Olympiad problems

1997 May Olympiad, 2

In the rectangle $ABCD, M, N, P$ and $Q$ are the midpoints of the sides. If the area of the shaded triangle is $1$, calculate the area of the rectangle $ABCD$. [img]https://2.bp.blogspot.com/-9iyKT7WP5fc/XNYuXirLXSI/AAAAAAAAKK4/10nQuSAYypoFBWGS0cZ5j4vn_hkYr8rcwCK4BGAYYCw/s400/may3.gif[/img]

2008 Indonesia TST, 1

Tags: geometry , combinatorics , combinatorial geometry , 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 $1/10$.

2011 Kosovo National Mathematical Olympiad, 4

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

Let $ a$, $ b$, $ c$ be the sides of a triangle, and $ S$ its area. Prove: \[ a^{2} \plus{} b^{2} \plus{} c^{2}\geq 4S \sqrt {3} \] In what case does equality hold?

1988 Mexico National Olympiad, 3

Tags: common tangent , tangent circle , area of a triangle , geometry

Two externally tangent circles with different radii are given. Their common tangents form a triangle. Find the area of this triangle in terms of the radii of the two circles.

1989 IMO Longlists, 87

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

Consider in a plane $ P$ the points $ O,A_1,A_2,A_3,A_4$ such that \[ \sigma(OA_iA_j) \geq 1 \quad \forall i, j \equal{} 1, 2, 3, 4, i \neq j.\] where $ \sigma(OA_iA_j)$ is the area of triangle $ OA_iA_j.$ Prove that there exists at least one pair $ i_0, j_0 \in \{1, 2, 3, 4\}$ such that \[ \sigma(OA_iA_j) \geq \sqrt{2}.\]

1999 Junior Balkan Team Selection Tests - Moldova, 4

Tags: area of a triangle , geometry , equilateral , area , ratio

Let $ABC$ be an equilateral triangle of area $1998$ cm$^2$. Points $K, L, M$ divide the segments $[AB], [BC] ,[CA]$, respectively, in the ratio $3:4$ . Line $AL$ intersects the lines $CK$ and $BM$ respectively at the points $P$ and $Q$, and the line $BM$ intersects the line $CK$ at point $R$. Find the area of the triangle $PQR$.

2016 Bundeswettbewerb Mathematik, 2

Tags: area of a triangle , game theory , area , geometry

A triangle $ABC$ with area $1$ is given. Anja and Bernd are playing the following game: Anja chooses a point $X$ on side $BC$. Then Bernd chooses a point $Y$ on side $CA$ und at last Anja chooses a point $Z$ on side $AB$. Also, $X,Y$ and $Z$ cannot be a vertex of triangle $ABC$. Anja wants to maximize the area of triangle $XYZ$ and Bernd wants to minimize that area. What is the area of triangle $XYZ$ at the end of the game, if both play optimally?

2014 Math Prize For Girls Problems, 8

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

A triangle has sides of length $\sqrt{13}$, $\sqrt{17}$, and $2 \sqrt{5}$. Compute the area of the triangle.

2010 Stanford Mathematics Tournament, 2

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

Find the radius of a circle inscribed in a triangle with side lengths $4$, $5$, and $6$

1976 IMO, 1

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

In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.

2010 Abels Math Contest (Norwegian MO) Final, 1b

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

The edges of the square in the figure have length $1$. Find the area of the marked region in terms of $a$, where $0 \le a \le 1$. [img]https://cdn.artofproblemsolving.com/attachments/2/2/f2b6ca973f66c50e39124913b3acb56feff8bb.png[/img]

1966 IMO Shortlist, 47

Tags: geometry , area of a triangle , counting , combinatorics , trigonometry

Consider all segments dividing the area of a triangle $ABC$ in two equal parts. Find the length of the shortest segment among them, if the side lengths $a,$ $b,$ $c$ of triangle $ABC$ are given. How many of these shortest segments exist ?

2013 Stanford Mathematics Tournament, 2

Tags: trig identities , area of a triangle , geometry , quadratic , law of sines , trigonometry

Points $A$, $B$, and $C$ lie on a circle of radius $5$ such that $AB=6$ and $AC=8$. Find the smaller of the two possible values of $BC$.

