Olympiad problems

2009 Oral Moscow Geometry Olympiad, 2

A square and a rectangle of the same perimeter have a common corner. Prove that the intersection point of the diagonals of the rectangle lies on the diagonal of the square. (Yu. Blinkov)

1961 Czech and Slovak Olympiad III A, 2

Tags: geometry , construction , square , isosceles right triangle , geometric construction

Let a right isosceles triangle $APQ$ with the hypotenuse $AP$ be given in plane. Construct such a square $ABCD$ that the lines $BC, CD$ contain points $P, Q,$ respectively. Compute the length of side $AB = b$ in terms of $AQ=a$.

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?

1998 Croatia National Olympiad, Problem 3

Tags: geometry , square , triangle , right triangle

Points $E$ and $F$ are chosen on the sides $AB$ and $BC$ respectively of a square $ABCD$ such that $BE=BF$. Let $BN$ be an altitude of the triangle $BCE$. Prove that the triangle $DNF$ is right-angled.

2017 Sharygin Geometry Olympiad, 5

Tags: geometry , midpoint , square , tangent

A square $ABCD$ is given. Two circles are inscribed into angles $A$ and $B$, and the sum of their diameters is equal to the sidelength of the square. Prove that one of their common tangents passes through the midpoint of $AB$.

2006 Sharygin Geometry Olympiad, 5

Tags: paper , folding , square , combinatorics , combinatorial geometry , geometry

a) Fold a $10 \times 10$ square from a $1 \times 118$ rectangular strip. b) Fold a $10 \times 10$ square from a $1 \times (100+9\sqrt3)$ rectangular strip (approximately $1\times 115.58$). The strip can be bent, but not torn.

1998 Tuymaada Olympiad, 7

Tags: algebra , arithmetic sequence , sequence , square , sum

All possible sequences of numbers $-1$ and $+1$ of length $100$ are considered. For each of them, the square of the sum of the terms is calculated. Find the arithmetic average of the resulting values.

Novosibirsk Oral Geo Oly VIII, 2020.1

Tags: square , geometry

Three squares of area $4, 9$ and $36$ are inscribed in the triangle as shown in the figure. Find the area of the big triangle [img]https://cdn.artofproblemsolving.com/attachments/9/7/3e904a9c78307e1be169ec0b95b1d3d24c1aa2.png[/img]

2019 Puerto Rico Team Selection Test, 1

Tags: combinatorics , combinatorial geometry , square

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?

2012 Denmark MO - Mohr Contest, 2

Tags: geometry , rectangle , combinatorial geometry , square

It is known about a given rectangle that it can be divided into nine squares which are situated relative to each other as shown. The black rectangle has side length $1$. Are there more than one possibility for the side lengths of the rectangle? [img]https://cdn.artofproblemsolving.com/attachments/1/0/af6bc5b867541c04586e4b03db0a7f97f8fe87.png[/img]

1954 Putnam, A2

Tags: square , distance

Consider any five points in the interior of square $S$ of side length $1$. Prove that at least one of the distances between these points is less than $\sqrt{2} \slash 2.$ Can this constant be replaced by a smaller number?

2004 Paraguay Mathematical Olympiad, 4

Tags: square , trisection , geometry

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.

2010 Saudi Arabia BMO TST, 2

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

Let $ABC$ be an acute triangle and let $MNPQ$ be a square inscribed in the triangle such that $M ,N \in BC$, $P \in AC$, $Q \in AB$. Prove that $area \, [MNPQ] \le \frac12 area\, [ABC]$.

1998 Chile National Olympiad, 6

Tags: combinatorial geometry , combinatorics , square , equilateral

Given an equilateral triangle, cut it into four polygonal shapes so that, reassembled appropriately, these figures form a square.

2019 Novosibirsk Oral Olympiad in Geometry, 7

Tags: acute , square , geometry

Cut a square into eight acute-angled triangles.

2023 Novosibirsk Oral Olympiad in Geometry, 3

Tags: geometry , rectangle , square

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]

2001 Chile National Olympiad, 2

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

2021 Iranian Geometry Olympiad, 2

Tags: inscribed , geometry , square , area

Points $K, L, M, N$ lie on the sides $AB, BC, CD, DA$ of a square $ABCD$, respectively, such that the area of $KLMN$ is equal to one half of the area of $ABCD$. Prove that some diagonal of $KLMN$ is parallel to some side of $ABCD$. [i]Proposed by Josef Tkadlec - Czech Republic[/i]

1991 All Soviet Union Mathematical Olympiad, 555

Tags: geometry , square , circumcircle

$ABCD$ is a square. The points $X$ on the side $AB$ and $Y$ on the side $AD$ are such that $AX\cdot AY = 2 BX\cdot DY$. The lines $CX$ and $CY$ meet the diagonal $BD$ in two points. Show that these points lie on the circumcircle of $AXY$.

2023 India Regional Mathematical Olympiad, 1

Tags: number theory , divisibility , square

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 $(a, b, c, d) \in S$.

2007 German National Olympiad, 2

Tags: square , set , product , number theory

Let $A$ be the set of odd integers $\leq 2n-1.$ For a positive integer $m$, let $B=\{a+m\,|\, a\in A \}.$ Determine for which positive integers $n$ there exists a positive integer $m$ such that the product of all elements in $A$ and $B$ is a square.

Kyiv City MO Juniors 2003+ geometry, 2018.9.51

Tags: geometry , square , angle

Given a circle $\Gamma$ with center at point $O$ and diameter $AB$. $OBDE$ is square, $F$ is the second intersection point of the line $AD$ and the circle $\Gamma$, $C$ is the midpoint of the segment $AF$. Find the value of the angle $OCB$.

2016 Auckland Mathematical Olympiad, 2

Tags: geometry , equal angles , square

In square $ABCD$, $\overline{AC}$ and $\overline{BD}$ meet at point $E$. Point $F$ is on $\overline{CD}$ and $\angle CAF = \angle FAD$. If $\overline{AF}$ meets $\overline{ED}$ at point $G$, and if $\overline{EG} = 24$ cm, then find the length of $\overline{CF}$.

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]

1996 Portugal MO, 1

Tags: geometry , square , right triangle

Consider a square on the hypotenuse of a right triangle $[ABC]$ (right at $B$). Prove that the line segment that joins vertex $B$ with the center of the square makes $45^o$ angles with legs of the triangle.