Found problems: 467
Is it possible to cut a square into five squares?
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 $DG$.
Triangle $XYZ$ is inside square $KLMN$ shown below so that its vertices each lie on three different sides of the square. It is known that:
$\bullet$ The area of square $KLMN$ is $1$.
$\bullet$ The vertices of the triangle divide three sides of the square up into these ratios:
$KX : XL = 3 : 2$
$KY : YN = 4 : 1$
$NZ : ZM = 2 : 3$
What is the area of the triangle $XYZ$? (Note that the sketch is not drawn to scale).
[img]https://cdn.artofproblemsolving.com/attachments/8/0/38e76709373ba02346515f9949ce4507ed4f8f.png[/img]
Is it possible to fill a rectangular table with black and white squares (only) so, that the number of black squares will equal to the number of white squares, and each row and each column will have more than $75\%$ squares of the same colour?
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.
The square $ABCD$ is divided into $8$ equal right triangles and the square $KLMN$, as shown in the figure. Find the area of the square $ABCD$ if $KL = 5, PS = 8$.
[img]https://1.bp.blogspot.com/-B2QIHvPcIx0/X4BhUTMDhSI/AAAAAAAAMj4/4h0_q1P6drskc5zSvtfTZUskarJjRp5LgCLcBGAsYHQ/s0/Yasinsky%2B2020%2Bp1.png[/img]
Let $ABCD$ be a square. Let $E, Z$ be points on the sides $AB, CD$ of the square respectively, such that $DE\parallel BZ$. Assume that the triangles $\triangle EAD, \triangle ZCB$ and the parallelogram $BEDZ$ have the same area.
If the distance between the parallel lines $DE$ and $BZ$ is equal to $1$, determine the area of the square.
Points $K$ and $L$ are taken on the sides $BC$ and $CD$ of a square $ABCD$ so that $\angle AKB = \angle AKL$. Find $\angle KAL$.
The center of a square of side length $ 1$ is placed uniformly at random inside a circle of radius $ 1$. Given that we are allowed to rotate the square about its center, what is the probability that the entire square is contained within the circle for some orientation of the square?
Can You fill the squares of the infinite cross-lined paper with integers so, that the sum of the numbers in every $4\times 6$ fields rectangle would be
a) $10$?
b) $1$?
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 $x=CX$.
On the sides $AB$, $BC$, $CD$, $DA$ of the square $ABCD$ points $P, Q, R, T$ are chosen such that $$\frac{AP}{PB}=\frac{BQ}{QC}=\frac{CR}{RD}=\frac{DT}{TA}=\frac12.$$
The segments $AR$, $BT$, $CP$, $DQ$ in the intersection form the quadrilateral $KLMN$ (see figure). [img]https://cdn.artofproblemsolving.com/attachments/f/c/587a2358734c300fe7082c520f90c91f872b49.png[/img]
a) Prove that $KLMN$ is a square.
b) Find the ratio of the areas of the squares $KLMN$ and $ABCD$.
(Alexander Shkolny)
Each diagonal of a convex quadrilateral divides it into two isosceles triangles. The two diagonals of the same quadrilateral divide it into four isosceles triangles. Must this quadrilateral be a square?
In the square $ABCD$ we consider $ E \in (AB)$, $ F \in (AD)$ and $EF \cap AC = \{P\}$. Show that:
a) $\frac{1}{AE} + \frac{1}{AF} = \frac{\sqrt2}{AP}$
b) $AP^2 \le \frac{AE \cdot AF}{2}$
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 $3,5$ .
Prove that the squares with sides $\frac{1}{1}, \frac{1}{2}, \frac{1}{3},\ldots$ may be put into the square with side $\frac{3}{2} $ in such a way that no two of them have any interior point in common.
The vertices of the square $ABCD$ are the centers of four circles, all of which pass through the center of the square. Prove that the intersections of the circles on the square $ABCD$ sides are vertices of a regular octagon.
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]
Cut a square into eight acute-angled triangles.
In the figure, $[ABCD]$ is a square of side $1$. The points $E, F, G$ and $H$ are such that $[AFB], [BGC], [CHD]$ and $[DEA]$ are right-angled triangles. Knowing that the circles inscribed in each of these triangles and the circle inscribed in the square $[EFGH]$ has all the same radius, what is the measure of the radius of the circles?
[img]https://1.bp.blogspot.com/-l37AEXa7_-c/X4KaJwe6HQI/AAAAAAAAMk4/14wvIipf26cRge_GqKSRwH32bp291vX4QCLcBGAsYHQ/s0/2018%2Bportugal%2Bp2.png[/img]
On the sides $BC$ and $CD$ of the square $ABCD$, the points $M$ and $N$ are selected in such a way that $\angle MAN= 45^o$. Using the segment $MN$, as the diameter, we constructed a circle $w$, which intersects the segments $AM$ and $AN$ at points $P$ and $Q$, respectively. Prove that the points $B, P$ and $Q$ lie on the same line.
Seven distinct points are given inside a square with side length $1.$ Together with the square's vertices, they form a set of $11$ points. Consider all triangles with vertices in $M.$
a) Show that at least one of these triangles has an area not exceeding $1\slash 16.$
b) Give an example in which no four of the seven points are on a line and none of the considered triangles has an area of less than $1\slash 16.$
Given an equilateral triangle, cut it into four polygonal shapes so that, reassembled appropriately, these figures form a square.
For the real number $a$ it holds that there is exactly one square whose vertices are all on the graph with the equation $y = x^3 + ax$. Find the side length of this square.
Construct a square knowing the sum of the diagonal and the side.