Olympiad problems

1947 Moscow Mathematical Olympiad, 129

How many squares different in size or location can be drawn on an $8 \times 8$ chess board? Each square drawn must consist of whole chess board’s squares.

2002 Denmark MO - Mohr Contest, 2

Tags: combinatorics , geometry , combinatorial geometry , Squares

Prove that for any integer $n$ greater than $5$, a square can be divided into $n$ squares.

1941 Moscow Mathematical Olympiad, 076

Tags: Squares , parallelogram , geometry

On the sides of a parallelogram, squares are constructed outwards. Prove that the centers of these squares are vertices of a square.

1979 Austrian-Polish Competition, 7

Tags: analytic geometry , grid , hexagon , Squares , combinatorial geometry , combinatorics

Let $n$ and $m$ be fixed positive integers. The hexagon $ABCDEF$ with vertices $A = (0,0)$, $B = (n,0)$, $C = (n,m)$, $D = (n-1,m)$, $E = (n-1,1)$, $F = (0,1)$ has been partitioned into $n+m-1$ unit squares. Find the number of paths from $A$ to $C$ along grid lines, passing through every grid node at most once.

1955 Moscow Mathematical Olympiad, 291

Tags: rectangle , cut , Squares , combinatorial geometry , geometry

Find all rectangles that can be cut into $13$ equal squares.

2023 Novosibirsk Oral Olympiad in Geometry, 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]

2023 India Regional Mathematical Olympiad, 1

Tags: number theory , RMO 2023 , Squares , Divisibility

Let $\mathbb{N}$ be the set of all positive integers and $S=\left\{(a, b, c, d) \in \mathbb{N}^4: a^2+b^2+c^2=d^2\right\}$. Find the largest positive integer $m$ such that $m$ divides abcd for all $(a, b, c, d) \in S$.

2022 Yasinsky Geometry Olympiad, 2

Tags: geometry , Squares , square , ratio , areas

On the sides $AB$, $BC$, $CD$, $DA$ of the square $ABCD$ points $P, Q, R, T$ are chosen such that $$\frac{AP}{PB}=\frac{BQ}{QC}=\frac{CR}{RD}=\frac{DT}{TA}=\frac12.$$ The segments $AR$, $BT$, $CP$, $DQ$ in the intersection form the quadrilateral $KLMN$ (see figure). [img]https://cdn.artofproblemsolving.com/attachments/f/c/587a2358734c300fe7082c520f90c91f872b49.png[/img] a) Prove that $KLMN$ is a square. b) Find the ratio of the areas of the squares $KLMN$ and $ABCD$. (Alexander Shkolny)

2021 Malaysia IMONST 1, 1

Tags: geometry , Squares , pentagons

Dinesh has several squares and regular pentagons, all with side length $ 1$. He wants to arrange the shapes alternately to form a closed loop (see diagram). How many pentagons would Dinesh need to do so? [img]https://cdn.artofproblemsolving.com/attachments/8/9/6345d7150298fe26cfcfba554656804ed25a6d.jpg[/img]

1994 Argentina National Olympiad, 3

Tags: geometry , Squares

Given in the plane the square $ABCD$, the square $A_1B_1C_1D_1$, smaller than the first, and a quadrilateral $PQRS$ that satisfy the following conditions $\bullet$ $ABCD$ and $A_1B_1C_1D_1$ have a common center and respectively parallel sides. $\bullet$$P$, $Q$, $R$, $S$ belong one to each side of the square $ABCD$. $\bullet$ $A_1$, $B_1$, $C_1$, $D_1$ belong one to each side of the quadrilateral $PQRS$. Prove that $PQRS$ is a square.

