Olympiad problems

1999 Austrian-Polish Competition, 4

Three lines $k, l, m$ are drawn through a point $P$ inside a triangle $ABC$ such that $k$ meets $AB$ at $A_1$ and $AC$ at $A_2 \ne A_1$ and $PA_1 = PA_2$, $l $ meets $BC$ at $B_1$ and $BA$ at $B_2 \ne B_1$ and $PB_1 = PB_2$, $m$ meets $CA$ at $C_1$ and $CB$ at $C_2\ne C_1$ and $PC_1=PC_2$. Prove that the lines $k,l,m$ are uniquely determined by these conditions. Find point $P$ for which the triangles $AA_1A_2, BB_1B_2, CC_1C_2$ have the same area and show that this point is unique.

1998 India National Olympiad, 5

Tags: quadratic , algebra , polynomial , inequalities , sum of roots , area of a triangle

Suppose $a,b,c$ are three rela numbers such that the quadratic equation \[ x^2 - (a +b +c )x + (ab +bc +ca) = 0 \] has roots of the form $\alpha + i \beta$ where $\alpha > 0$ and $\beta \not= 0$ are real numbers. Show that (i) The numbers $a,b,c$ are all positive. (ii) The numbers $\sqrt{a}, \sqrt{b} , \sqrt{c}$ form the sides of a triangle.

1967 Dutch Mathematical Olympiad, 1

Tags: equal area , geometry , area of a triangle

In this exercise we only consider convex quadrilaterals. (a) For such a quadrilateral $ABCD$, determine the set of points $P$ contained within that quadrilateral for which $PA$ and $PC$ divide the quadrilateral into two pieces of equal areas. (b) Prove that there is a point $P$ inside such a quadrilateral, such that the triangles $PAB$ and $PCD$ have equal areas, as well as the triangles $PBC$ and $PAD$. (c) Find out which quadrilaterals $ABCD$ contains a point $P$, so that the triangles $PAB$, $PBC$, $PCD$ and $PDA$ have equal areas.

2017 Ecuador NMO (OMEC), 2

Tags: geometry , area of a triangle

Let $ABC$ be a triangle with $AC = 18$ and $D$ is the point on the segment $AC$ such that $AD = 5$. Draw perpendiculars from $D$ on $AB$ and $BC$ which have lengths $4$ and $5$ respectively. Find the area of the triangle $ABC$.

2016 IFYM, Sozopol, 4

Tags: geometry , area of a triangle

Circle $k$ passes through $A$ and intersects the sides of $\Delta ABC$ in $P,Q$, and $L$. Prove that: $\frac{S_{PQL}}{S_{ABC}}\leq \frac{1}{4} (\frac{PL}{AQ})^2$.

1988 China National Olympiad, 4

Tags: area of a triangle , heron's formula , n-variable inequality , inequalities , geometry

(1) Let $a,b,c$ be positive real numbers satisfying $(a^2+b^2+c^2)^2>2(a^4+b^4+c^4)$. Prove that $a,b,c$ can be the lengths of three sides of a triangle respectively. (2) Let $a_1,a_2,\dots ,a_n$ be $n$ ($n>3$) positive real numbers satisfying $(a_1^2+a_2^2+\dots +a_n^2)^2>(n-1)(a_1^4+ a_2^4+\dots +a_n^4)$. Prove that any three of $a_1,a_2,\dots ,a_n$ can be the lengths of three sides of a triangle respectively.

2010 Oral Moscow Geometry Olympiad, 3

Tags: geometry , fixed , fixed point , area of a triangle , area

On the sides $AB$ and $BC$ of triangle $ABC$, points $M$ and $K$ are taken, respectively, so that $S_{KMC} + S_{KAC}=S_{ABC}$. Prove that all such lines $MK$ pass through one point.

1994 India Regional Mathematical Olympiad, 2

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

In a triangle $ABC$, the incircle touches the sides $BC, CA, AB$ at $D, E, F$ respectively. If the radius if the incircle is $4$ units and if $BD, CE , AF$ are consecutive integers, find the sides of the triangle $ABC$.

1996 AMC 12/AHSME, 28

Tags: geometry , 3d geometry , pyramid , vector , pythagorean theorem , area of a triangle , heron's formula

On a $4 \times 4 \times 3$ rectangular parallelepiped, vertices $A$, $B$, and $C$ are adjacent to vertex $D$. The perpendicular distance from $D$ to the plane containing $A$, $B$, and $C$ is closest to $\text{(A)}\ 1.6 \qquad \text{(B)}\ 1.9 \qquad \text{(C)}\ 2.1 \qquad \text{(D)}\ 2.7 \qquad \text{(E)}\ 2.9$

2008 Bulgarian Autumn Math Competition, Problem 10.2

Tags: geometry , midpoint , area of a triangle

Let $\triangle ABC$ have $M$ as the midpoint of $BC$ and let $P$ and $Q$ be the feet of the altitudes from $M$ to $AB$ and $AC$ respectively. Find $\angle BAC$ if $[MPQ]=\frac{1}{4}[ABC]$ and $P$ and $Q$ lie on the segments $AB$ and $AC$.

