Found problems: 467
2020 Yasinsky Geometry Olympiad, 2
Let $ABCD$ be a square, point $E$ be the midpoint of the side $BC$. On the side $AB$ mark a point $F$ such that $FE \perp DE$. Prove that $AF + BE = DF$.
(Ercole Suppa, Italy)
2018 Romania National Olympiad, 2
In the square $ABCD$ the point $E$ is located on the side $[AB]$, and $F$ is the foot of the perpendicular from $B$ on the line $DE$. The point $L$ belongs to the line $DE$, such that $F$ is between $E$ and $L$, and $FL = BF$. $N$ and $P$ are symmetric of the points $A , F$ with respect to the lines $DE, BL$, respectively. Prove that:
a) The quadrilateral $BFLP$ is square and the quadrilateral $ALND$ is rhombus.
b) The area of the rhombus $ALND$ is equal to the difference between the areas of the squares $ABCD$ and $BFLP$.
Ukraine Correspondence MO - geometry, 2016.11
Inside the square $ABCD$ mark the point $P$, for which $\angle BAP = 30^o$ and $\angle BCP = 15^o$. The point $Q$ was chosen so that $APCQ$ is an isosceles trapezoid ($PC\parallel AQ$). Find the angles of the triangle $CAM$, where $M$ is the midpoint of $PQ$.
May Olympiad L2 - geometry, 1996.4
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 $\angle FQC$?
2005 Chile National Olympiad, 1
In the center of the square of side $1$ shown in the figure is an ant. At one point the ant starts walking until it touches the left side $(a)$, then continues walking until it reaches the bottom side $(b)$, and finally returns to the starting point. Show that, regardless of the path followed by the ant, the distance it travels is greater than the square root of $2$.
[asy]
unitsize(2 cm);
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
label("$a$", (0,0.5), W);
label("$b$", (0.5,0), S);
dot((0.5,0.5));
[/asy]
2019 Tuymaada Olympiad, 6
Prove that the expression
$$ (1^4+1^2+1)(2^4+2^2+1)\dots(n^4+n^2+1)$$
is not square for all $n \in \mathbb{N}$
1999 Tournament Of Towns, 5
A square is cut into $100$ rectangles by $9$ straight lines parallel to one of the sides and $9$ lines parallel to another. If exactly $9$ of the rectangles are actually squares, prove that at least two of these $9$ squares are of the same size .
(V Proizvolov)
2008 Switzerland - Final Round, 5
Let $ABCD$ be a square with side length $1$.
Find the locus of all points $P$ with the property $AP\cdot CP + BP\cdot DP = 1$.
2020 New Zealand MO, 2
Let $ABCD$ be a square and let $X$ be any point on side $BC$ between $B$ and $C$. Let $Y$ be the point on line $CD$ such that $BX = YD$ and $D$ is between $C$ and $Y$ . Prove that the midpoint of $XY$ lies on diagonal $BD$.
1998 Chile National Olympiad, 6
Given an equilateral triangle, cut it into four polygonal shapes so that, reassembled appropriately, these figures form a square.
1983 Bulgaria National Olympiad, Problem 4
Find the smallest possible side of a square in which five circles of radius $1$ can be placed, so that no two of them have a common interior point.
2017 Romania National Olympiad, 3
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.
1942 Putnam, B1
A square of side $2a$, lying always in the first quadrant of the $xy$-plane, moves so that two consecutive vertices
are always on the $x$- and $y$-axes respectively. Prove that a point within or on the boundary of the square will in general describe a portion of a conic. For what points of the square does this locus degenerate?
2011 Bundeswettbewerb Mathematik, 1
Prove that you can't split a square into finitely many hexagons, whose inner angles are all less than $180^o$.
1991 All Soviet Union Mathematical Olympiad, 548
A polygon can be transformed into a new polygon by making a straight cut, which creates two new pieces each with a new edge. One piece is then turned over and the two new edges are reattached. Can repeated transformations of this type turn a square into a triangle?
1950 Polish MO Finals, 2
We are given two concentric circles, Construct a square whose two vertices lie on one circle and the other two on the other circle.
1958 February Putnam, B2
Prove that the product of four consecutive positive integers cannot be a perfect square or cube.
1998 Switzerland Team Selection Test, 4
Find all numbers $n$ for which it is possible to cut a square into $n$ smaller squares.
Novosibirsk Oral Geo Oly VIII, 2020.1
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]
1986 All Soviet Union Mathematical Olympiad, 426
Find all the natural numbers equal to the square of its divisors number.
2021 Iranian Geometry Olympiad, 2
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]
1998 Tuymaada Olympiad, 7
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 IX, 2016.4
The two angles of the squares are adjacent, and the extension of the diagonals of one square intersect the diagonal of another square at point $O$ (see figure). Prove that $O$ is the midpoint of $AB$.
[img]https://cdn.artofproblemsolving.com/attachments/7/8/8daaaa55c38e15c4a8ac7492c38707f05475cc.png[/img]
Oliforum Contest V 2017, 2
Find all quadrilaterals which can be covered (without overlappings) with squares with side $ 1$ and equilateral triangles with side $ 1$.
(Emanuele Tron)
1988 Dutch Mathematical Olympiad, 4
Given is an isosceles triangle $ABC$ with $AB = 2$ and $AC = BC = 3$. We consider squares where $A, B$ and $C$ lie on the sides of the square (so not on the extension of such a side). Determine the maximum and minimum value of the area of such a square. Justify the answer.