Let $S$ denote a square of the side-length $7$, and let eight squares of the side-length $3$ be given. Show that $S$ can be covered by those eight small squares.

Given is the square $ABCD$. Point $E$ lies on the diagonal $AC$, where $AE> EC$. On the side $AB$, a different point from $B$ has been selected for which $EF = DE$. Prove that $\angle DEF = 90^o$.

Let $ABCD$ be a square. The points $N$ and $P$ are chosen on the sides $AB$ and $AD$ respectively, such that $NP=NC$. The point $Q$ on the segment $AN$ is such that that $\angle QPN=\angle NCB$. Prove that $\angle BCQ=\frac{1}{2}\angle AQP$.

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]

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)

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]

Four different points $A,B,C$ and $D$ lie in a plane. No three of these points lie on a single straight line. Describe the construction of a square $PQRS$ such that on each of the sides of $PQRS$, or the extensions , lies one of the points $A, B, C$ and $D$.

Let $ABC$ be a right triangle with right angle at $B$. Let $ACDE$ be a square drawn exterior to triangle $ABC$. If $M$ is the center of this square, find the measure of $\angle MBC$.

Let $ABCD$ be a square, and let $E$ be an internal point on side $AD$. Let $F$ be the foot of the perpendicular from $B$ to $CE$. Suppose $G$ is a point such that $BG = FG$, and the line through $G$ parallel to $BC$ passes through the midpoint of $EF$. Prove that $AC < 2 \cdot FG$.

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]

A square is given. Lines divide it into $n$ polygons. What is he the largest possible sum of the internal angles of all polygons?

The points lie on three parallel lines with distances as indicated in the figure $A, B$ and $C$ such that square $ABCD$ is a square. Find the area of this square. [img]https://1.bp.blogspot.com/-xeFvahqPVyM/XzcFfB0-NfI/AAAAAAAAMYA/SV2XU59uBpo_K99ZBY43KSSOKe-veOdFQCLcBGAsYHQ/s0/1998%2BMohr%2Bp3.png[/img]

Let $S$ denote a square of side-length $7$, and let eight squares with side-length $3$ be given. Show that it is impossible to cover $S$ by those eight small squares with the condition: an arbitrary side of those (eight) squares is either coincided, parallel, or perpendicular to others of $S$ .

A square is dissected into rectangles with sides parallel to the sides of the square. For each of these rectangles, the ratio of its shorter side to its longer side is considered. Show that the sum of all these ratios is at least $1$.

Give a square of side $1$. Show that for each finite set of points of the sides of the square you can find a vertex of the square with the following property: the arithmetic mean of the squares of the distances from this vertex to the points of the set is greater than or equal to $3/4$.

Show that for any non-negative integer $n$, the number $2^{2n+1}$ cannot be expressed as a sum of four non-zero square numbers.

Construct a square, if you know its center and two points that lie on adjacent sides.

Points $E$ and $F$ are the midpoints of sides $BC$ and $CD$ of square $ABCD$, respectively. Lines $AE$ and $BF$ meet at point $P$. Prove that $\angle PDA = \angle AED$.

Prove that there exist infinitely many integers $n$ such that $n, n + 1, n + 2$ are each the sum of two squares of integers.

Let $E$ be an arbitrary point on the side $BC$ of the square $ABCD$. Prove that the inscribed circles of triangles $ABE$, $CDE$, $ADE$ have a common tangent.

Given the rectangle $ABCD$. The points $E ,F$ lie on the segments $(BC) , (DC)$ respectively, such that $\angle DAF = \angle FAE$. Proce that if $DF + BE = AE$ then $ABCD$ is square.

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]

Is it possible to place within a square an equilateral triangle whose area is larger than $9/ 20$ of the area of the square?

Suppose we have a square $ABCD$ and a point $S$ in the interior of this square. Under homothety with centre $S$ and ratio of magnification $k > 1$, this square becomes another square $A'B'C'D'$. Prove that the sum of the areas of the two quadrilaterals $A'ABB'$ and $C'CDD'$ are equal to the sum of the areas of the two quadrilaterals $B'BCC'$ and $D'DAA'$. [asy] unitsize(3 cm); pair[] A, B, C, D; pair S; A[1] = (0,1); B[1] = (0,0); C[1] = (1,0); D[1] = (1,1); S = (0.3,0.6); A[0] = interp(S,A[1],2/3); B[0] = interp(S,B[1],2/3); C[0] = interp(S,C[1],2/3); D[0] = interp(S,D[1],2/3); draw(A[0]--B[0]--C[0]--D[0]--cycle); draw(A[1]--B[1]--C[1]--D[1]--cycle); draw(A[1]--S, dashed); draw(B[1]--S, dashed); draw(C[1]--S, dashed); draw(D[1]--S, dashed); dot("$A$", A[0], N); dot("$B$", B[0], SE); dot("$C$", C[0], SW); dot("$D$", D[0], SE); dot("$A'$", A[1], NW); dot("$B'$", B[1], SW); dot("$C'$", C[1], SE); dot("$D'$", D[1], NE); dot("$S$", S, dir(270)); [/asy]

Let a square $ABCD$ with sides of length $1$ be given. A point $X$ on $BC$ is at distance $d$ from $C$, and a point $Y$ on $CD$ is at distance $d$ from $C$. The extensions of: $AB$ and $DX$ meet at $P$, $AD$ and $BY$ meet at $Q, AX$ and $DC$ meet at $R$, and $AY$ and $BC$ meet at $S$. If points $P, Q, R$ and $S$ are collinear, determine $d$.