Olympiad problems

2017 OMMock - Mexico National Olympiad Mock Exam, 3

Let $x, y, z$ be positive integers such that $xy=z^2+2$. Prove that there exist integers $a, b, c, d$ such that the following equalities are satisfied: \begin{eqnarray*} x=a^2+2b^2\\ y=c^2+d^2\\ z=ac+2bd\\ \end{eqnarray*} [i]Proposed by Isaac Jiménez[/i]

Novosibirsk Oral Geo Oly VIII, 2020.1

Tags: geometry , Squares

Three squares of area $4, 9$ and $36$ are inscribed in the triangle as shown in the figure. Find the area of the big triangle [img]https://cdn.artofproblemsolving.com/attachments/9/7/3e904a9c78307e1be169ec0b95b1d3d24c1aa2.png[/img]

1983 All Soviet Union Mathematical Olympiad, 362

Tags: infinite board , Squares , Sum , combinatorial geometry , combinatorics

Can You fill the squares of the infinite cross-lined paper with integers so, that the sum of the numbers in every $4\times 6$ fields rectangle would be a) $10$? b) $1$?

2000 Tuymaada Olympiad, 6

Tags: geometry , circumcircle , right triangle , Squares

Let $O$ be the center of the circle circumscribed around the the triangle $ABC$. The centers of the circles circumscribed around the squares $OAB,OBC,OCA$ lie at the vertices of a regular triangle. Prove that the triangle $ABC$ is right.

2009 Swedish Mathematical Competition, 1

Tags: geometry , Squares , areas

Five square carpets have been bought for a square hall with a side of $6$ m , two with the side $2$ m, one with the side $2.1$ m and two with the side $2.5$ m. Is it possible to place the five carpets so that they do not overlap in any way each other? The edges of the carpets do not have to be parallel to the cradles in the hall.

1999 Tournament Of Towns, 5

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

A square is cut into $100$ rectangles by $9$ straight lines parallel to one of the sides and $9$ lines parallel to another. If exactly $9$ of the rectangles are actually squares, prove that at least two of these $9$ squares are of the same size . (V Proizvolov)

2019 Romania National Olympiad, 2

Tags: geometry , Squares , sqauare , equal segments , perpendicular

Let $ABCD$ be a square and $E$ a point on the side $(CD)$. Squares $ENMA$ and $EBQP$ are constructed outside the triangle $ABE$. Prove that: a) $ND = PC$ b) $ND\perp PC$.

2003 Junior Balkan Team Selection Tests - Romania, 4

Tags: geometry , Squares , area , min , maximum value , distance

Two unit squares with parallel sides overlap by a rectangle of area $1/8$. Find the extreme values of the distance between the centers of the squares.

2017 Oral Moscow Geometry Olympiad, 1

Tags: geometry , angle , Squares , circle

One square is inscribed in a circle, and another square is circumscribed around the same circle so that its vertices lie on the extensions of the sides of the first (see figure). Find the angle between the sides of these squares. [img]https://3.bp.blogspot.com/-8eLBgJF9CoA/XTodHmW87BI/AAAAAAAAKY0/xsHTx71XneIZ8JTn0iDMHupCanx-7u4vgCK4BGAYYCw/s400/sharygin%2Boral%2B2017%2B10-11%2Bp1.png[/img]

2023 Novosibirsk Oral Olympiad in Geometry, 7

Tags: geometry , college , Squares

Squares $ABCD$ and $BEFG$ are located as shown in the figure. It turned out that points $A, G$ and $E$ lie on the same straight line. Prove that then the points $D, F$ and $E$ also lie on the same line. [img]https://cdn.artofproblemsolving.com/attachments/4/2/9faf29a399d3a622c84f5d4a3cfcf5e99539c0.png[/img]

Novosibirsk Oral Geo Oly VII, 2023.3

Tags: geometry , rectangle , Squares

The rectangle is cut into $10$ squares as shown in the figure on the right. Find its sides if the side of the smallest square is $3$.[img]https://cdn.artofproblemsolving.com/attachments/e/5/1fe3a0e41b2d3182338a557d3d44ff5ef9385d.png[/img]

1975 Chisinau City MO, 111

Tags: Squares , geometry , Concyclic

Three squares are constructed on the sides of the triangle to the outside. What should be the angles of the triangle so that the six vertices of these squares, other than the vertices of the triangle, lie on the same circle?

