Olympiad problems

1976 Bundeswettbewerb Mathematik, 2

Two congruent squares $Q$ and $Q'$ are given in the plane. Show that they can be divided into parts $T_1, T_2, \ldots , T_n$ and $T'_1 , T'_2 , \ldots , T'_n$, respectively, such that $T'_i$ is the image of $T_i$ under a translation for $i=1,2, \ldots, n.$

2024 Junior Balkan Team Selection Tests - Romania, P2

Tags: square , equilateral triangle , geometry

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]

1987 Tournament Of Towns, (157) 1

Tags: area , perpendicular , square , geometry

From vertex $A$ in square $ABCD$ (of side length $1$ ) two lines are drawn , one intersecting side $BC$ and the other intersecting side $CD$. The angle between these lines is $\theta$. From vertices $B$ and $D$ we construct perpendiculars to each of these lines . Find the area of the quadrilateral whose vertices are the four feet of these perpendiculars.

2012 Tournament of Towns, 6

Tags: square , combinatorics , combinatorial geometry , 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 $N$?

Novosibirsk Oral Geo Oly VIII, 2023.7

Tags: geometry , square , parallel lines

A square with side $1$ is intersected by two parallel lines as shown in the figure. Find the sum of the perimeters of the shaded triangles if the distance between the lines is also $1$. [img]https://cdn.artofproblemsolving.com/attachments/9/e/4e70610b80871325a72e923a0909eff06aebfa.png[/img]

2018 Adygea Teachers' Geometry Olympiad, 1

Tags: geometry , square , distance

Can the distances from a certain point on the plane to the vertices of a certain square be equal to $1, 4, 7$, and $8$ ?

Estonia Open Senior - geometry, 1997.2.3

Tags: geometry , equal circles , circles , square , tangent circle

The figure shows a square and three circles of equal radius tangent to each other and square passes. Find the radius of the circles if the square length is $1$. [img]http://3.bp.blogspot.com/-iIjwupkz7DQ/XnrIRhKIJnI/AAAAAAAALhA/clERrIDqEtcujzvZk_qu975wsTjKaxCLQCK4BGAYYCw/s400/97%2Bestonia%2Bopen%2Bs2.3.png[/img]

1956 Moscow Mathematical Olympiad, 330

Tags: geometric inequality , geometry , inradius , square , inscribed

A square of side $a$ is inscribed in a triangle so that two of the square’s vertices lie on the base, and the other two lie on the sides of the triangle. Prove that if $r$ is the radius of the circle inscribed in the triangle, then $r\sqrt2 < a < 2r$.

2000 Denmark MO - Mohr Contest, 1

Tags: square , area , midpoint

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]

1976 Spain Mathematical Olympiad, 1

Tags: construction , square , geometry

In a plane there are four fixed points $A, B, C, D$, no $3$ collinear. Construct a square with sides $a, b, c, d$ such that $A \in a$, $B \in b$, $C \in c$, $D \in d$.

2016 Junior Balkan Team Selection Tests - Moldova, 7

Tags: geometry , perpendicular , square , equal segments

Let $ABCD$ ba a square and let point $E$ be the midpoint of side $AD$. Points $G$ and $F$ are located on the segment $(BE)$ such that the lines $AG$ and $CF$ are perpendicular on the line $BE$. Prove that $DF= CG$.

1985 IMO Longlists, 26

Tags: square , geometry , dissection

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

Kyiv City MO Seniors Round2 2010+ geometry, 2016.10.2

Tags: geometry , collinear , construction , square

On the horizontal line from left to right are the points $P, \, \, Q, \, \, R, \, \, S$. Construct a square $ABCD$, for which on the line $AD$ lies lies the point $P$, on the line $BC$ lies the point $Q$, on the line $AB$ lies the point $R$, on the line $CD$ lies the point $S $.

2015 BMT Spring, 9

Tags: geometry , 3d geometry , cube , geometric inequality , square

Find the side length of the largest square that can be inscribed in the unit cube.

2009 Postal Coaching, 4

Tags: square , geometry , area

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 $[1/a, a]$.

