Determine all positive integers $n$ such that $n$ equals the square of the sum of the digits of $n$.

To obtain a square $P$ of side length $2$ cm divided into $4$ unit squares it is sufficient to draw $3$ squares: $P$ and another $2$ unit squares with a common vertex, as shown below: [img]https://cdn.artofproblemsolving.com/attachments/1/d/827516518871ec8ff00a66424f06fda9812193.png[/img] Find the minimum number of squares sufficient to obtain a square.of side length $n$ cm divided into $n^2$ unit squares ($n \ge 3$ is an integer).

One square is inscribed in a circle, and another square is circumscribed around the same circle so that its vertices lie on the extensions of the sides of the first (see figure). Find the angle between the sides of these squares. [img]https://3.bp.blogspot.com/-8eLBgJF9CoA/XTodHmW87BI/AAAAAAAAKY0/xsHTx71XneIZ8JTn0iDMHupCanx-7u4vgCK4BGAYYCw/s400/sharygin%2Boral%2B2017%2B10-11%2Bp1.png[/img]

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]

a) Given two squares $ABCD$ and $DEFG$, with point $E$ lying on the segment $CD$, and points$ F,G$ outside the square $ABCD$. Find the angle between lines $AE$ and $BF$. b) Two regular pentagons $OKLMN$ and $OPRST$ are given, and the point $P$ lies on the segment $ON$, and the points $R, S, T$ are outside the pentagon $OKLMN$. Find the angle between straight lines $KP$ and $MS$.

On the sides $AB$ and $AD$ of the square $ABCD$, the points $N$ and $P$ are selected, respectively, so that $PN = NC$, the point $Q$ Is a point on the segment $AN$ for which $\angle NCB = \angle QPN$. Prove that $\angle BCQ = \tfrac {1} {2} \angle PQA$.

On sides $AB$ and $BC$ of a square $ABCD$ the respective points $E$ and $F$ have been chosen so that $BE = BF$. Let $BN$ be the altitude in triangle $BCE$. Prove that $\angle DNF = 90$.

$ABCD$ is a square. $E$ is a point on the side $BC$ such that $BE =1/3 BC$, and $F$ is a point on the ray $DC$ such that $CF =1/2 DC$. Prove that the lines $AE$ and $BF$ intersect on the circumcircle of the square. [img]https://cdn.artofproblemsolving.com/attachments/e/d/09a8235d0748ce4479e21a3bb09b0359de54b5.png[/img]

In unit square $ABCD,$ the inscribed circle $\omega$ intersects $\overline{CD}$ at $M,$ and $\overline{AM}$ intersects $\omega$ at a point $P$ different from $M.$ What is $AP?$ $\textbf{(A) } \frac{\sqrt5}{12} \qquad \textbf{(B) } \frac{\sqrt5}{10} \qquad \textbf{(C) } \frac{\sqrt5}{9} \qquad \textbf{(D) } \frac{\sqrt5}{8} \qquad \textbf{(E) } \frac{2\sqrt5}{15}$

Circle inscribed in square $ABCD$ , is tangent to sides $AB$ and $CD$ at points $M$ and $K$ respectively. Line $BK$ intersects this circle at the point $L, X$ is the midpoint of $KL$. Find the angle $\angle MXK $.

$ABCD$ is any convex quadrilateral. Squares center $E, F, G, H$ are constructed on the outside of the edges $AB, BC, CD$ and $DA$ respectively. Show that $EG$ and $FH$ are equal and perpendicular.

A square $ABCD$ lies in the coordinate plane with its vertices $A$ and $C$ lying on different coordinate axes. Prove that one of the vertices $B$ or $D$ lies on the line $y = x$ and the other one on $y = -x$.

Prove that the number \[ 3 \underbrace{99\ldots9}_{2025} \underbrace{60\ldots01}_{2025} \] is a square of a positive integer.

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 $1$.

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.

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]

The squares of a chessboard have side $4$. What is the circumference of the largest circle that can be drawn entirely on the black squares of the board?

A point $O$ is chosen inside the square $ABCD$. The square $A'B'C'D'$ is the image of the square $ABCD$ under the homothety with center at point $O$ and coefficient $k> 1$ (points $A', B', C', D' $ are images of points $A, B, C, D$ respectively). Prove that the sum of the areas of the quadrilaterals $A'ABB'$ and $C'CDD'$ is equal to the sum of the areas quadrilaterals $B'BCC'$ and $D'DAA'$.

In a triangle $ABC$, right in $A$ and isosceles, let $D$ be a point on the side $AC$ ($A \ne D \ne C$) and $E$ be the point on the extension of $BA$ such that the triangle $ADE$ is isosceles. Let $P$ be the midpoint of segment $BD$, $R$ be the midpoint of the segment $CE$ and $Q$ the intersection point of $ED$ and $BC$. Prove that the quadrilateral $ARQP$ is a square.

Let $ABCD$ be a square and let $M$ be the midpoint of $BC$. Let $C ^ \prime$ be the reflection of $C$ wrt to $DM$. The parallel to $AB$ passing through $C ^ \prime$ intersects $AD$ at $R$ and $BC$ at $S$. Show that $$\frac {RC ^ \prime} {C ^\prime S} = \frac {3} {2}$$

Four masts stand on a flat horizontal piece of land at the vertices of a square $ABCD$. The height of the mast on $A$ is $7$ meters, of the mast on $B$ $13$ meters, and of the mast on $C$ $15$ meters. Within the square there is a point $P$ on the ground equidistant from each of the tops of these three masts. (a) What length must the sides of the square be at least for this to be possible? (b) The distance from $P$ to the top of the mast on $D$ is equal to the distance from$ P$ to each of the tops of the three other masts. Calculate the height of the mast at $D$.

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) $AP^2 \le \frac{AE \cdot AF}{2}$

A circle $K_1$ of radius $r_1 = 1\slash 2$ is inscribed in a semi-circle $H$ with diameter $AB$ and radius $1.$ A sequence of different circles $K_2, K_3, \ldots$ with radii $r_2, r_3, \ldots$ respectively are drawn so that for each $n\geq 1$, the circle $K_{n+1}$ is tangent to $H$, $K_n$ and $AB.$ Prove that $a_n = 1\slash r_n$ is an integer for each $n$, and that it is a perfect square for $n$ even and twice a perfect square for $n$ odd.

A square of side $5$ is divided into $25$ unit squares. Let $A$ be the set of the $16$ interior points of the initial square which are vertices of the unit squares. What is the largest number of points of $A$ no three of which form an isosceles right triangle?

Consider the square $ABCD$ and the point $E$ inside the angle $CAB$, such that $\angle BAE =15^o$, and the lines $BE$ and $BD$ are perpendicular. Prove that $AE = BD$.