2022 Paraguay Mathematical Olympiad, 5

Tags: geometry , areas , Squares

In the figure, there is a circle of radius $1$ such that the segment $AG$ is diameter and that line $AF$ is perpendicular to line $DC$. There are also two squares $ABDC$ and $DEGF$, where $B$ and $E$ are points on the circle, and the points $A$, $D$ and $E$ are collinear. What is the area of square $DEGF$? [img]https://cdn.artofproblemsolving.com/attachments/1/e/794da3bc38096ef5d5daaa01d9c0f8c41a6f84.png[/img]

2019 Costa Rica - Final Round, LR3

Tags: geometry , Squares , square , areas

Consider the following sequence of squares (side $1$), in each step the central square is divided into equal parts and colored as shown in the figure: [img]https://cdn.artofproblemsolving.com/attachments/9/0/6874ab5aecadf2112fbe4a196ab3091ab8b31a.png[/img] Square 1 Square 2 Square 3 Let $A_n$ with $n \in N$, $n> 1$ be the shaded area of square $n$, show that $A_n <\frac23$

1986 All Soviet Union Mathematical Olympiad, 419

Tags: Squares , octagon , Sum , perimeter , geometry

Two equal squares, one with red sides, another with blue ones, give an octagon in intersection. Prove that the sum of red octagon sides lengths is equal to the sum of blue octagon sides lengths.

2016 Novosibirsk Oral Olympiad in Geometry, 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]

2025 Israel National Olympiad (Gillis), P2

Tags: geometry , rhombus , Squares , collinearity

Let $ABCD$ be a rhombus. Eight additional points $X_1$, $X_2$, $Y_1$, $Y_2$, $Z_1$, $Z_2$, $W_1$, $W_2$ were chosen so that the quadrilaterals $AX_1BX_2$, $BY_1CY_2$, $CZ_1DZ_2$, $DW_1AW_2$ are squares. Prove that the eight new points lie on two straight lines.

1985 IMO Longlists, 26

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 \[K'=\bigcup_{i=1}^{p} t_i(T_i) \ ? \]

Ukrainian TYM Qualifying - geometry, 2020.12

Tags: geometry , Squares , Concyclic , equal circles , Ukrainian TYM

On the side $CD$ of the square $ABCD$, the point $F$ is chosen and the equal squares $DGFE$ and $AKEH$ are constructed ($E$ and $H$ lie inside the square). Let $M$ be the midpoint of $DF$, $J$ is the incenter of the triangle $CFH$. Prove that: a) the points $D, K, H, J, F$ lie on the same circle; b) the circles inscribed in triangles $CFH$ and $GMF$ have the same radii.

2012 BMT Spring, 3

Tags: ratio , geometry , Squares , areas

Let $ABC$ be a triangle with side lengths $AB = 2011$, $BC = 2012$, $AC = 2013$. Create squares $S_1 =ABB'A''$, $S_2 = ACC''A'$ , and $S_3 = CBB''C'$ using the sides $AB$, $AC$, $BC$ respectively, so that the side $B'A''$ is on the opposite side of $AB$ from $C$, and so forth. Let square $S_4$ have side length $A''A' $, square $S_5$ have side length $C''C'$, and square $S_6$ have side length $B''B'$. Let $A(S_i)$ be the area of square $S_i$ . Compute $\frac{A(S_4)+A(S_5)+A(S_6)}{A(S_1)+A(S_2)+A(S_3)}$?

2018 Junior Regional Olympiad - FBH, 5

Tags: ratio , Squares , circumcircle

It is given square $ABCD$ which is circumscribed by circle $k$. Let us construct a new square so vertices $E$ and $F$ lie on side $ABCD$ and vertices $G$ and $H$ on arc $AB$ of circumcircle. Find out the ratio of area of squares

Durer Math Competition CD 1st Round - geometry, 2008.C3

Tags: geometry , Squares , square , Durer

Given the squares $ABCD$ and $DEFG$, whose only common point is $D$. Let the midpoints of segments $AG$, $GE$, $EC$, and $CA$ be $H, I, J$, and $K$ respectively . Prove that $HIJK$ is a square. [img]https://cdn.artofproblemsolving.com/attachments/f/d/c3313e5bbf581977a74ea2b114d14950e38605.png[/img]