Found problems: 467
The edges of the square in the figure have length $1$.
Find the area of the marked region in terms of $a$, where $0 \le a \le 1$.
[img]https://cdn.artofproblemsolving.com/attachments/2/2/f2b6ca973f66c50e39124913b3acb56feff8bb.png[/img]
The center $M$ of the square $ABCD$ is reflected wrt $C$. This gives point $E$. The intersection of the circumcircle of the triangle $BDE$ with the line $AM$ is denoted by $S$. Show that $S$ bisects the distance $AM$.
(W. Janous, WRG Ursulinen, Innsbruck)
Given a square $ABCD$. Points $P$ and $Q$ are in the sides $[AB]$ and $[BC]$ respectively. $|BP|=|BQ|$. Let $H$ be the foot of the perpendicular from the point $B$ to the segment $[PC]$. Prove that the $\angle DHQ =90^o$ .
Each of the ${4n^2}$ unit squares of a ${2n \times 2n}$ board ${(n \ge 1) }$ has been colored blue or red. A set of four different unit squares of the board is called [i]pretty [/i]if these squares can be labeled ${A,B,C,D}$ in such a way that ${A}$ and ${B}$ lie in the same row, ${C}$ and ${D}$ lie in the same row, ${A}$ and ${C}$ lie in the same column, ${B}$ and ${D}$ lie in the same column, ${A}$ and ${D}$ are blue, and ${B}$ and ${C}$ are red. Determine the largest possible number of different [i]pretty [/i]sets on such a board.
(Poland)
Let $A_1$ be the center of the square inscribed in acute triangle $ABC$ with two vertices of the square on side $BC$. Thus one of the two remaining vertices of the square is on side $AB$ and the other is on $AC$. Points $B_1,\ C_1$ are defined in a similar way for inscribed squares with two vertices on sides $AC$ and $AB$, respectively. Prove that lines $AA_1,\ BB_1,\ CC_1$ are concurrent.
The squares $ABCD$ and $AXYZ$ are given. It turns out that $CDXY$ is a cyclic quadrilateral inscribed in the circle $\Omega$, and the points $A, B$ and $Z{}$ lie inside this circle. Prove that either $AB = AX$ or $AC\perp{}XY$.
Cut a square with three straight lines into three triangles and four quadrilaterals.
The rectangle is cut into $10$ squares as shown in the figure on the right. Find its sides if the side of the smallest square is $3$.[img]https://cdn.artofproblemsolving.com/attachments/e/5/1fe3a0e41b2d3182338a557d3d44ff5ef9385d.png[/img]
Consider the following sequence of squares (side $1$), in each step the central square is divided into equal parts and colored as shown in the figure:
[img]https://cdn.artofproblemsolving.com/attachments/9/0/6874ab5aecadf2112fbe4a196ab3091ab8b31a.png[/img]
Square 1 Square 2 Square 3
Let $A_n$ with $n \in N$, $n> 1$ be the shaded area of square $n$, show that $A_n <\frac23$
Consider an equilateral triangle and square, both with area $1$. What is the product of their perimeters?
To the exterior of side $AB$ of square $ABCD$, we have drawn the regular triangle $ABE$. Point $A$ reflected on line $BE$ is $F$, and point $E$ reflected on line $BF$ is $G$. Let the perpendicular bisector of segment $FG$ meet segment $AD$ at $X$. Show that the circle centered at $X$ with radius $XA$ touches line$ FB$.
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 $y = -x$.
Let $P$ be the set of points in the plane, and let $f : P \to P$ be a function such that the image through $f$ of any triangle is a square (any polygon is considered to be formed by the reunion of the points on its sides). Prove that $f(P)$ is a square.
We call a positive integer [i]heinersch[/i] if it can be written as the sum of a positive square and positive cube.
Prove: There are infinitely many heinersch numbers $h$, such that $h-1$ and $h+1$ are also heinersch.
Given a square $ABCD$ with side $10$. On sides BC and $AD$ of this square are selected respectively points $E$ and $F$ such that formed a rectangle $ABEF$. Rectangle $KLMN$ is located so that its the vertices $K, L, M$ and $N$ lie one on each segments $CD, DF, FE$ and $EC$, respectively. It turned out that the rectangles $ABEF$ and $KLMN$ are equal with $AB = MN$. Find the length of segment $AL$.
Given is a square with side length $a$. A distance $PQ$ of length $p$, where $p < a$, moves so that its end points are always on the sides of the square. What is the geometric locus of the midpoints of the segments $PQ$?
$E$ is the midpoint of the side $AB$ of the square $ABCD$, and $F, G$ are any points on the sides $BC$, $CD$ such that $EF$ is parallel to $AG$. Show that $FG$ touches the inscribed circle of the square.
From a square of side length $1$, four identical triangles are removed, one at each corner, leaving a regular octagon. What is the area of the octagon?
Consider the square $ABCD$ and $E \in (AB)$. The diagonal $AC$ intersects the segment $[DE]$ at point $P$. The perpendicular taken from point $P$ on $DE$ intersects the side $BC$ at point $F$. Prove that $EF = AE + FC$.
Find the locus of centroids of equilateral triangles whose vertices lie on sides of a given square $ABCD$.
You have to divide a square paper into three parts, by two straight cuts, so that by locating these parts properly, without gaps or overlaps, an obtuse triangle is formed. Indicate how to cut the square and how to assemble the triangle with the three parts.
If a square is intersected by another square equal to it but rotated by $45^o$ around its centre, each side is divided into three parts in a certain ratio $a : b : a$ (which one can compute). Make the following construction for an arbitrary convex quadrilateral: divide each of its sides into three parts in this same ratio $a : b : a$, and draw a line through the two division points neighbouring each vertex. Prove that the new quadrilateral bounded by the four drawn lines has the same area as the original one.
(A. Savin, Moscow)
Given a square whose side measures $a$, consider the set of all points of its plane through which passes a circumference of radius whose circle contains to the quoted square. You are asked to prove that the contour of the figure formed by the points with this property is formed by arcs of circumference, and determine the positions, their centers, their radii and their lengths.
$4$ 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?
a) Prove that it is impossible to divide a rectangle into five squares of distinct sizes.
b) Prove that it is impossible to divide a rectangle into six squares of distinct sizes.