Olympiad problems

Estonia Open Senior - geometry, 1996.2.4

Tags: geometry , area , square , trigonometry , circles

The figure shows a square and a circle with a common center $O$, with equal areas of striped shapes. Find the value of $\cos a$. [img]https://2.bp.blogspot.com/-7uwa0H42ELg/XnmsSoPMgcI/AAAAAAAALgk/pHNBqtbsdKgMhcvIRYLm_8JRpOeIYcUeACK4BGAYYCw/s400/96%2Bestonia%2Bopen%2Bs2.4.png[/img]

1981 Swedish Mathematical Competition, 5

Tags: area , geometry , geometric inequalities

$ABC$ is a triangle. $X$, $Y$, $Z$ lie on $BC$, $CA$, $AB$ respectively. Show that area $XYZ$ cannot be smaller than each of area $AYZ$, area $BZX$, area $CXY$.

1986 Tournament Of Towns, (107) 1

Tags: equal area , geometry , area

Through vertices $A$ and $B$ of triangle $ABC$ are constructed two lines which divide the triangle into four regions (three triangles and one quadrilateral). It is known that three of them have equal area. Prove that one of these three regions is the quadrilateral . (G . Galperin , A . Savin, Moscow)

2018 Auckland Mathematical Olympiad, 3

Tags: area , paper folding , folding , rectangle , geometry

A rectangular sheet of paper whose dimensions are $12 \times 18$ is folded along a diagonal, creating the $M$-shaped region drawn in the picture (see below). Find the area of the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/4/7/d82cde3e91ab83fa14cd6cefe9bba28462dde1.png[/img]

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?

1984 All Soviet Union Mathematical Olympiad, 384

Tags: geometry , perimeter , polygon , disc , circles , area

The centre of the coin with radius $r$ is moved along some polygon with the perimeter $P$, that is circumscribed around the circle with radius $R$ ($R>r$). Find the coin trace area (a sort of polygon ring).

1996 Estonia National Olympiad, 4

Tags: geometry , triangle area , square , area

Let $K, L, M$, and $N$ be the midpoints of $CD,DA,AB$ and $BC$ of a square $ABCD$ respectively. Find the are of the triangles $AKB, BLC, CMD$ and $DNA$ if the square $ABCD$ has area $1$.

2017 India PRMO, 25

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

Let $ABCD$ be a rectangle and let $E$ and $F$ be points on $CD$ and $BC$ respectively such that area $(ADE) = 16$, area $(CEF) = 9$ and area $(ABF) = 25$. What is the area of triangle $AEF$ ?

2016 Sharygin Geometry Olympiad, 4

Tags: area , game , combinatorics , game strategy , geometry

The Devil and the Man play a game. Initially, the Man pays some cash $s$ to the Devil. Then he lists some $97$ triples $\{i,j,k\}$ consisting of positive integers not exceeding $100$. After that, the Devil draws some convex polygon $A_1A_2...A_{100}$ with area $100$ and pays to the Man, the sum of areas of all triangles $A_iA_jA_k$. Determine the maximal value of $s$ which guarantees that the Man receives at least as much cash as he paid. [i]Proposed by Nikolai Beluhov, Bulgaria[/i]

1978 Chisinau City MO, 167

Tags: area , geometric inequality , geometry

Prove that the largest area of a triangle with sides $a, b, c$ satisfying the relation $a^2 +b^2 c^2 = 3m^2$, equals to $\frac{\sqrt3}{4}m^2$.

2020 AMC 12/AHSME, 18

Tags: area , geometry , quadrilateral

Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC = 20$, and $CD = 30$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E$, and $AE = 5$. What is the area of quadrilateral $ABCD$? $\textbf{(A) } 330 \qquad\textbf{(B) } 340 \qquad\textbf{(C) } 350 \qquad\textbf{(D) } 360 \qquad\textbf{(E) } 370$

2006 Sharygin Geometry Olympiad, 21

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

On the sides $AB, BC, CA$ of triangle $ABC$, points $C', A', B'$ are taken. Prove that for the areas of the corresponding triangles, the inequality holds: $$S_{ABC}S^2_{A'B'C'}\ge 4S_{AB'C'}S_{BC'A'}S_{CA'B'}$$ and equality is achieved if and only if the lines $AA', BB', CC'$ intersect at one point.

2016 Chile National Olympiad, 2

Tags: area , geometry

For a triangle $\vartriangle ABC$, determine whether or not there exists a point $P$ on the interior of $\vartriangle ABC$ in such a way that every straight line through $P$ divides the triangle $\vartriangle ABC$ in two polygons of equal area.

2016 ASMT, 4

Tags: geometry , area

Let $ABCD$ be a convex quadrilateral with $BA = BC$ and $DA = DC$. Let $E$ and $F$ be the midpoints of $BC$ and $CD$ respectively, and let$ BF$ and $DE$ intersect at $G$. If the area of $CEGF$ is $50$, what is the area of $ABGD$?

