circle. It turned out that the area of the triangle $AQB$ is four times smaller than the area of the triangle $APC$. Find the coordinates of $Q$.

2014 Harvard-MIT Mathematics Tournament, 6

Tags: geometry , ratio , trigonometry , area of a triangle , heron's formula , trig identities , law of cosines

In quadrilateral $ABCD$, we have $AB = 5$, $BC = 6$, $CD = 5$, $DA = 4$, and $\angle ABC = 90^\circ$. Let $AC$ and $BD$ meet at $E$. Compute $\dfrac{BE}{ED}$.

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?

2005 All-Russian Olympiad Regional Round, 11.6

Tags: inequalities , geometric transformation , vector , trigonometry , area of a triangle , geometry , reflection

11.6 Construct for each vertex of the quadrilateral of area $S$ a symmetric point wrt to the diagonal, which doesn't contain this vertex. Let $S'$ be an area of the obtained quadrilateral. Prove that $\frac{S'}{S}<3$. ([i]L. Emel'yanov[/i])

2005 Sharygin Geometry Olympiad, 14

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

Let $P$ be an arbitrary point inside the triangle $ABC$. Let $A_1, B_1$ and $C_1$ denote the intersection points of the straight lines $AP, BP$ and $CP$, respectively, with the sides $BC, CA$ and $AB$. We order the areas of the triangles $AB_1C_1,A_1BC_1,A_1B_1C$. Denote the smaller by $S_1$, the middle by $S_2$, and the larger by $S_3$. Prove that $\sqrt{S_1 S_2} \le S \le \sqrt{S_2 S_3}$ ,where $S$ is the area of the triangle $A_1B_1S_1$.

1989 IMO Longlists, 14

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

For a triangle $ ABC,$ let $ k$ be its circumcircle with radius $ r.$ The bisectors of the inner angles $ A, B,$ and $ C$ of the triangle intersect respectively the circle $ k$ again at points $ A', B',$ and $ C'.$ Prove the inequality \[ 16Q^3 \geq 27 r^4 P,\] where $ Q$ and $ P$ are the areas of the triangles $ A'B'C'$ and $ABC$ respectively.

2005 Sharygin Geometry Olympiad, 6

Tags: area of a triangle , area , geometry

Side $AB$ of triangle $ABC$ was divided into $n$ equal parts (dividing points $B_0 = A, B_1, B_2, ..., B_n = B$), and side $AC$ of this triangle was divided into $(n + 1)$ equal parts (dividing points $C_0 = A, C_1, C_2, ..., C_{n+1} = C$). Colored are the triangles $C_iB_iC_{i+1}$ (where $i = 1,2, ..., n$). What part of the area of the triangle is painted over?

2012 Czech-Polish-Slovak Junior Match, 3

Tags: geometry , circumcircle , projection , equilateral , fixed , area of a triangle

Different points $A, B, C, D$ lie on a circle with a center at the point $O$ at such way that $\angle AOB$ $= \angle BOC =$ $\angle COD =$ $60^o$. Point $P$ lies on the shorter arc $BC$ of this circle. Points $K, L, M$ are projections of $P$ on lines $AO, BO, CO$ respectively . Show that (a) the triangle $KLM$ is equilateral, (b) the area of triangle $KLM$ does not depend on the choice of the position of point $P$ on the shorter arc $BC$

2005 Junior Tuymaada Olympiad, 2

Tags: geometry , area , area of a triangle

Points $ X $ and $ Y $ are the midpoints of the sides $ AB $ and $ AC $ of the triangle $ ABC $, $ I $ is the center of its inscribed circle, $ K $ is the point of tangency of the inscribed circles with side $ BC $. The external angle bisector at the vertex $ B $ intersects the line $ XY $ at the point $ P $, and the external angle bisector at the vertex of $ C $ intersects $ XY $ at $ Q $. Prove that the area of the quadrilateral $ PKQI $ is equal to half the area of the triangle $ ABC $.

1977 IMO Longlists, 58

Tags: inequalities , trigonometry , area of a triangle , heron's formula , geometry

Prove that for every triangle the following inequality holds: \[\frac{ab+bc+ca}{4S} \geq \cot \frac{\pi}{6}.\] where $a, b, c$ are lengths of the sides and $S$ is the area of the triangle.

1999 Bundeswettbewerb Mathematik, 3

Tags: geometry , parallelogram , convex quadrilateral , area of a triangle , geometric inequality

Let $P$ be a point inside a convex quadrilateral $ABCD$. Points $K,L,M,N$ are given on the sides $AB,BC,CD,DA$ respectively such that $PKBL$ and $PMDN$ are parallelograms. Let $S,S_1$, and $S_2$ be the areas of $ABCD, PNAK$, and $PLCM$. Prove that $\sqrt{S}\ge \sqrt{S_1} +\sqrt{S_2}$.

2013 Purple Comet Problems, 17

Tags: rectangle , number theory , geometry , probability , inradius , relatively prime , area of a triangle

A rectangle has side lengths $6$ and $8$. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. Find $m + n$.

2020 Princeton University Math Competition, 7

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

Let $X, Y$ , and $Z$ be concentric circles with radii $1$, $13$, and $22$, respectively. Draw points $A, B$, and $C$ on $X$, $Y$ , and $Z$, respectively, such that the area of triangle $ABC$ is as large as possible. If the area of the triangle is $\Delta$, find $\Delta^2$.

2007 China Northern MO, 4

Tags: inequalities , area of a triangle , algebra , geometry , inradius , heron's formula , triangle inequality

The inradius of triangle $ ABC$ is $ 1$ and the side lengths of $ ABC$ are all integers. Prove that triangle $ ABC$ is right-angled.

2002 AMC 12/AHSME, 23

Tags: angle bisector , geometry , perpendicular bisector , trigonometry , trig identities , law of sines , area of a triangle

In triangle $ ABC$, side $ AC$ and the perpendicular bisector of $ BC$ meet in point $ D$, and $ BD$ bisects $ \angle ABC$. If $ AD \equal{} 9$ and $ DC \equal{} 7$, what is the area of triangle $ ABD$? $ \textbf{(A)}\ 14 \qquad \textbf{(B)}\ 21 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 14\sqrt5 \qquad \textbf{(E)}\ 28\sqrt5$

2012 AMC 12/AHSME, 20

Tags: parallelogram , trapezoid , geometry , area of a triangle , heron's formula

A trapezoid has side lengths $3, 5, 7,$ and $11$. The sum of all the possible areas of the trapezoid can be written in the form of $r_1 \sqrt{n_1} + r_2 \sqrt{n_2} + r_3$, where $r_1, r_2,$ and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of a prime. What is the greatest integer less than or equal to \[r_1 + r_2 + r_3 + n_1 + n_2?\] $ \textbf{(A)}\ 57\qquad\textbf{(B)}\ 59\qquad\textbf{(C)}\ 61\qquad\textbf{(D)}\ 63\qquad\textbf{(E)}\ 65 $

2008 Indonesia TST, 1

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

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 $1/10$.