Olympiad problems

Novosibirsk Oral Geo Oly IX, 2016.4

The two angles of the squares are adjacent, and the extension of the diagonals of one square intersect the diagonal of another square at point $O$ (see figure). Prove that $O$ is the midpoint of $AB$. [img]https://cdn.artofproblemsolving.com/attachments/7/8/8daaaa55c38e15c4a8ac7492c38707f05475cc.png[/img]

2015 Hanoi Open Mathematics Competitions, 10

Tags: geometry , Squares , Concyclic , right triangle

A right-angled triangle has property that, when a square is drawn externally on each side of the triangle, the six vertices of the squares that are not vertices of the triangle are concyclic. Assume that the area of the triangle is $9$ cm$^2$. Determine the length of sides of the triangle.

1974 IMO Longlists, 23

Tags: combinatorics , Squares , packing , combinatorial geometry , IMO Shortlist

Prove that the squares with sides $\frac{1}{1}, \frac{1}{2}, \frac{1}{3},\ldots$ may be put into the square with side $\frac{3}{2} $ in such a way that no two of them have any interior point in common.

Indonesia Regional MO OSP SMA - geometry, 2020.1

Tags: geometry , rectangle , Squares

In the figure, point $P, Q,R,S$ lies on the side of the rectangle $ABCD$. [img]https://1.bp.blogspot.com/-Ff9rMibTuHA/X9PRPbGVy-I/AAAAAAAAMzA/2ytG0aqe-k0fPL3hbSp_zHrMYAfU-1Y_ACLcBGAsYHQ/s426/2020%2BIndonedia%2BMO%2BProvince%2BP2%2Bq1.png[/img] If it is known that the area of the small square is $1$ unit, determine the area of the rectangle $ABCD$.

2017 Hanoi Open Mathematics Competitions, 11

Tags: geometry , Squares

Let $S$ denote a square of the side-length $7$, and let eight squares of the side-length $3$ be given. Show that $S$ can be covered by those eight small squares.

2000 Poland - Second Round, 6

Tags: algebra , polynomial , Squares , Poland

Polynomial $w(x)$ of second degree with integer coefficients takes for integer arguments values, which are squares of integers. Prove that polynomial $w(x)$ is a square of a polynomial.

2010 Saudi Arabia IMO TST, 2

Tags: geometry , concurrency , concurrent , Squares , square

The squares $OABC$ and $OA_1B_1C_1$ are situated in the same plane and are directly oriented. Prove that the lines $AA_1$ , $BB_1$, and $CC_1$ are concurrent.

2002 Portugal MO, 5

Tags: geometry , Squares , Perfect Square

Consider the three squares indicated in the figure. Show that if the lengths of the sides of the smaller square and the square greater are integers, then adding to the area of the smallest square the area of the inclined square, a perfect square is obtained. [img]https://1.bp.blogspot.com/-B0QdvZIjOLw/X4URvs3C0ZI/AAAAAAAAMmw/S5zMpPBXBn8Jj39d-OZVtMRUDn3tXbyWgCLcBGAsYHQ/s0/2002%2Bportugal%2Bp5.png[/img]

Estonia Open Junior - geometry, 2002.1.1.

Tags: geometry , rectangle , Squares

A figure consisting of five equal-sized squares is placed as shown in a rectangle of size $7\times 8$ units. Find the side length of the squares. [img]https://cdn.artofproblemsolving.com/attachments/e/e/cbc2b7b0693949790c1958fb1449bdd15393d8.png[/img]

Novosibirsk Oral Geo Oly VIII, 2016.4

Tags: geometry , Squares , midpoint

The two angles of the squares are adjacent, and the extension of the diagonals of one square intersect the diagonal of another square at point $O$ (see figure). Prove that $O$ is the midpoint of $AB$. [img]https://cdn.artofproblemsolving.com/attachments/7/8/8daaaa55c38e15c4a8ac7492c38707f05475cc.png[/img]

1994 Austrian-Polish Competition, 3

Tags: rectangle , Squares , Tiling , combinatorial geometry , combinatorics

A rectangular building consists of $30$ square rooms situated like the cells of a $2 \times 15$ board. In each room there are three doors, each of which leads to another room (not necessarily different). How many ways are there to distribute the doors between the rooms so that it is possible to get from any room to any other one without leaving the building?

