Olympiad problems

2024 CIIM, 4

Given the points $O = (0, 0)$ and $A = (2024, -2024)$ in the plane. For any positive integer $n$, Damian draws all the points with integer coordinates $B_{i,j} = (i, j)$ with $0 \leq i, j \leq n$ and calculates the area of each triangle $OAB_{i,j}$. Let $S(n)$ denote the sum of the $(n+1)^2$ areas calculated above. Find the following limit: \[ \lim_{n \to \infty} \frac{S(n)}{n^3}. \]

2006 Sharygin Geometry Olympiad, 21

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

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.

1989 Nordic, 2

Tags: geometry , tetrahedron , area

Three sides of a tetrahedron are right-angled triangles having the right angle at their common vertex. The areas of these sides are $A, B$, and $C$. Find the total surface area of the tetrahedron.

2012 May Olympiad, 3

Tags: geometry , area , paper

From a paper quadrilateral like the one in the figure, you have to cut out a new quadrilateral whose area is equal to half the area of the original quadrilateral.You can only bend one or more times and cut by some of the lines of the folds. Describe the folds and cuts and justify that the area is half. [img]https://2.bp.blogspot.com/-btvafZuTvlk/XNY8nba0BmI/AAAAAAAAKLo/nm4c21A1hAIK3PKleEwt6F9cd6zv4XffwCK4BGAYYCw/s400/may%2B2012%2Bl1.png[/img]

2003 Paraguay Mathematical Olympiad, 5

Tags: geometry , square , area

In a square $ABCD$, $E$ is the midpoint of side $BC$. Line $AE$ intersects line $DC$ at $F$ and diagonal $BD$ at $G$. If the area $(EFC) = 8$, determine the area $(GBE)$.

1980 Polish MO Finals, 1

Tags: geometry , octagon , area

Compute the area of an octagon inscribed in a circle, whose four sides have length $1$ and the other four sides have length $2$.

Estonia Open Senior - geometry, 1999.2.5

Tags: square , equal area , geometry , area

Inside the square $ABCD$ there is the square $A'B' C'D'$ so that the segments $AA', BB', CC'$ and $DD'$ do not intersect each other neither the sides of the smaller square (the sides of the larger and the smaller square do not need to be parallel). Prove that the sum of areas of the quadrangles $AA'B' B$ and $CC'D'D$ is equal to the sum of areas of the quadrangles $BB'C'C$ and $DD'A'A$.

1996 Tournament Of Towns, (498) 5

Tags: square , geometry , area

The squares $ABMN$, $BCKL$ and $ACPQ$ are constructed outside triangle $ABC$. The difference between the areas of $AB MN$ and $BCKL$ is $d$. Find the difference between the areas of the squares with sides $NQ$ and $PK$ respectively, if $\angle ABC$ is (a) a right angle; (b) not necessarily a right angle. (A Gerko)

V Soros Olympiad 1998 - 99 (Russia), 9.9

Tags: geometry , geometric inequality , area

What is the largest area of a right triangle, the vertices of which are located at distances $a$, $b$ and $c$ from a certain point (where $a$ is the distance to the vertex of the right angle)?

1955 Poland - Second Round, 4

Tags: construction , area , geometry

Inside the triangle $ ABC $ a point $ P $ is given; find a point $ Q $ on the perimeter of this triangle such that the broken line $ APQ $ divides the triangle into two parts with equal areas.

2000 Czech And Slovak Olympiad IIIA, 3

Tags: combinatorial geometry , combinatorics , triangle area , area , geometry , translation

In the plane are given $2000$ congruent triangles of area $1$, which are all images of one triangle under translations. Each of these triangles contains the centroid of every other triangle. Prove that the union of these triangles has area less than $22/9$.

Kyiv City MO 1984-93 - geometry, 1985.7.3

Tags: geometry , area

$O$ is the point of intersection of the diagonals of the convex quadrilateral $ABCD$. It is known that the areas of triangles $AOB, BOC, COD$ and $DOA$ are expressed in natural numbers. Prove that the product of these areas cannot end in $1985$.

2002 Junior Balkan Team Selection Tests - Romania, 4

Tags: area , geometric inequalities , combinatorial geometry , geometry , combinatorics

Five points are given in the plane that each of $10$ triangles they define has area greater than $2$. Prove that there exists a triangle of area greater than $3$.

