This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 104

2016 Oral Moscow Geometry Olympiad, 3
Tags: geometry , Squares , square , areas , equal area
Two squares are arranged as shown in the picture. Prove that the areas of shaded quadrilaterals are equal. [img]https://3.bp.blogspot.com/-W50DOuizFvY/XT6wh3-L6sI/AAAAAAAAKaw/pIW2RKmttrwPAbrKK3bpahJz7hfIZwM8QCK4BGAYYCw/s400/Oral%2BSharygin%2B2016%2B10.11%2Bp3.png[/img]

2018 Hanoi Open Mathematics Competitions, 6
Tags: geometry , perimeter , Squares , hexagon
In the below figure, there is a regular hexagon and three squares whose sides are equal to $4$ cm. Let $M,N$, and $P$ be the centers of the squares. The perimeter of the triangle $MNP$ can be written in the form $a + b\sqrt3$ (cm), where $a, b$ are integers. Compute the value of $a + b$. [img]https://cdn.artofproblemsolving.com/attachments/e/8/5996e994d4bbed8d3b3269d3e38fc2ec5d2f0b.png[/img]

1998 Switzerland Team Selection Test, 4
Tags: combinatorial geometry , Squares , combinatorics
Find all numbers $n$ for which it is possible to cut a square into $n$ smaller squares.

Denmark (Mohr) - geometry, 1997.2
Tags: geometry , Squares , square , area
Two squares, both with side length $1$, are arranged so that one has one vertex in the center of the other. Determine the area of the gray area. [img]https://1.bp.blogspot.com/-xt3pe0rp1SI/XzcGLgEw1EI/AAAAAAAAMYM/vFKxvvVuLvAJ5FO_yX315X3Fg_iFaK2fACLcBGAsYHQ/s0/1997%2BMohr%2Bp2.png[/img]

2019 Novosibirsk Oral Olympiad in Geometry, 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]

2017 Czech-Polish-Slovak Match, 2
Tags: Squares , board , combinatorics , maximum
Each of the ${4n^2}$ unit squares of a ${2n \times 2n}$ board ${(n \ge 1) }$ has been colored blue or red. A set of four different unit squares of the board is called [i]pretty [/i]if these squares can be labeled ${A,B,C,D}$ in such a way that ${A}$ and ${B}$ lie in the same row, ${C}$ and ${D}$ lie in the same row, ${A}$ and ${C}$ lie in the same column, ${B}$ and ${D}$ lie in the same column, ${A}$ and ${D}$ are blue, and ${B}$ and ${C}$ are red. Determine the largest possible number of different [i]pretty [/i]sets on such a board. (Poland)

1996 Bundeswettbewerb Mathematik, 1
Tags: combinatorial geometry , combinatorics , geometry , Squares
Can a square of side length $5$ be covered by three squares of side length $4$?

Novosibirsk Oral Geo Oly VIII, 2023.2
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]

2020 Novosibirsk Oral Olympiad in Geometry, 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]

1997 All-Russian Olympiad Regional Round, 8.5
Tags: geometry , Squares , geometric inequality
Segments $AB$, $BC$ and $CA$ are, respectively, diagonals of squares $K_1$, $K_2$, $K3$. Prove that if triangle $ABC$ is acute, then it completely covered by squares $K_1$, $K_2$ and $K_3$.

2017 Sharygin Geometry Olympiad, 5
Tags: geometry , Tangents , midpoints , Squares
A square $ABCD$ is given. Two circles are inscribed into angles $A$ and $B$, and the sum of their diameters is equal to the sidelength of the square. Prove that one of their common tangents passes through the midpoint of $AB$.

2020 Adygea Teachers' Geometry Olympiad, 2
Tags: ratio , geometry , circle , Squares
The square $ABCD$ is inscribed in a circle. Points $E$ and $F$ are located on the side of the square, and points $G$ and $H$ are located on the smaller arc $AB$ of the circle so that the $EFGH$ is a square. Find the area ratio of these squares.

1972 All Soviet Union Mathematical Olympiad, 164
Tags: geometry , combinatorial geometry , areas , Squares
Given several squares with the total area $1$. Prove that you can pose them in the square of the area $2$ without any intersections.

1981 All Soviet Union Mathematical Olympiad, 314
Tags: combinatorial geometry , combinatorics , Squares , Coloring , table
Is it possible to fill a rectangular table with black and white squares (only) so, that the number of black squares will equal to the number of white squares, and each row and each column will have more than $75\%$ squares of the same colour?