Novosibirsk Oral Geo Oly IX, 2021.4

Tags: geometry , square , tangent circle

A semicircle of radius $5$ and a quarter of a circle of radius $8$ touch each other and are located inside the square as shown in the figure. Find the length of the part of the common tangent, enclosed in the same square. [img]https://cdn.artofproblemsolving.com/attachments/f/2/010f501a7bc1d34561f2fe585773816f168e93.png[/img]

1985 Tournament Of Towns, (088) 4

Tags: equal area , geometry , rectangle , square

A square is divided into $5$ rectangles in such a way that its $4$ vertices belong to $4$ of the rectangles , whose areas are equal , and the fifth rectangle has no points in common with the side of the square (see diagram) . Prove that the fifth rectangle is a square. [img]https://3.bp.blogspot.com/-TQc1v_NODek/XWHHgmONboI/AAAAAAAAKi4/XES55OJS5jY9QpNmoURp4y80EkanNzmMwCK4BGAYYCw/s1600/TOT%2B1985%2BSpring%2BJ4.png[/img]

2024 Polish Junior MO Finals, 2

Tags: square , combinatorics

Determine the smallest integer $n \ge 1$ such that a $n \times n$ square can be cut into square pieces of size $1 \times 1$ and $2 \times 2$ with both types occuring the same number of times.

May Olympiad L1 - geometry, 1998.4

Tags: geometry , area of a triangle , equilateral , square

$ABCD$ is a square of center $O$. On the sides $DC$ and $AD$ the equilateral triangles DAF and DCE have been constructed. Decide if the area of the $EDF$ triangle is greater, less or equal to the area of the $DOC$ triangle. [img]https://4.bp.blogspot.com/-o0lhdRfRxl0/XNYtJgpJMmI/AAAAAAAAKKg/lmj7KofAJosBZBJcLNH0JKjW3o17CEMkACK4BGAYYCw/s1600/may4_2.gif[/img]

1949-56 Chisinau City MO, 41

Tags: rectangle , angle bisector , square , geometry

Prove that the bisectors of the angles of a rectangle, extended to their mutual intersection, form a square.

2001 IMO Shortlist, 1

Tags: geometry , triangle , square

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.

2024 Kyiv City MO Round 1, Problem 2

Tags: arranging , square , parity , combinatorics

Is it possible to write the numbers from $1$ to $100$ in the cells of a of a $10 \times 10$ square so that: 1. Each cell contains exactly one number; 2. Each number is written exactly once; 3. For any two cells that are symmetrical with respect to any of the perpendicular bisectors of sides of the original $10 \times 10$ square, the numbers in them must have the same parity. The figure below shows examples of such pairs of cells, in which the numbers must have the same parity. [img]https://i.ibb.co/b3P8t36/Kyiv-MO-2024-7-2.png[/img] [i]Proposed by Mykhailo Shtandenko[/i]

2020 Yasinsky Geometry Olympiad, 1

Tags: square , area , right triangle , geometry

The square $ABCD$ is divided into $8$ equal right triangles and the square $KLMN$, as shown in the figure. Find the area of the square $ABCD$ if $KL = 5, PS = 8$. [img]https://1.bp.blogspot.com/-B2QIHvPcIx0/X4BhUTMDhSI/AAAAAAAAMj4/4h0_q1P6drskc5zSvtfTZUskarJjRp5LgCLcBGAsYHQ/s0/Yasinsky%2B2020%2Bp1.png[/img]

2017 BMT Spring, 6

Tags: geometry , square , probability , rotation

The center of a square of side length $ 1$ is placed uniformly at random inside a circle of radius $ 1$. Given that we are allowed to rotate the square about its center, what is the probability that the entire square is contained within the circle for some orientation of the square?

1996 May Olympiad, 4

Tags: square , angle , geometry

Let $ABCD$ be a square and let point $F$ be any point on side $BC$. Let the line perpendicular to $DF$, that passes through $B$, intersect line $DC$ at $Q$. What is value of $\angle FQC$?