Olympiad problems

1974 Czech and Slovak Olympiad III A, 6

Tags: rotation , geometry , square , locus , area , geometric transformation

Let a unit square $\mathcal D$ be given in the plane. For any point $X$ in the plane denote $\mathcal D_X$ the image of $\mathcal D$ in rotation with respect to origin $X$ by $+90^\circ.$ Find the locus of all $X$ such that the area of union $\mathcal D\cup\mathcal D_X$ is at most 1.5.

2022 Cyprus JBMO TST, 2

Tags: parallelogram , geometry , square , area

Let $ABCD$ be a square. Let $E, Z$ be points on the sides $AB, CD$ of the square respectively, such that $DE\parallel BZ$. Assume that the triangles $\triangle EAD, \triangle ZCB$ and the parallelogram $BEDZ$ have the same area. If the distance between the parallel lines $DE$ and $BZ$ is equal to $1$, determine the area of the square.

2020 Yasinsky Geometry Olympiad, 2

Tags: geometry , perpendicular , square

Let $ABCD$ be a square, point $E$ be the midpoint of the side $BC$. On the side $AB$ mark a point $F$ such that $FE \perp DE$. Prove that $AF + BE = DF$. (Ercole Suppa, Italy)

2016 Oral Moscow Geometry Olympiad, 3

Tags: geometry , equal area , area , square

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]

1982 Tournament Of Towns, (021) 2

Tags: geometry , minimum , square , broken line , combinatorial 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)

1999 Tournament Of Towns, 5

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

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)

2023 Yasinsky Geometry Olympiad, 6

Tags: geometry , collinear , square

Given a square $ABCD$, point $E$ is the midpoint of $AD$. Let $F$ be the foot of the perpendicular drawn from point $B$ on $EC$. Point $K$ on $AB$ is such that $\angle DFK = 90^o$. The point $N$ on the $CE$ is such that $\angle NKB = 90^o$. Prove that the point $N$ lies on the segment $BD$. (Matvii Kurskyi) [img]https://cdn.artofproblemsolving.com/attachments/4/2/d42b8c8117ec1d5e5c5b981904779b156fce93.png[/img]

Novosibirsk Oral Geo Oly VIII, 2016.4

Tags: midpoint , geometry , square

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]

2016 Oral Moscow Geometry Olympiad, 6

Tags: paper , segment , folding , square , geometry

Given a square sheet of paper with a side of $2016$. Is it possible to bend its not more than ten times, construct a segment of length $1$?

Cono Sur Shortlist - geometry, 1993.5

Tags: geometry , square

A block of houses is a square. There is a courtyard there in which a gold medal has fallen. Whoever calculates how long the side of said apple is, knowing that the distances from the medal to three consecutive corners of the apple are, respectively, $40$ m, $60$ m and $80$ m, will win the medal.

2017 BMT Spring, 13

Tags: equilateral , area , geometry , square

$4$ equilateral triangles of side length $1$ are drawn on the interior of a unit square, each one of which shares a side with one of the $4$ sides of the unit square. What is the common area enclosed by all $4$ equilateral triangles?

2011 Bundeswettbewerb Mathematik, 1

Tags: combinatorial geometry , angle , geometry , hexagon , tiling , tiles , square

Prove that you can't split a square into finitely many hexagons, whose inner angles are all less than $180^o$.

2024 Junior Balkan Team Selection Tests - Romania, P2

Tags: geometry , square , equilateral triangle

Let $M$ be the midpoint of the side $AD$ of the square $ABCD.$ Consider the equilateral triangles $DFM{}$ and $BFE{}$ such that $F$ lies in the interior of $ABCD$ and the lines $EF$ and $BC$ are concurrent. Denote by $P{}$ the midpoint of $ME.$ Prove that" [list=a] [*]The point $P$ lies on the line $AC.$ [*]The halfline $PM$ is the bisector of the angle $APF.$ [/list] [i]Adrian Bud[/i]

2002 Estonia National Olympiad, 1

Tags: geometry , angle , square , equal angles

Points $K$ and $L$ are taken on the sides $BC$ and $CD$ of a square $ABCD$ so that $\angle AKB = \angle AKL$. Find $\angle KAL$.

2020 Malaysia IMONST 1, 3

Tags: circles , geometry , square

Given a square with area $A$. A circle lies inside the square, such that the circle touches all sides of the square. Another square with area $B$ lies inside the circle, such that all its vertices lie on the circle. Find the value of $\frac{A}{B}.$

Novosibirsk Oral Geo Oly VIII, 2019.3

Tags: square , paper folding , folding , ratio , geometry

A square sheet of paper $ABCD$ is folded straight in such a way that point $B$ hits to the midpoint of side $CD$. In what ratio does the fold line divide side $BC$?

2002 Germany Team Selection Test, 2

Tags: square , geometry , triangle

Let $A_1$ be the center of the square inscribed in acute triangle $ABC$ with two vertices of the square on side $BC$. Thus one of the two remaining vertices of the square is on side $AB$ and the other is on $AC$. Points $B_1,\ C_1$ are defined in a similar way for inscribed squares with two vertices on sides $AC$ and $AB$, respectively. Prove that lines $AA_1,\ BB_1,\ CC_1$ are concurrent.

1993 Tournament Of Towns, (385) 3

Tags: midpoint , geometry , square

Three angles of a non-convex, non-self-intersecting quadrilateral are equal to $45$ degrees (i.e. the last equals $225$ degrees). Prove that the midpoints of its sides are vertices of a square. (V Proizvolov)

1997 Denmark MO - Mohr Contest, 2

Tags: area , square , geometry

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]

1941 Moscow Mathematical Olympiad, 081

Tags: combinatorial geometry , square , rectangle , impossible , geometry

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.

2009 Postal Coaching, 4

Tags: area , square , geometry

Determine the least real number $a > 1$ such that for any point $P$ in the interior of a square $ABCD$, the ratio of the areas of some two triangle $PAB, PBC, PCD, PDA$ lies in the interval $[1/a, a]$.

1994 Portugal MO, 2

Tags: equal angles , square , geometry

Consider in a square $[ABCD]$ a point $E$ on the side $AB$, different from $A$ and $B$. On the side $BC$ consider the point $F$ such that $\angle AED = \angle DEF$ . Prove that $EF = AE + FC$.

1947 Moscow Mathematical Olympiad, 129

Tags: combinatorics , combinatorial geometry , chessboard , square

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.

2000 Denmark MO - Mohr Contest, 1

Tags: area , midpoint , square

The quadrilateral $ABCD$ is a square of sidelength $1$, and the points $E, F, G, H$ are the midpoints of the sides. Determine the area of quadrilateral $PQRS$. [img]https://1.bp.blogspot.com/--fMGH2lX6Go/XzcDqhgGKfI/AAAAAAAAMXo/x4NATcMDJ2MeUe-O0xBGKZ_B4l_QzROjACLcBGAsYHQ/s0/2000%2BMohr%2Bp1.png[/img]

2003 German National Olympiad, 3

Tags: combinatorics , weight , board , grid , square

Consider a $N\times N$ square board where $N\geq 3$ is an odd integer. The caterpillar Carl sits at the center of the square; all other cells contain distinct positive integers. An integer $n$ weights $1\slash n$ kilograms. Carl wants to leave the board but can eat at most $2$ kilograms. Determine whether Carl can always find a way out when a) $N=2003.$ b) $N$ is an arbitrary odd integer.