Consider the three squares indicated in the figure. Show that if the lengths of the sides of the smaller square and the square greater are integers, then adding to the area of the smallest square the area of the inclined square, a perfect square is obtained. [img]https://1.bp.blogspot.com/-B0QdvZIjOLw/X4URvs3C0ZI/AAAAAAAAMmw/S5zMpPBXBn8Jj39d-OZVtMRUDn3tXbyWgCLcBGAsYHQ/s0/2002%2Bportugal%2Bp5.png[/img]

Let the diagonals of the square $ABCD$ intersect at $S$ and let $P$ be the midpoint of $AB$. Let $M$ be the intersection of $AC$ and $PD$ and $N$ the intersection of $BD$ and $PC$. A circle is incribed in the quadrilateral $PMSN$. Prove that the radius of the circle is $MP- MS$.

Given in the plane the square $ABCD$, the square $A_1B_1C_1D_1$, smaller than the first, and a quadrilateral $PQRS$ that satisfy the following conditions $\bullet$ $ABCD$ and $A_1B_1C_1D_1$ have a common center and respectively parallel sides. $\bullet$$P$, $Q$, $R$, $S$ belong one to each side of the square $ABCD$. $\bullet$ $A_1$, $B_1$, $C_1$, $D_1$ belong one to each side of the quadrilateral $PQRS$. Prove that $PQRS$ is a square.

An equilateral triangle $BDE$ is constructed on the diagonal $BD$ of the square $ABCD$, and the point $C$ is located inside the triangle $BDE$. Let $M$ be the midpoint of $BE$. Find the angle between the lines $MC$ and $DE$. (Dmitry Shvetsov)

Prove that for any integer $n$ greater than $5$, a square can be divided into $n$ squares.

$ABC$ 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)

Let $ABCD$ be a rhombus. Eight additional points $X_1$, $X_2$, $Y_1$, $Y_2$, $Z_1$, $Z_2$, $W_1$, $W_2$ were chosen so that the quadrilaterals $AX_1BX_2$, $BY_1CY_2$, $CZ_1DZ_2$, $DW_1AW_2$ are squares. Prove that the eight new points lie on two straight lines.

In a square of side $10$ cm , the vertices are joined to the midpoints on the opposite sides, as shown in the figure. How much does the area of the colored region measure? [img]https://1.bp.blogspot.com/-bHrc1Nu0PQI/X4KaJysLAcI/AAAAAAAAMk0/LLGv1fotQO0Tk1AXqQymG_nNdpyWcbjyACLcBGAsYHQ/s109/2019%2BPortugal%2Bp1.png[/img]

Consider in a square $[ABCD]$ a point $E$ on the side $AB$, different from $A$ and $B$. On the side $BC$ consider the point $F$ such that $\angle AED = \angle DEF$ . Prove that $EF = AE + FC$.

The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ meet at $O$. Let $m$ be the measure of the acute angle formed by these diagonals. A variable angle $xOy$ of measure $m$ intersects the quadrilateral by a convex quadrilateral of constant area. Prove that $ABCD$ is a square.

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 $DQ$.

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]

Miguel has a square piece of paper $ABCD$ that he folded along a line $EF$, $E$ on $AB$, and $F$ on $CD$. This fold sent $A$ to point $A'$ on $BC$, distinct from $B$ and $C$. Also, it brought $D$ to point $D'$. $G$ is the intersection of $A'D'$ and $DC$. Prove that the inradius of $GCA'$ is equal to the sum of the inradius of $D'GF$ and $A'BE$.

The numbers $1$ to $n^2$ are written in an n×n squared paper in the usual ordering. Any sequence of right and downwards steps from a square to an adjacent one (by side) starting at square $1$ and ending at square $n^2$ is called a path. Denote by $L(C)$ the sum of the numbers through which path $C$ goes. (a) For a fixed $n$, let $M$ and $m$ be the largest and smallest $L(C)$ possible. Prove that $M-m$ is a perfect cube. (b) Prove that for no $n$ can one find a path $C$ with $L(C ) = 1996$.

The side $AD$ of a square $ABCD$ is the base of an obtuse-angled isosceles triangle $AED$ with vertex $E$ lying inside the square. Let $AF$ be a diameter of the circumcircle of this triangle, and $G$ be a point on $CD$ such that $CG = DF$. Prove that angle $BGE$ is less than half of angle $AED$.