2019 Kosovo Team Selection Test, 4

Tags: similar triangles , perpendicular , area of a triangle , rectangle , geometry

Given a rectangle $ABCD$ such that $AB = b > 2a = BC$, let $E$ be the midpoint of $AD$. On a line parallel to $AB$ through point $E$, a point $G$ is chosen such that the area of $GCE$ is $$(GCE)= \frac12 \left(\frac{a^3}{b}+ab\right)$$ Point $H$ is the foot of the perpendicular from $E$ to $GD$ and a point $I$ is taken on the diagonal $AC$ such that the triangles $ACE$ and $AEI$ are similar. The lines $BH$ and $IE$ intersect at $K$ and the lines $CA$ and $EH$ intersect at $J$. Prove that $KJ \perp AB$.

2003 All-Russian Olympiad, 1

Tags: algebra , polynomial , vieta , area of a triangle , heron's formula , geometry

The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.

1995 AIME Problems, 14

Tags: trigonometry , asymptote , heron's formula , function , area of a triangle , geometry , vector

In a circle of radius 42, two chords of length 78 intersect at a point whose distance from the center is 18. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lenghts, and the area of either of them can be expressed uniquley in the form $m\pi-n\sqrt{d},$ where $m, n,$ and $d$ are positive integers and $d$ is not divisible by the square of any prime number. Find $m+n+d.$

1967 IMO, 4

Tags: area of a triangle , geometry , triangle , similar triangles

$A_0B_0C_0$ and $A_1B_1C_1$ are acute-angled triangles. Describe, and prove, how to construct the triangle $ABC$ with the largest possible area which is circumscribed about $A_0B_0C_0$ (so $BC$ contains $B_0, CA$ contains $B_0$, and $AB$ contains $C_0$) and similar to $A_1B_1C_1.$

1997 Brazil National Olympiad, 4

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

Let $V_n=\sqrt{F_n^2+F_{n+2}^2}$, where $F_n$ is the Fibonacci sequence ($F_1=F_2=1,F_{n+2}=F_{n+1}+F_{n}$) Show that $V_n,V_{n+1},V_{n+2}$ are the sides of a triangle with area $1/2$

2005 Harvard-MIT Mathematics Tournament, 8

Tags: area of a triangle , heron's formula , perimeter , ratio , inradius , geometry

Let $T$ be a triangle with side lengths $26$, $51$, and $73$. Let $S$ be the set of points inside $T$ which do not lie within a distance of $5$ of any side of $T$. Find the area of $S$.

2016 India Regional Mathematical Olympiad, 5

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

Given a rectangle $ABCD$, determine two points $K$ and $L$ on the sides $BC$ and $CD$ such that the triangles $ABK, AKL$ and $ADL$ have same area.

1989 Mexico National Olympiad, 1

Tags: geometry , area of a triangle , median , perpendicular

In a triangle $ABC$ the area is $18$, the length $AB$ is $5$, and the medians from $A$ and $B$ are orthogonal. Find the lengths of the sides $BC,AC$.

Brazil L2 Finals (OBM) - geometry, 2010.6

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

The three sides and the area of a triangle are integers. What is the smallest value of the area of this triangle?

2003 District Olympiad, 4

Tags: construction , geometry , symmetric , area of a triangle

Let $ABC$ be a triangle. Let $B'$ be the symmetric of $B$ with respect to $C, C'$ the symmetry of $C$ with respect to $A$ and $A'$ the symmetry of $A$ with respect to $B$. a) Prove that the area of triangle $AC'A'$ is twice the area of triangle $ABC$. b) If we delete points $A, B, C$, how can they be reconstituted? Justify your reasoning.

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 $AMN$.

2017 BMT Spring, 13

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

Two points are located $10$ units apart, and a circle is drawn with radius $ r$ centered at one of the points. A tangent line to the circle is drawn from the other point. What value of $ r$ maximizes the area of the triangle formed by the two points and the point of tangency?