Olympiad problems

2019 May Olympiad, 4

You have to divide a square paper into three parts, by two straight cuts, so that by locating these parts properly, without gaps or overlaps, an obtuse triangle is formed. Indicate how to cut the square and how to assemble the triangle with the three parts.

1969 Spain Mathematical Olympiad, 4

Tags: geometry , octagon , square , area

A circle of radius $R$ is divided into $8$ equal parts. The points of division are denoted successively by $A, B, C, D, E, F , G$ and $H$. Find the area of the square formed by drawing the chords $AF$ , $BE$, $CH$ and $DG$.

2022 Kurschak Competition, 2

Tags: number theory , prime number , square

Let $p$ and $q$ be prime numbers of the form $4k+3$. Suppose that there exist integers $x$ and $y$ such that $x^2-pqy^2=1$. Prove that there exist positive integers $a$ and $b$ such that $|pa^2-qb^2|=1$.

Novosibirsk Oral Geo Oly VII, 2022.1

Tags: geometry , square

Cut a square with three straight lines into three triangles and four quadrilaterals.

Novosibirsk Oral Geo Oly IX, 2020.4

Tags: equal angles , square

Points $E$ and $F$ are the midpoints of sides $BC$ and $CD$ of square $ABCD$, respectively. Lines $AE$ and $BF$ meet at point $P$. Prove that $\angle PDA = \angle AED$.

1975 Chisinau City MO, 111

Tags: square , geometry , concyclic

Three squares are constructed on the sides of the triangle to the outside. What should be the angles of the triangle so that the six vertices of these squares, other than the vertices of the triangle, lie on the same circle?

2019 Puerto Rico Team Selection Test, 1

Tags: square , combinatorics , combinatorial geometry

A square is divided into $25$ unit squares by drawing lines parallel to the sides of the square. Some diagonals of unit squares are drawn from such that two diagonals do not share points. What is the maximum number diagonals that can be drawn with this property?

2014 Swedish Mathematical Competition, 4

Tags: combinatorial geometry , square , geometric inequalities , geometry

A square is cut into a finitely number of triangles in an arbitrary way. Show the sum of the diameters of the inscribed circles in these triangles is greater than the side length of the square.

2001 Chile National Olympiad, 2

Tags: square , circles , combinatorics , combinatorial geometry , geometry

Prove that the only way to cover a square of side $1$ with a finite number of circles that do not overlap, it is with only one circle of radius greater than or equal to $\frac{1}{\sqrt2}$. Circles can occupy part of the outside of the square and be of different radii.

2010 IFYM, Sozopol, 4

Tags: geometry , square

Let $ABCD$ be a square with side 1. On the sides $BC$ and $CD$ are chosen points $P$ and $Q$ where $AP$ and $AQ$ intersect the diagonal $BD$ in points $M$ and $N$ respectively. If $DQ\neq BP$ and the line through $A$ and the intersection point of $MQ$ and $NP$ is perpendicular to $PQ$, prove that $\angle MAN=45^\circ$.

Ukraine Correspondence MO - geometry, 2011.3

Tags: geometry , rectangle , square

The kid cut out of grid paper with the side of the cell $1$ rectangle along the grid lines and calculated its area and perimeter. Carlson snatched his scissors and cut out of this rectangle along the lines of the grid a square adjacent to the boundary of the rectangle. - My rectangle ... - kid sobbed. - There is something strange about this figure! - Nonsense, do not mention it - Carlson said - waving his hand carelessly. - Here you see, in this figure the perimeter is the same as the area of the rectangle was, and the area is the same as was the perimeter! What size square did Carlson cut out?

1972 All Soviet Union Mathematical Olympiad, 164

Tags: square , combinatorial geometry , area , geometry

Given several squares with the total area $1$. Prove that you can pose them in the square of the area $2$ without any intersections.

2015 Danube Mathematical Competition, 4

Tags: geometry , rectangle , perpendicular bisector , square , angle

Let $ABCD$ be a rectangle with $AB\ge BC$ Point $M$ is located on the side $(AD)$, and the perpendicular bisector of $[MC]$ intersects the line $BC$ at the point $N$. Let ${Q} =MN\cup AB$ . Knowing that $\angle MQA= 2\cdot \angle BCQ $, show that the quadrilateral $ABCD$ is a square.

2012 District Olympiad, 4

Tags: geometry , perpendicular , square

Consider the square $ABCD$ and the point $E$ on the side $AB$. The line $DE$ intersects the line $BC$ at point $F$, and the line $CE$ intersects the line $AF$ at point $G$. Prove that the lines $BG$ and $DF$ are perpendicular.

Novosibirsk Oral Geo Oly IX, 2019.3

Tags: area , square , geometry

The circle touches the square and goes through its two vertices as shown in the figure. Find the area of the square. (Distance in the picture is measured horizontally from the midpoint of the side of the square.) [img]https://cdn.artofproblemsolving.com/attachments/7/5/ab4b5f3f4fb4b70013e6226ce5189f3dc2e5be.png[/img]

2019 Romania National Olympiad, 2

Tags: square , geometry , equal segments , perpendicular

Let $ABCD$ be a square and $E$ a point on the side $(CD)$. Squares $ENMA$ and $EBQP$ are constructed outside the triangle $ABE$. Prove that: a) $ND = PC$ b) $ND\perp PC$.

VI Soros Olympiad 1999 - 2000 (Russia), 9.9

Tags: geometry , square

On the plane there are two isosceles non-intersecting right triangles $ABC$ and $DEC$ ($AB$ and $DE$ are the hypotenuses,$ ABDE$ is a convex quadrilateral), and $AB = 2 DE$. Let's construct two more isosceles right triangles: $BDF$ (with hypotenuse $BF$ located outside triangle $BDC$) and $AEG$ (with hypotenuse $AG$ located outside triangle $AEC$). Prove that the line $FG$ passes through a point $N$ such that $DCEN$ is a square.

1998 May Olympiad, 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]

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 $\angle EKF.$

1994 Austrian-Polish Competition, 3

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

A rectangular building consists of $30$ square rooms situated like the cells of a $2 \times 15$ board. In each room there are three doors, each of which leads to another room (not necessarily different). How many ways are there to distribute the doors between the rooms so that it is possible to get from any room to any other one without leaving the building?

2013 Argentina National Olympiad Level 2, 6

Tags: geometry , rectangle , square , cover

Is there a square with side lenght $\ell < 1$ that can completely cover any rectangle of diagonal $1$?

1967 German National Olympiad, 1

Tags: geometry , locus , equilateral , square

In a plane, a square $ABCD$ and a point $P$ located inside it are given. Let a point $ Q$ pass through all sides of the square. Describe the set of all those points $R$ in for which the triangle $PQR$ is equilateral.

2010 Oral Moscow Geometry Olympiad, 2

Tags: geometry , paper , folding , square , distance

Given a square sheet of paper with side $1$. Measure on this sheet a distance of $ 5/6$. (The sheet can be folded, including, along any segment with ends at the edges of the paper and unbend back, after unfolding, a trace of the fold line remains on the paper).

2009 Bundeswettbewerb Mathematik, 1

Tags: number theory , square

Determine all possible digits $z$ for which $\underbrace{9...9}_{100}z\underbrace{0...0}_{100}9$ is a square number.

1992 Austrian-Polish Competition, 2

Tags: coloring , combinatorics , right triangle , square

Each point on the boundary of a square has to be colored in one color. Consider all right triangles with the vertices on the boundary of the square. Determine the least number of colors for which there is a coloring such that no such triangle has all its vertices of the same color.