$4$ equilateral triangles of side length $1$ are drawn on the interior of a unit square, each one of which shares a side with one of the $4$ sides of the unit square. What is the common area enclosed by all $4$ equilateral triangles?

Let $R$, $S$, and $T$ be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the x-axis. The left edge of $R$ and the right edge of $S$ are on the $y$-axis, and $R$ contains $\frac{9}{4}$ as many lattice points as does $S$. The top two vertices of $T$ are in $R \cup S$, and $T$ contains $\frac{1}{4}$ of the lattice points contained in $R \cup S$. See the figure (not drawn to scale). [asy] //kaaaaaaaaaante314 size(8cm); import olympiad; label(scale(.8)*"$y$", (0,60), N); label(scale(.8)*"$x$", (60,0), E); filldraw((0,0)--(55,0)--(55,55)--(0,55)--cycle, yellow+orange+white+white); label(scale(1.3)*"$R$", (55/2,55/2)); filldraw((0,0)--(0,28)--(-28,28)--(-28,0)--cycle, green+white+white); label(scale(1.3)*"$S$",(-14,14)); filldraw((-10,0)--(15,0)--(15,25)--(-10,25)--cycle, red+white+white); label(scale(1.3)*"$T$",(3.5,25/2)); draw((0,-10)--(0,60),EndArrow(TeXHead)); draw((-34,0)--(60,0),EndArrow(TeXHead));[/asy] The fraction of lattice points in $S$ that are in $S \cap T$ is 27 times the fraction of lattice points in $R$ that are in $R \cap T$. What is the minimum possible value of the edge length of $R$ plus the edge length of $S$ plus the edge length of $T$? $\textbf{(A) }336\qquad\textbf{(B) }337\qquad\textbf{(C) }338\qquad\textbf{(D) }339\qquad\textbf{(E) }340$

Let $ABCD$ be a unit square and let $E$ and $F$ be points inside $ABCD$ such that the line containing $\overline{EF}$ is parallel to $\overline{AB}$. Point $E$ is closer to $\overline{AD}$ than point $F$ is to $\overline{AD}$. The line containing $\overline{EF}$ also bisects the square into two rectangles of equal area. Suppose $[AEF B] = [DEFC] = 2[AED] = 2[BFC]$. The length of segment $\overline{EF}$ can be expressed as $m/n$ , where m and $n$ are relatively prime positive integers. Compute $m + n$.

Let $ABCD$ be a convex quadrilateral. Suppose that similar isosceles triangles $APB, BQC, CRD, DSA$ with the bases on the sides of $ABCD$ are constructed in the exterior of the quadrilateral such that $PQRS$ is a rectangle but not a square. Show that $ABCD$ is a rhombus.

Let $ABCD$ be a square with side length $1$. Find the locus of all points $P$ with the property $AP\cdot CP + BP\cdot DP = 1$.

Let $E$ and $F$ be the midpoints of the lines $AB$ and $DA$ of a square $ABCD$, respectively and let $G$ be the intersection of $DE$ with $CF$. Find the aspect ratio of sidelengths of the triangle $EGC$, $| EG | : | GC | : | CE |$.

Square $ABCD$ and circle $O$ intersect in eight points, forming four curvilinear triangles, $AEF , BGH , CIJ$ and $DKL$ ($EF , GH, IJ$ and $KL$ are arcs of the circle) . Prove that (a) The sum of lengths of $EF$ and $IJ$ equals the sum of the lengths of $GH$ and $KL$. (b) The sum of the perimeters of curvilinear triangles $AEF$ and $CIJ$ equals the sum of the perimeters of the curvilinear triangles $BGH$ and $DKL$. ( V . V . Proizvolov , Moscow)

Cut a square with three straight lines into three triangles and four quadrilaterals.

We consider a square $ABCD$ and a point $E$ on the segment $CD$. The bisector of $\angle EAB$ cuts the segment $BC$ in $F$. Prove that $BF + DE = AE$.

Consider a shape obtained from two equal squares with the same center. Prove that the ratio of the area of this shape to the perimeter does not change when the squares are rotated around their center. [img]http://4.bp.blogspot.com/-1AI4FxsNSr4/XovZWkvAwiI/AAAAAAAALvY/-kIzOgXB5rk3iIqGbpoKRCW9rwJPcZ3uQCK4BGAYYCw/s400/estonia%2B2000%2Bo.j.1.3.png[/img]