Estonia Open Junior - geometry, 2000.1.3

Tags: ratio , areas , Squares

Consider a shape obtained from two equal squares with the same center. Prove that the ratio of the area of this shape to the perimeter does not change when the squares are rotated around their center. [img]http://4.bp.blogspot.com/-1AI4FxsNSr4/XovZWkvAwiI/AAAAAAAALvY/-kIzOgXB5rk3iIqGbpoKRCW9rwJPcZ3uQCK4BGAYYCw/s400/estonia%2B2000%2Bo.j.1.3.png[/img]

1977 All Soviet Union Mathematical Olympiad, 249

Tags: combinatorial geometry , Squares

Given $1000$ squares on the plane with their sides parallel to the coordinate axes. Let $M$ be the set of those squares centres. Prove that you can mark some squares in such a way, that every point of $M$ will be contained not less than in one and not more than in four marked squares

Novosibirsk Oral Geo Oly VII, 2019.4

Tags: geometry , Squares , isosceles , collinear

Two squares and an isosceles triangle are positioned as shown in the figure (the up left vertex of the large square lies on the side of the triangle). Prove that points $A, B$ and $C$ are collinear. [img]https://cdn.artofproblemsolving.com/attachments/d/c/03515e40f74ced1f8243c11b3e610ef92137ac.png[/img]

1990 Tournament Of Towns, (248) 2

Tags: ratio , Squares , equal areas , geometry

If a square is intersected by another square equal to it but rotated by $45^o$ around its centre, each side is divided into three parts in a certain ratio $a : b : a$ (which one can compute). Make the following construction for an arbitrary convex quadrilateral: divide each of its sides into three parts in this same ratio $a : b : a$, and draw a line through the two division points neighbouring each vertex. Prove that the new quadrilateral bounded by the four drawn lines has the same area as the original one. (A. Savin, Moscow)

Revenge EL(S)MO 2024, 3

Tags: Diophantine equation , Squares , algebra

Find all solutions to \[ (abcde)^2 = a^2+b^2+c^2+d^2+e^2+f^2. \] in integers. Proposed by [i]Seongjin Shim[/i]

1978 All Soviet Union Mathematical Olympiad, 259

Tags: Squares , combinatorial geometry , combinatorics

Prove that there exists such a number $A$ that you can inscribe $1978$ different size squares in the plot of the function $y = A sin(x)$. (The square is inscribed if all its vertices belong to the plot.)

2013 International Zhautykov Olympiad, 3

Tags: rectangle , combinatorics , Extremal combinatorics , covering , Squares , square grid

A $10 \times 10$ table consists of $100$ unit cells. A [i]block[/i] is a $2 \times 2$ square consisting of $4$ unit cells of the table. A set $C$ of $n$ blocks covers the table (i.e. each cell of the table is covered by some block of $C$ ) but no $n -1$ blocks of $C$ cover the table. Find the largest possible value of $n$.

Oliforum Contest V 2017, 2

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

Find all quadrilaterals which can be covered (without overlappings) with squares with side $ 1$ and equilateral triangles with side $ 1$. (Emanuele Tron)

1984 Brazil National Olympiad, 5

Tags: geometry , Squares , convex quadrilateral

$ABCD$ is any convex quadrilateral. Squares center $E, F, G, H$ are constructed on the outside of the edges $AB, BC, CD$ and $DA$ respectively. Show that $EG$ and $FH$ are equal and perpendicular.

2017 Oral Moscow Geometry Olympiad, 5

Tags: geometry , Squares , areas

Two squares are arranged as shown. Prove that the area of the black triangle equal to the sum of the gray areas. [img]https://2.bp.blogspot.com/-byhWqNr1ras/XTq-NWusg2I/AAAAAAAAKZA/1sxEZ751v_Evx1ij7K_CGiuZYqCjhm-mQCK4BGAYYCw/s400/Oral%2BSharygin%2B2017%2B8.9%2Bp5.png[/img]

1963 Swedish Mathematical Competition., 2

Tags: geometry , Chessboard , Squares , max

The squares of a chessboard have side $4$. What is the circumference of the largest circle that can be drawn entirely on the black squares of the board?

2017 Hanoi Open Mathematics Competitions, 15

Tags: Squares , combinatorial geometry , geometry

Let $S$ denote a square of side-length $7$, and let eight squares with side-length $3$ be given. Show that it is impossible to cover $S$ by those eight small squares with the condition: an arbitrary side of those (eight) squares is either coincided, parallel, or perpendicular to others of $S$ .

1941 Moscow Mathematical Olympiad, 081

Tags: geometry , rectangle , Squares , combinatorial geometry , Impossible

a) Prove that it is impossible to divide a rectangle into five squares of distinct sizes. b) Prove that it is impossible to divide a rectangle into six squares of distinct sizes.