2008 Argentina National Olympiad, 3

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

On a circle of center $O$, let $A$ and $B$ be points on the circle such that $\angle AOB = 120^o$. Point $C$ lies on the small arc $AB$ and point $D$ lies on the segment $AB$. Let also $AD = 2, BD = 1$ and $CD = \sqrt2$. Calculate the area of triangle $ABC$.

2015 Israel National Olympiad, 2

Tags: altitude , geometry , area of a triangle

A triangle is given whose altitudes' lengths are $\frac{1}{5},\frac{1}{5},\frac{1}{8}$. Evaluate the triangle's area.

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]

2014 Harvard-MIT Mathematics Tournament, 3

Tags: geometry , angle bisector , 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.

May Olympiad L1 - geometry, 2011.3

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

In the rectangle $ABCD, BC = 5, EC = 1/3 CD$ and $F$ is the point where $AE$ and $BD$ are cut. The triangle $DFE$ has area $12$ and the triangle $ABF$ has area $27$. Find the area of the quadrilateral $BCEF$ . [img]https://1.bp.blogspot.com/-4w6e729AF9o/XNY9hqHaBaI/AAAAAAAAKL0/eCaNnWmgc7Yj9uV4z29JAvTcWCe21NIMgCK4BGAYYCw/s400/may%2B2011%2Bl1.png[/img]

1986 AIME Problems, 15

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

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 $ABC$.

Estonia Open Senior - geometry, 1995.1.3

Tags: geometry , 3d geometry , tetrahedron , area of a triangle , area

We call a tetrahedron a "trirectangular " if it has a vertex (we call this is called a "right-angled" vertex) in which the planes of the three sides of the tetrahedron intersect at right angles. Prove the "three-dimensional Pythagorean theorem": The square of the area of the opposite face of the "right-angled" vertex of the ""trirectangular " tetrahedron is equal to the sum of the squares of the areas of three other sides of the tetrahedron .

1995 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: geometry , area of a triangle , orthocenter , inradius

In a circle of center $O$ and radius $r$, a triangle $ABC$ of orthocenter $H$ is inscribed. It is considered a triangle $A'B'C'$ whose sides have by length the measurements of the segments $AB, CH$ and $2r$. Determine the triangle $ABC$ so that the area of the triangle $A'B'C'$ is maximum.

2012 Czech And Slovak Olympiad IIIA, 2

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

Find out the maximum possible area of the triangle $ABC$ whose medians have lengths satisfying inequalities $m_a \le 2, m_b \le 3, m_c \le 4$.

1991 Nordic, 2

Tags: trapezoid , area of a triangle , geometry

In the trapezium $ABCD$ the sides $AB$ and $CD$ are parallel, and $E$ is a fixed point on the side $AB$. Determine the point $F$ on the side $CD$ so that the area of the intersection of the triangles $ABF$ and $CDE$ is as large as possible.

2019 BMT Spring, 16

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

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 $A$.

2003 India National Olympiad, 5

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

Let a, b, c be the sidelengths and S the area of a triangle ABC. Denote $x=a+\frac{b}{2}$, $y=b+\frac{c}{2}$ and $z=c+\frac{a}{2}$. Prove that there exists a triangle with sidelengths x, y, z, and the area of this triangle is $\geq\frac94 S$.

1962 Putnam, A3

Tags: ratio , triangle , area of a triangle

In a triangle $ABC$, let $A'$ be a point on the segment $BC$, $B'$ be a point on the segment $CA$ and $C'$ a point on the segment $AB$ such that $$ \frac{AB'}{B'C}= \frac{BC'}{C'A} =\frac{CA'}{A'B}=k,$$ where $k$ is a positive constant. Let $\triangle$ be the triangle formed by the interesctions of $AA'$, $BB'$ and $CC'$. Prove that the areas of $\triangle $ and $ABC$ are in the ratio $$\frac{(k-1)^{2}}{k^2 +k+1}.$$

1978 IMO Shortlist, 4

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

Let $T_1$ be a triangle having $a, b, c$ as lengths of its sides and let $T_2$ be another triangle having $u, v,w$ as lengths of its sides. If $P,Q$ are the areas of the two triangles, prove that \[16PQ \leq a^2(-u^2 + v^2 + w^2) + b^2(u^2 - v^2 + w^2) + c^2(u^2 + v^2 - w^2).\] When does equality hold?

1998 Argentina National Olympiad, 5

Tags: geometry , right triangle , isosceles , construction , inscribed , minimum , area of a triangle

Let $ABC$ a right isosceles triangle with hypotenuse $AB=\sqrt2$ . Determine the positions of the points $X,Y,Z$ on the sides $BC,CA,AB$ respectively so that the triangle $XYZ$ is isosceles, right, and with minimum area.