Olympiad problems

2009 Kyiv Mathematical Festival, 4

Two convex polygons can be placed into a square with the side $1$ without intersection. Prove that at least one polygon has the perimeter that is less than or equal to $3,5$ .

Estonia Open Junior - geometry, 2002.1.1.

Tags: rectangle , square , geometry

A figure consisting of five equal-sized squares is placed as shown in a rectangle of size $7\times 8$ units. Find the side length of the squares. [img]https://cdn.artofproblemsolving.com/attachments/e/e/cbc2b7b0693949790c1958fb1449bdd15393d8.png[/img]

2016 Novosibirsk Oral Olympiad in Geometry, 3

Tags: square , geometry , angle

A square is drawn on a sheet of grid paper on the sides of the cells $ABCD$ with side $8$. Point $E$ is the midpoint of side $BC$, $Q$ is such a point on the diagonal $AC$ such that $AQ: QC = 3: 1$. Find the angle between straight lines $AE$ and $DQ$.

2023 Novosibirsk Oral Olympiad in Geometry, 7

Tags: geometry , square , college

Squares $ABCD$ and $BEFG$ are located as shown in the figure. It turned out that points $A, G$ and $E$ lie on the same straight line. Prove that then the points $D, F$ and $E$ also lie on the same line. [img]https://cdn.artofproblemsolving.com/attachments/4/2/9faf29a399d3a622c84f5d4a3cfcf5e99539c0.png[/img]

1995 Tournament Of Towns, (443) 3

Tags: geometry , square , angle

Suppose $L$ is the circle inscribed in the square $T_1$, and $T_2$ is the square inscribed in $L$, so that vertices of $T_1$ lie on the straight lines containing the sides of $T_2$. Find the angles of the convex octagon whose vertices are at the tangency points of $L$ with the sides of $T_1$ and at the vertices of $T_2$. (S Markelov)

1983 Tournament Of Towns, (044) 1

Tags: square , geometry

Inside square $ABCD$ consider a point $M$. Prove that the points of intersection of the medians of triangles $ABM, BCM, CDM$ and $DAM$ form a square. (V Prasolov)

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

2017 Romania National Olympiad, 3

Tags: geometry , parallelogram , square

In the square $ABCD$ denote by $M$ the midpoint of the side $[AB]$, with $P$ the projection of point $B$ on the line $CM$ and with $N$ the midpoint of the segment $[CP]$, Bisector of the angle $DAN$ intersects the line $DP$ at point $Q$. Show that the quadrilateral $BMQN$ is a parallelogram.

Oliforum Contest V 2017, 2

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

Find all quadrilaterals which can be covered (without overlappings) with squares with side $ 1$ and equilateral triangles with side $ 1$. (Emanuele Tron)

2017 Bundeswettbewerb Mathematik, 4

Tags: square , cube , number theory

We call a positive integer [i]heinersch[/i] if it can be written as the sum of a positive square and positive cube. Prove: There are infinitely many heinersch numbers $h$, such that $h-1$ and $h+1$ are also heinersch.

1990 All Soviet Union Mathematical Olympiad, 514

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

2010 Contests, 1b

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

The edges of the square in the figure have length $1$. Find the area of the marked region in terms of $a$, where $0 \le a \le 1$. [img]https://cdn.artofproblemsolving.com/attachments/2/2/f2b6ca973f66c50e39124913b3acb56feff8bb.png[/img]

2019 Novosibirsk Oral Olympiad in Geometry, 4

Tags: isosceles , collinear , square , geometry

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]

2018 Hanoi Open Mathematics Competitions, 6

Tags: geometry , perimeter , hexagon , square

In the below figure, there is a regular hexagon and three squares whose sides are equal to $4$ cm. Let $M,N$, and $P$ be the centers of the squares. The perimeter of the triangle $MNP$ can be written in the form $a + b\sqrt3$ (cm), where $a, b$ are integers. Compute the value of $a + b$. [img]https://cdn.artofproblemsolving.com/attachments/e/8/5996e994d4bbed8d3b3269d3e38fc2ec5d2f0b.png[/img]

2017 Greece Junior Math Olympiad, 1

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

Let $ABCD$ be a square of side $a$. On side $AD$ consider points $E$ and $Z$ such that $DE=a/3$ and $AZ=a/4$. If the lines $BZ$ and $CE$ intersect at point $H$, calculate the area of the triangle $BCH$ in terms of $a$.

2024 Polish Junior MO Finals, 2

Tags: combinatorics , square

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.

1991 IMO Shortlist, 22

Tags: algebra , polynomial , square , cubic , cubic equation

Real constants $ a, b, c$ are such that there is exactly one square all of whose vertices lie on the cubic curve $ y \equal{} x^3 \plus{} ax^2 \plus{} bx \plus{} c.$ Prove that the square has sides of length $ \sqrt[4]{72}.$

Durer Math Competition CD Finals - geometry, 2022.C3

Tags: tangent , equilateral , square , geometry

To the exterior of side $AB$ of square $ABCD$, we have drawn the regular triangle $ABE$. Point $A$ reflected on line $BE$ is $F$, and point $E$ reflected on line $BF$ is $G$. Let the perpendicular bisector of segment $FG$ meet segment $AD$ at $X$. Show that the circle centered at $X$ with radius $XA$ touches line$ FB$.

Novosibirsk Oral Geo Oly VIII, 2020.1

Tags: geometry , square

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]

2002 Junior Balkan Team Selection Tests - Romania, 2

Tags: square , geometry , fixed , area , diagonal

The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ meet at $O$. Let $m$ be the measure of the acute angle formed by these diagonals. A variable angle $xOy$ of measure $m$ intersects the quadrilateral by a convex quadrilateral of constant area. Prove that $ABCD$ is a square.

1967 IMO Shortlist, 4

Tags: geometry , perimeter , square , dissection , triangulation

The square $ABCD$ has to be decomposed into $n$ triangles (which are not overlapping) and which have all angles acute. Find the smallest integer $n$ for which there exist a solution of that problem and for such $n$ construct at least one decomposition. Answer whether it is possible to ask moreover that (at least) one of these triangles has the perimeter less than an arbitrarily given positive number.

1962 Poland - Second Round, 5

Tags: geometry , square , equilateral , locus

In the plane there is a square $ Q $ and a point $ P $. The point $ K $ runs through the perimeter of the square $ Q $. Find the locus of the vertex $ M $ of the equilateral triangle $ KPM $.

Denmark (Mohr) - geometry, 2000.1

Tags: midpoint , square , area

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]

2008 District Olympiad, 2

Tags: geometry , square , perpendicular

Consider the square $ABCD$ and $E \in (AB)$. The diagonal $AC$ intersects the segment $[DE]$ at point $P$. The perpendicular taken from point $P$ on $DE$ intersects the side $BC$ at point $F$. Prove that $EF = AE + FC$.

2003 Paraguay Mathematical Olympiad, 5

Tags: geometry , square , area

In a square $ABCD$, $E$ is the midpoint of side $BC$. Line $AE$ intersects line $DC$ at $F$ and diagonal $BD$ at $G$. If the area $(EFC) = 8$, determine the area $(GBE)$.