2012 Denmark MO - Mohr Contest, 2
Tags: geometry , rectangle , combinatorial geometry , Squares
It is known about a given rectangle that it can be divided into nine squares which are situated relative to each other as shown. The black rectangle has side length $1$. Are there more than one possibility for the side lengths of the rectangle? [img]https://cdn.artofproblemsolving.com/attachments/1/0/af6bc5b867541c04586e4b03db0a7f97f8fe87.png[/img]

2003 Bosnia and Herzegovina Team Selection Test, 2
Tags: geometry , Squares , concurrent
Upon sides $AB$ and $BC$ of triangle $ABC$ are constructed squares $ABB_{1}A_{1}$ and $BCC_{1}B_{2}$. Prove that lines $AC_{1}$, $CA_{1}$ and altitude from $B$ to side $AC$ are concurrent.

1997 Bundeswettbewerb Mathematik, 3
Tags: geometry , Squares , congruent
A square $S_a$ is inscribed in an acute-angled triangle $ABC$ with two vertices on side $BC$ and one on each of sides $AC$ and $AB$. Squares $S_b$ and $S_c$ are analogously inscribed in the triangle. For which triangles are the squares $S_a,S_b$, and $S_c$ congruent?

2018 Portugal MO, 2
Tags: geometry , incircle , Squares , square
In the figure, $[ABCD]$ is a square of side $1$. The points $E, F, G$ and $H$ are such that $[AFB], [BGC], [CHD]$ and $[DEA]$ are right-angled triangles. Knowing that the circles inscribed in each of these triangles and the circle inscribed in the square $[EFGH]$ has all the same radius, what is the measure of the radius of the circles? [img]https://1.bp.blogspot.com/-l37AEXa7_-c/X4KaJwe6HQI/AAAAAAAAMk4/14wvIipf26cRge_GqKSRwH32bp291vX4QCLcBGAsYHQ/s0/2018%2Bportugal%2Bp2.png[/img]

1997 Denmark MO - Mohr Contest, 2
Tags: geometry , Squares , square , area
Two squares, both with side length $1$, are arranged so that one has one vertex in the center of the other. Determine the area of the gray area. [img]https://1.bp.blogspot.com/-xt3pe0rp1SI/XzcGLgEw1EI/AAAAAAAAMYM/vFKxvvVuLvAJ5FO_yX315X3Fg_iFaK2fACLcBGAsYHQ/s0/1997%2BMohr%2Bp2.png[/img]

1994 ITAMO, 1
Tags: partition , Tiling , combinatorics , combinatorial geometry , Squares
Show that there exists an integer $N$ such that for all $n \ge N$ a square can be partitioned into $n$ smaller squares.

2010 QEDMO 7th, 4
Tags: geometry , areas , Squares
Let $ABCD$ and $A'B'C'D'$ be two squares, both are oriented clockwise. In addition, it is assumed that all points are arranged as shown in the figure.Then it has to be shown that the sum of the areas of the quadrilaterals $ABB'A'$ and $CDD'C'$ equal to the sum of the areas of the quadrilaterals $BCC'B'$ and $DAA'D'$. [img]https://cdn.artofproblemsolving.com/attachments/0/2/6f7f793ded22fe05a7b0a912ef6c4e132f963e.png[/img]

2009 Junior Balkan Team Selection Tests - Romania, 4
Tags: combinatorial geometry , combinatorics , Squares
To obtain a square $P$ of side length $2$ cm divided into $4$ unit squares it is sufficient to draw $3$ squares: $P$ and another $2$ unit squares with a common vertex, as shown below: [img]https://cdn.artofproblemsolving.com/attachments/1/d/827516518871ec8ff00a66424f06fda9812193.png[/img] Find the minimum number of squares sufficient to obtain a square.of side length $n$ cm divided into $n^2$ unit squares ($n \ge 3$ is an integer).

1949 Moscow Mathematical Olympiad, 172
Tags: Impossible , Squares , geometry , combinatorial geometry
Two squares are said to be [i]juxtaposed [/i] if their intersection is a point or a segment. Prove that it is impossible to [i]juxtapose [/i] to a square more than eight non-overlapping squares of the same size.

2015 NIMO Summer Contest, 11
Tags: prime , Squares , NIMO , summer contest
We say positive integer $n$ is $\emph{metallic}$ if there is no prime of the form $m^2-n$. What is the sum of the three smallest metallic integers? [i] Proposed by Lewis Chen [/i]

2015 JBMO Shortlist, C1
Tags: combinatorics , grid , table , Squares
A board $ n \times n$ ($n \ge 3$) is divided into $n^2$ unit squares. Integers from $O$ to $n$ included, are written down: one integer in each unit square, in such a way that the sums of integers in each $2\times 2$ square of the board are different. Find all $n$ for which such boards exist.