Let $ABCD$ be a rectangle with $AB\ge BC$ Point $M$ is located on the side $(AD)$, and the perpendicular bisector of $[MC]$ intersects the line $BC$ at the point $N$. Let ${Q} =MN\cup AB$ . Knowing that $\angle MQA= 2\cdot \angle BCQ $, show that the quadrilateral $ABCD$ is a square.

Let $ABCD$ be a quadrilateral that can be inscribed in a circle whose diagonals are perpendicular. Denote by $P$ and $Q$ the feet of the perpendiculars through $D$ and $C$ respectively on the line $AB$, $X$ is the intersection point of the lines $AC$ and $DP$, $Y$ is the intersection point of the lines $BD$ and $CQ$. Show that $XY CD$ is a rhombus.

Cut a square into eight acute-angled triangles.

In the interior of a square $ABCD$ we construct the equilateral triangles $ABK, BCL, CDM, DAN.$ Prove that the midpoints of the four segments $KL, LM, MN, NK$ and the midpoints of the eight segments $AK, BK, BL, CL, CM, DM, DN, AN$ are the 12 vertices of a regular dodecagon.

$ABC$ is a right-angled triangle. A square $ABDE$ is constructed on the opposite side of the hypothenuse $AB$ from $C$. The bisector of $\angle C$ cuts $DE$ at $F$. If $AC = 1$ and $BC = 3$, compute $\frac{DF}{EF}$. (A Blinkov)

In the square $ABCD$ the points $E$ and $F$ are marked on the sides $AB$ and $BC$ respectively, in such a way that $EB = 2AE$ and $BF = FC$. Let $G$ be the intersection between $DF$ and $EC$. If the side of the square equals $10$, what is the distance from point $G$ to side $AB$?

A square with sides $1$ and four circles of radius $1$ considered each having a vertex of have the square as the center. Find area of the shaded part (see figure). [img]https://cdn.artofproblemsolving.com/attachments/b/6/6e28d94094d69bac13c2702853ac2c906a80a1.png[/img]

Let $ABCD$ be a square. Point $P, Q, R, S$ are chosen on the sides $AB$, $BC$, $CD$, $DA$, respectively, such that $AP + CR \ge AB \ge BQ + DS$. Prove that $$area \,\, (PQRS) \le \frac12 \,\, area \,\, (ABCD)$$ and determine all cases when equality holds.

Each point on the boundary of a square has to be colored in one color. Consider all right triangles with the vertices on the boundary of the square. Determine the least number of colors for which there is a coloring such that no such triangle has all its vertices of the same color.

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?

Let $P$ be a point inside a square $ABCD,$ such that $\angle BPC = 135^\circ $ and the area of triangle $ADP$ is twice as much as the area of triangle $PCD.$ Find $\frac {AP}{DP}.$ [i]Proposed by Andria Gvaramia, Georgia [/i]

An interior $P$ point to a square $ABCD$ is such that $PA = a, PB = b$ and $PC = b + c$, where the numbers $a, b$ and $c$ satisfy the relationship $a^2 = b^2 + c^2$. Prove that the angle $BPC$ is right.

Cut a square into eight acute-angled triangles.

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 $n \in \mathbb{N}$

Consider $ABCD$ a square of side $1$. Points $P,Q,R,S$ are chosen on sides $AB$, $BC$, $CD$ and $DA$ respectively such that $|AP| = |BQ| =|CR| =|DS| = a$, with $a < 1$. The segments $AQ$, $BR$, $CS$ and $DP$ are drawn. Calculate the area of the quadrilateral that is formed in the center of the figure. [asy] unitsize(1 cm); pair A, B, C, D, P, Q, R, S; A = (0,3); B = (0,0); C = (3,0); D = (3,3); P = (0,2); Q = (1,0); R = (3,1); S = (2,3); draw(A--B--C--D--cycle); draw(A--Q); draw(B--R); draw(C--S); draw(D--P); label("$A$", A, NW); label("$B$", B, SW); label("$C$", C, SE); label("$D$", D, NE); label("$P$", P, W); label("$Q$", Q, dir(270)); label("$R$", R, E); label("$S$", S, N); label("$a$", (A + P)/2, W); label("$a$", (B + Q)/2, dir(270)); label("$a$", (C + R)/2, E); label("$a$", (D + S)/2, N); [/asy]

Let $x, y, z$ be positive integers such that $xy=z^2+2$. Prove that there exist integers $a, b, c, d$ such that the following equalities are satisfied: \begin{eqnarray*} x=a^2+2b^2\\ y=c^2+d^2\\ z=ac+2bd\\ \end{eqnarray*} [i]Proposed by Isaac Jiménez[/i]

In the following figure, $ABCD$ is a square of unit length and $P, Q$ are points on $AD$ and $AB$ respectively. Find $\angle PCQ$ if $|AP| + |AQ| + |PQ| = 2$. [img]https://cdn.artofproblemsolving.com/attachments/2/c/2f40db978c1d3fcbc0161f874b5cbec926058e.png[/img]

Consider a square on the hypotenuse of a right triangle $[ABC]$ (right at $B$). Prove that the line segment that joins vertex $B$ with the center of the square makes $45^o$ angles with legs of the triangle.

Seven polygons of area $1$ lie in the interior of a square with side length $2$. Show that there are two of these polygons whose intersection has an area of at least $1\slash 7.$

The figure shows a square and a circle with a common center $O$, with equal areas of striped shapes. Find the value of $\cos a$. [img]https://2.bp.blogspot.com/-7uwa0H42ELg/XnmsSoPMgcI/AAAAAAAALgk/pHNBqtbsdKgMhcvIRYLm_8JRpOeIYcUeACK4BGAYYCw/s400/96%2Bestonia%2Bopen%2Bs2.4.png[/img]

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. Find the locus of the midpoint of the square.

Square $ABCD$ with sides of length $4$ is a base of a cuboid $ABCDA'B'C'D'$. Side edges $AA'$, $BB'$, $CC'$, $DD'$ of this cuboid have length $7$. Points $K, L, M$ lie respectively on line segments $AA'$, $BB'$, $CC'$, and $AK = 3$, $BL = 2$, $CM = 5$. Plane passing through points $K, L, M$ cuts cuboid on two blocks. Calculate volumes of these blocks.

In a square $ABCD$, $K$ is a point on the side $BC$ and the bisector of $\angle KAD$ cuts the side $CD$ at the point $M$. Prove that the length of segment $AK$ is equal to the sum of the lengths of segments $DM$ and $BK$. (Folklore)

Determine all triplets of prime numbers $p<q<r$, such that $p+q=r$ and $(r-p)(q-p)-27p$ is a square.

Let $K, L, M$, and $N$ be the midpoints of $CD,DA,AB$ and $BC$ of a square $ABCD$ respectively. Find the are of the triangles $AKB, BLC, CMD$ and $DNA$ if the square $ABCD$ has area $1$.