Found problems: 467
Novosibirsk Oral Geo Oly VIII, 2016.3
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 $DQ$.
2016 Saint Petersburg Mathematical Olympiad, 2
On a $300 \times 300$ board, several rooks are placed that beat the entire board. Within this case, each rook beats no more than one other rook. At what least $k$, it is possible to state that there is at least one rook in each $k\times k$ square ?
Novosibirsk Oral Geo Oly VIII, 2019.7
The square was cut into acute -angled triangles. Prove that there are at least eight of them.
1935 Moscow Mathematical Olympiad, 005
Given three parallel straight lines. Construct a square three of whose vertices belong to these lines.
2001 Cuba MO, 2
Let $ABCD$ be a square. On the sides $BC$ and $CD$ the points $M$ and $K$ respectively, so that $MC = KD$. Let $P$ the intersection point of of segments $MD$ and $BK$. Prove that $AP \perp MK$.
Denmark (Mohr) - geometry, 1997.2
Two squares, both with side length $1$, are arranged so that one has one vertex in the center of the other. Determine the area of the gray area.
[img]https://1.bp.blogspot.com/-xt3pe0rp1SI/XzcGLgEw1EI/AAAAAAAAMYM/vFKxvvVuLvAJ5FO_yX315X3Fg_iFaK2fACLcBGAsYHQ/s0/1997%2BMohr%2Bp2.png[/img]
2021 Brazil National Olympiad, 4
A set \(A\) of real numbers is framed when it is bounded and, for all \(a, b \in A\), not necessarily distinct, \((a-b)^{2} \in A\). What is the smallest real number that belongs to some framed set?
2016 Regional Olympiad of Mexico Northeast, 4
Let $ABCD$ be a square. Let $P$ be a point on the semicircle of diameter $AB$ outside the square. Let $M$ and $N$ be the intersections of $PD$ and $PC$ with $AB$, respectively. Prove that $MN^2 = AM \cdot BN$.
2002 All-Russian Olympiad Regional Round, 8.6
Each side of the convex quadrilateral was continued into both sides and on all eight extensions set aside equal segments. It turned out that the resulting $8$ points are the outer ends of the construction the given segments are different and lie on the same circle. Prove that the original quadrilateral is a square.
2013 May Olympiad, 3
Let $ABCD$ be a square of side paper $10$ and $P$ a point on side $BC$. By folding the paper along the $AP$ line, point $B$ determines the point $Q$, as seen in the figure. The line $PQ$ cuts the side $CD$ at $R$. Calculate the perimeter of the triangle $ PCR$
[img]https://3.bp.blogspot.com/-ZSyCUznwutE/XNY7cz7reQI/AAAAAAAAKLc/XqgQnjm8DQYq6Q7fmCAKJwKt3ihoL8AuQCK4BGAYYCw/s400/may%2B2013%2Bl1.png[/img]
2022 Durer Math Competition Finals, 1
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$.
1973 Chisinau City MO, 66
If $A$ and $B$ are points of the plane, then by $A * B$ we denote a point symmetric to $A$ with respect to $B$. Is it possible, by applying the operation $*$ several times, to obtain from the three vertices of a given square its fourth vertex?
2011 Junior Balkan Team Selection Tests - Romania, 2
We consider an $n \times n$ ($n \in N, n \ge 2$) square divided into $n^2$ unit squares. Determine all the values of $k \in N$ for which we can write a real number in each of the unit squares such that the sum of the $n^2$ numbers is a positive number, while the sum of the numbers from the unit squares of any $k \times k$ square is a negative number.
2016 Novosibirsk Oral Olympiad in Geometry, 3
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 $DQ$.
1981 Czech and Slovak Olympiad III A, 3
Let $ABCD$ be a unit square. Consider an equilateral triangle $XYZ$ with $X,Y$ as (inner or boundary) points of the square. Determine the locus $M$ of vertices $Z$ of all these triangles $XYZ$ and compute the area of $M.$
2015 Estonia Team Selection Test, 2
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$^2$
1966 Czech and Slovak Olympiad III A, 3
A square $ABCD,AB=s=1$ is given in the plane with its center $S$. Furthermore, points $E,F$ are given on the rays opposite to $CB,DA$, respectively, $CE=a,DF=b$. Determine all triangles $XYZ$ such that $X,Y,Z$ lie in this order on segments $CD,AD,BC$ and $E,S,F$ lie on lines $XY,YZ,ZX$ respectively. Discuss conditions of solvability in terms of $a,b,s$ and unknown $x=CX$.
1984 Tournament Of Towns, (068) T2
A village is constructed in the form of a square, consisting of $9$ blocks , each of side length $\ell$, in a $3 \times 3$ formation . Each block is bounded by a bitumen road . If we commence at a corner of the village, what is the smallest distance we must travel along bitumen roads , if we are to pass along each section of bitumen road at least once and finish at the same corner?
(Muscovite folklore)
2017 Czech-Polish-Slovak Match, 2
Each of the ${4n^2}$ unit squares of a ${2n \times 2n}$ board ${(n \ge 1) }$ has been colored blue or red. A set of four different unit squares of the board is called [i]pretty [/i]if these squares can be labeled ${A,B,C,D}$ in such a way that ${A}$ and ${B}$ lie in the same row, ${C}$ and ${D}$ lie in the same row, ${A}$ and ${C}$ lie in the same column, ${B}$ and ${D}$ lie in the same column, ${A}$ and ${D}$ are blue, and ${B}$ and ${C}$ are red. Determine the largest possible number of different [i]pretty [/i]sets on such a board.
(Poland)
2019 Novosibirsk Oral Olympiad in Geometry, 7
The square was cut into acute -angled triangles. Prove that there are at least eight of them.
2017 Bundeswettbewerb Mathematik, 4
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.
May Olympiad L1 - geometry, 2019.4
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.
2010 Saudi Arabia BMO TST, 2
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 $area \, [MNPQ] \le \frac12 area\, [ABC]$.
Novosibirsk Oral Geo Oly IX, 2020.2
A $2 \times 2$ square was cut out of a sheet of grid paper. Using only a ruler without divisions and without going beyond the square, divide the diagonal of the square into $6$ equal parts.
1996 Cono Sur Olympiad, 1
In the following figure, the largest square is divided into two squares and three rectangles, as shown:
The area of each smaller square is equal to $a$ and the area of each small rectangle is equal to $b$. If $a+b=24$ and the root square of $a$ is a natural number, find all possible values for the area of the largest square.
[img]https://cdn.artofproblemsolving.com/attachments/f/a/0b424d9c293889b24d9f31b1531bed5081081f.png[/img]