1982 Tournament Of Towns, (021) 2

Tags: minimum , broken line , Squares , combinatorial geometry , geometry

A square is subdivided into $K^2$ equal smaller squares. We are given a broken line which passes through the centres of all the smaller squares (such a broken line may intersect itself). Find the minimum number of links in this broken line. (A Andjans, Riga)

1995 Tournament Of Towns, (443) 3

Tags: geometry , Squares , angles

Suppose $L$ is the circle inscribed in the square $T_1$, and $T_2$ is the square inscribed in $L$, so that vertices of $T_1$ lie on the straight lines containing the sides of $T_2$. Find the angles of the convex octagon whose vertices are at the tangency points of $L$ with the sides of $T_1$ and at the vertices of $T_2$. (S Markelov)

2012 Junior Balkan Team Selection Tests - Romania, 2

Tags: combinatorics , Squares

From an $n \times n $ square, $n \ge 2,$ the unit squares situated on both odd numbered rows and odd numbers columns are removed. Determine the minimum number of rectangular tiles needed to cover the remaining surface.

1997 Spain Mathematical Olympiad, 2

Tags: combinatorics , grid , Squares , square grid , maximum

A square of side $5$ is divided into $25$ unit squares. Let $A$ be the set of the $16$ interior points of the initial square which are vertices of the unit squares. What is the largest number of points of $A$ no three of which form an isosceles right triangle?

2012 Tournament of Towns, 6

Tags: combinatorics , combinatorial geometry , Squares , Rectangles , geometry , rectangle

We attempt to cover the plane with an infinite sequence of rectangles, overlapping allowed. (a) Is the task always possible if the area of the $n$th rectangle is $n^2$ for each $n$? (b) Is the task always possible if each rectangle is a square, and for any number $N$, there exist squares with total area greater than $N$?

1979 Chisinau City MO, 177

Tags: combinatorial geometry , geometry , Squares , Tiling

Is it possible to cut a square into five squares?

2019 Puerto Rico Team Selection Test, 1

Tags: combinatorics , combinatorial geometry , Squares

A square is divided into $25$ unit squares by drawing lines parallel to the sides of the square. Some diagonals of unit squares are drawn from such that two diagonals do not share points. What is the maximum number diagonals that can be drawn with this property?

2006 Sharygin Geometry Olympiad, 4

Tags: geometry , Regular , pentagon , Squares

a) Given two squares $ABCD$ and $DEFG$, with point $E$ lying on the segment $CD$, and points$ F,G$ outside the square $ABCD$. Find the angle between lines $AE$ and $BF$. b) Two regular pentagons $OKLMN$ and $OPRST$ are given, and the point $P$ lies on the segment $ON$, and the points $R, S, T$ are outside the pentagon $OKLMN$. Find the angle between straight lines $KP$ and $MS$.

2003 Korea Junior Math Olympiad, 1

Tags: number theory , Squares , Integer

Show that for any non-negative integer $n$, the number $2^{2n+1}$ cannot be expressed as a sum of four non-zero square numbers.

2024 Mozambican National MO Selection Test, P2

Tags: power of a point , circles , Pythagorean Theorem , geometry , tangent , Squares

On a sheet divided into squares, each square measuring $2cm$, two circles are drawn such that both circles are inscribed in a square as in the figure below. Determine the minimum distance between the two circles.

Estonia Open Senior - geometry, 1999.2.5

Tags: Squares , areas , equal areas , geometry

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 $DD'A'A$.

1985 IMO Shortlist, 15

Tags: geometry , dissection , Squares , IMO Shortlist

Let $K$ and $K'$ be two squares in the same plane, their sides of equal length. Is it possible to decompose $K$ into a finite number of triangles $T_1, T_2, \ldots, T_p$ with mutually disjoint interiors and find translations $t_1, t_2, \ldots, t_p$ such that \[K'=\bigcup_{i=1}^{p} t_i(T_i) \ ? \]

2023 Yasinsky Geometry Olympiad, 3

Tags: geometry , Squares , concurrency , concurrent

Let $ABC$ be an acute triangle. Squares $AA_1A_2A_3$, $BB_1B_2B_3$ and $CC_1C_2C_3$ are located such that the lines $A_1A_2$, $B_1B_2$, $C_1C_2$ pass through the points $B$, $C$ and $A$ respectively and the lines $A_2A_3$, $B_2B_3$, $C_2C_3$ pass through the points $C$, $A$ and $B$ respectively. Prove that (a) the lines $AA_2$, $B_1B_2$ and $C_1C_3$ intersect at one point. (b) the lines $AA_2$, $BB_2$ and $CC_2$ intersect at one point. (Mykhailo Plotnikov) [img]https://cdn.artofproblemsolving.com/attachments/3/d/ad2fe12ae2c82d04b48f5e683b7d54e0764baf.png[/img]

2013 Tournament of Towns, 2

Tags: geometry , Squares , right angle , right triangle

Let $C$ be a right angle in triangle $ABC$. On legs $AC$ and$BC$ the squares $ACKL, BCMN$ are constructed outside of triangle. If $CE$ is an altitude of the triangle prove that $LEM$ is a right angle.