Olympiad problems

2011 Tournament of Towns, 4

Each diagonal of a convex quadrilateral divides it into two isosceles triangles. The two diagonals of the same quadrilateral divide it into four isosceles triangles. Must this quadrilateral be a square?

Russian TST 2015, P1

Tags: square , geometry

The points $A', B', C', D'$ are selected respectively on the sides $AB, BC, CD, DA$ of the cyclic quadrilateral $ABCD$. It is known that $AA' = BB' = CC' = DD'$ and \[\angle AA'D' =\angle BB'A' =\angle CC'B' =\angle DD'C'.\]Prove that $ABCD$ is a square.

1950 Polish MO Finals, 4

Tags: geometry , hexagon , geometric inequality , square , geometric inequalities

Someone wants to unscrew a square nut with side $a$, with a wrench whose hole has the form of a regular hexagon with side $b$. What condition should the lengths $a$ and $b$ meet to make this possible?

1986 Austrian-Polish Competition, 3

Tags: coloring , square , combinatorics

Each point in space is colored either blue or red. Show that there exists a unit square having exactly $0, 1$ or $4$ blue vertices.

2019 Novosibirsk Oral Olympiad in Geometry, 4

Tags: square , geometry , 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]

2012 Sharygin Geometry Olympiad, 4

Tags: locus , right triangle , square , geometry

Consider a square. Find the locus of midpoints of the hypothenuses of rightangled triangles with the vertices lying on three different sides of the square and not coinciding with its vertices. (B.Frenkin)

1988 Tournament Of Towns, (189) 2

Tags: geometry , equal angles , angle , square

A point $M$ is chosen inside the square $ABCD$ in such a way that $\angle MAC = \angle MCD = x$ . Find $\angle ABM$.

1970 Dutch Mathematical Olympiad, 3

Tags: square , symmetry , equilateral , geometry

The points $P,Q,R$ and $A,B,C,D$ lie on a circle (clockwise) such that $\vartriangle PQR$ is equilateral and $ABCD$ is a square. The points $A$ and $P$ coincide. Prove that the symmetric of $B$ and $D$ wrt $PQ$ and $PR$ respectively lie on the sidelines of the symmetric square wrt $QR$.

2010 NZMOC Camp Selection Problems, 2

Tags: geometry , square , chord

$AB$ is a chord of length $6$ in a circle of radius $5$ and centre $O$. A square is inscribed in the sector $OAB$ with two vertices on the circumference and two sides parallel to $ AB$. Find the area of the square.

1947 Moscow Mathematical Olympiad, 129

Tags: square , chessboard , combinatorics , combinatorial geometry

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.

2022 Bolivia Cono Sur TST, P3

Tags: combinatorics , arithmetic progression , number theory , square

Is it possible to complete the following square knowning that each row and column make an aritmetic progression?

2022 Centroamerican and Caribbean Math Olympiad, 4

Tags: geometry , rectangle , square , tangent circle

Let $A_1A_2A_3A_4$ be a rectangle and let $S_1,S_2,S_3,S_4$ four circumferences inside of the rectangle such that $S_k$ and $S_{k+1}$ are tangent to each other and tangent to the side $A_kA_{k+1}$ for $k=1,2,3,4$, where $A_5=A_1$ and $S_5=S_1$. Prove that $A_1A_2A_3A_4$ is a square.

1958 November Putnam, B3

Tags: square , diameter

Show that if a unit square is partitioned into two sets, then the diameter (least upper bound of the distances between pairs of points) of one of the sets is not less than $\sqrt{5} \slash 2.$ Show also that no larger number will do.

1993 Chile National Olympiad, 1

Tags: geometry , square , path , distance

There are four houses, located on the vertices of a square. You want to draw a road network, so that you can go from any house to any other. Prove that the network formed by the diagonals is not the shortest. Find a shorter network.

2016 Latvia Baltic Way TST, 11

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

Is it possible to cut a square with side $\sqrt{2015}$ into no more than five pieces so that these pieces can be rearranged into a rectangle with sides of integer length? (The cuts should be made using straight lines, and flipping of the pieces is disallowed.)

1949 Moscow Mathematical Olympiad, 172

Tags: impossible , geometry , combinatorial geometry , square

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.

2007 Estonia Team Selection Test, 4

Tags: angle chasing , square , geometry , angle

In square $ABCD,$ points $E$ and $F$ are chosen in the interior of sides $BC$ and $CD$, respectively. The line drawn from $F$ perpendicular to $AE$ passes through the intersection point $G$ of $AE$ and diagonal $BD$. A point $K$ is chosen on $FG$ such that $|AK|= |EF|$. Find $\angle EKF.$

2015 Estonia Team Selection Test, 2

Tags: rectangle , square , combinatorial geometry , combinatorics

A square-shaped pizza with side length $30$ cm is cut into pieces (not necessarily rectangular). All cuts are parallel to the sides, and the total length of the cuts is $240$ cm. Show that there is a piece whose area is at least $36$ cm$^2$

Denmark (Mohr) - geometry, 2000.1

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]

2018 Hanoi Open Mathematics Competitions, 8

Tags: square , angle , geometry

Let $P$ be a point inside the square $ABCD$ such that $\angle PAC = \angle PCD = 17^o$ (see Figure 1). Calculate $\angle APB$? [img]https://cdn.artofproblemsolving.com/attachments/d/0/0b20ebee1fe28e9c5450d04685ac8537acda07.png[/img]

2004 Paraguay Mathematical Olympiad, 4

Tags: geometry , square , trisection

In a square $ABCD$, $E$ is the midpoint of $BC$ and $F$ is the midpoint of $CD$. Prove that $AF$ and $AE$ divide the diagonal $BD$ in three equal segments.

1982 Spain Mathematical Olympiad, 5

Tags: construction , square , geometry

Construct a square knowing the sum of the diagonal and the side.

2014 Romania National Olympiad, 2

Tags: geometry , rhombus , square , perpendicular , isosceles , right triangle

Outside the square $ABCD$, the rhombus $BCMN$ is constructed with angle $BCM$ obtuse . Let $P$ be the intersection point of the lines $BM$ and $AN$ . Prove that $DM \perp CP$ and the triangle $DPM$ is right isosceles .

1982 Tournament Of Towns, (021) 2

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

1990 All Soviet Union Mathematical Olympiad, 514

Tags: tiling , congruent , rectangle , square , combinatorial geometry , geometry

Does there exist a rectangle which can be dissected into $15$ congruent polygons which are not rectangles? Can a square be dissected into $15$ congruent polygons which are not rectangles?