May Olympiad L2 - geometry, 2016.5

Tags: combinatorics , geometry , game , winning strategy , area

Rosa and Sara play with a triangle $ABC$, right at $B$. Rosa begins by marking two interior points of the hypotenuse $AC$, then Sara marks an interior point of the hypotenuse $AC$ different from those of Rosa. Then, from these three points the perpendiculars to the sides $AB$ and $BC$ are drawn, forming the following figure. [img]https://cdn.artofproblemsolving.com/attachments/9/9/c964bbacc4a5960bee170865cc43902410e504.png[/img] Sara wins if the area of the shaded surface is equal to the area of the unshaded surface, in other case wins Rosa. Determine who of the two has a winning strategy.

2016 Canadian Mathematical Olympiad Qualification, 2

Tags: area , geometry , algebra , perimeter

Let $P = (7, 1)$ and let $O = (0, 0)$. (a) If $S$ is a point on the line $y = x$ and $T$ is a point on the horizontal $x$-axis so that $P$ is on the line segment $ST$, determine the minimum possible area of triangle $OST$. (b) If $U$ is a point on the line $y = x$ and $V$ is a point on the horizontal $x$-axis so that $P$ is on the line segment $UV$, determine the minimum possible perimeter of triangle $OUV$.

2014 Tournament of Towns., 3

Tags: area , geometry , square

A square table is covered with a square cloth (may be of a different size) without folds and wrinkles. All corners of the table are left uncovered and all four hanging parts are triangular. Given that two adjacent hanging parts are equal prove that two other parts are also equal.

1980 Spain Mathematical Olympiad, 7

Tags: analytic geometry , locus , geometry , area

The point $M$ varies on the segment $AB$ that measures $2$ m. a) Find the equation and the graphical representation of the locus of the points of the plane whose coordinates, $x$, and $y$, are, respectively, the areas of the squares of sides $AM$ and $MB$ . b) Find out what kind of curve it is. (Suggestion: make a $45^o$ axis rotation). c) Find the area of the enclosure between the curve obtained and the coordinate axes.

2020 BMT Fall, 14

Tags: geometry , area

In the star shaped figure below, if all side lengths are equal to $3$ and the three largest angles of the figure are $210$ degrees, its area can be expressed as $\frac{a \sqrt{b}}{c}$ , where $a, b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime and that $b$ is square-free. Compute $a + b + c$. [img]https://cdn.artofproblemsolving.com/attachments/a/f/d16a78317b0298d6894c6bd62fbcd1a5894306.png[/img]

1996 Denmark MO - Mohr Contest, 1

Tags: right triangle , area , geometry

In triangle $ABC$, angle $C$ is right and the two catheti are both length $1$. For one given the choice of the point $P$ on the cathetus $BC$, the point $Q$ on the hypotenuse and the point $R$ are plotted on the second cathetus so that $PQ$ is parallel to $AC$ and $QR$ is parallel to $BC$. Thereby the triangle is divided into three parts. Determine the locations of point $P$ for which the rectangular part has a larger area than each of the other two parts.

2015 Romania National Olympiad, 3

Tags: geometry , median , area

Let be a point $ P $ in the interior of a triangle $ ABC. $ The lines $ AP,BP,CP $ meet $ BC,AC, $ respectively, $ AB $ at $ A_1,B_1, $ respectively, $ C_1. $ If $$ \mathcal{A}_{PBA_1} +\mathcal{A}_{PCB_1} +\mathcal{A}_{PAC_1} =\frac{1}{2}\mathcal{A}_{ABC} , $$ show that $ P $ lies on a median of $ ABC. $ $ \mathcal{A} $ [i]denotes area.[/i]

2008 Switzerland - Final Round, 1

Tags: geometry , area , equal area , circumcircle

Let $ABC$ be a triangle with $\angle BAC \ne 45^o$ and $\angle ABC \ne 135^o$. Let $P$ be the point on the line $AB$ with $\angle CPB = 45^o$. Let $O_1$ and $O_2$ be the centers of the circumcircles of the triangles $ACP$ and $BCP$ respectively. Show that the area of the square $CO_1P O_2$ is equal to the area of the triangle $ABC$.

1966 IMO Shortlist, 53

Tags: hexagon , diagonal , combinatorics , area , geometry

Prove that in every convex hexagon of area $S$ one can draw a diagonal that cuts off a triangle of area not exceeding $\frac{1}{6}S.$

2014 Czech-Polish-Slovak Junior Match, 4

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

Point $M$ is the midpoint of the side $AB$ of an acute triangle $ABC$. Circle with center $M$ passing through point $ C$, intersects lines $AC ,BC$ for the second time at points $P,Q$ respectively. Point $R$ lies on segment $AB$ such that the triangles $APR$ and $BQR$ have equal areas. Prove that lines $PQ$ and $CR$ are perpendicular.

1970 Spain Mathematical Olympiad, 3

Tags: geometry , equal area , area

An arbitrary triangle $ABC$ is given and a point $P$ lies on the side $AB$. It is requested to draw through $P$ a line that divides the triangle into two figures of the same area.