2024 Korea Junior Math Olympiad (First Round), 11.

Tags: geometry , area

There is a square $ ABCD. $ $ P $ is on $\bar{AB}$ , and $Q$ is on $ \bar{AD} $ They follow $ \bar{AP}=\bar{AQ}=\frac{\bar{AB}}{5} $ Let $ H $ be the foot of the perpendicular point from $ A $ to $ \bar{PD} $ If $ |\triangle APH|=20 $, Find the area of $ \triangle HCQ $.

1986 All Soviet Union Mathematical Olympiad, 438

Tags: square , geometry , geometric inequality , area , circles

A triangle and a square are circumscribed around the unit circle. Prove that the intersection area is more than $3.4$. Is it possible to assert that it is more than $3.5$?

2011 Sharygin Geometry Olympiad, 15

Tags: geometry , tangent , equal area , quadrilateral , area , circles

Given a circle with center $O$ and radius equal to $1$. $AB$ and $AC$ are the tangents to this circle from point $A$. Point $M$ on the circle is such that the areas of quadrilaterals $OBMC$ and $ABMC$ are equal. Find $MA$.

2017 India PRMO, 25

Tags: geometry , rectangle , 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 $AEF$ ?

1987 IMO Shortlist, 13

Tags: geometry , coordinate geometry , area

Is it possible to put $1987$ points in the Euclidean plane such that the distance between each pair of points is irrational and each three points determine a non-degenerate triangle with rational area? [i](IMO Problem 5)[/i] [i]Proposed by Germany, DR[/i]

2024 Singapore Junior Maths Olympiad, Q3

Tags: combinatorics , area

Seven triangles of area $7$ lie in a square of area $27$. Prove that among the $7$ triangles there are $2$ that intersect in a region of area not less than $1$.

2015 Portugal MO, 4

Tags: geometry , parallelogram , area

Let $[ABCD]$ be a parallelogram and $P$ a point between $C$ and $D$. The line parallel to $AD$ that passes through $P$ intersects the diagonal $AC$ in $Q$. Knowing that the area of $[PBQ]$ is $2$ and the area of $[ABP]$ is $6$, determine the area of $[PBC]$. [img]https://cdn.artofproblemsolving.com/attachments/0/8/664a00020065b7ad6300a062613fca4650b8d0.png[/img]

2020 Yasinsky Geometry Olympiad, 6

Tags: geometry , 3d geometry , cube , plane geometry , area

A cube whose edge is $1$ is intersected by a plane that does not pass through any of its vertices, and its edges intersect only at points that are the midpoints of these edges. Find the area of the formed section. Consider all possible cases. (Alexander Shkolny)

1997 Mexico National Olympiad, 5

Tags: ratio , geometry , area

Let $P,Q,R$ be points on the sides $BC,CA,AB$ respectively of a triangle $ABC$. Suppose that $BQ$ and $CR$ meet at $A', AP$ and $CR$ meet at $B'$, and $AP$ and $BQ$ meet at $C'$, such that $AB' = B'C', BC' =C'A'$, and $CA'= A'B'$. Compute the ratio of the area of $\triangle PQR$ to the area of $\triangle ABC$.

May Olympiad L1 - geometry, 2006.2

Tags: geometry , rectangle , area , pentagon , paper

A rectangle of paper of $3$ cm by $9$ cm is folded along a straight line, making two opposite vertices coincide. In this way a pentagon is formed. Calculate it's area.

1980 All Soviet Union Mathematical Olympiad, 287

Tags: geometry , locus , convex , midpoint , area

The points $M$ and $P$ are the midpoints of $[BC]$ and $[CD]$ sides of a convex quadrangle $ABCD$. It is known that $|AM| + |AP| = a$. Prove that $ABCD$ has area less than $\frac{a^2}{2}$.

2006 Spain Mathematical Olympiad, 3

Tags: diagonal , geometry , geometric inequality , convex quadrilateral , square root , area

The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ intersect at $E$. Denotes by $S_1,S_2$ and $S$ the areas of the triangles $ABE$, $CDE$ and the quadrilateral $ABCD$ respectively. Prove that $\sqrt{S_1}+\sqrt{S_2}\le \sqrt{S}$ . When equality is reached?