Found problems: 467
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]
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.
Barack is an equilateral triangle and Michelle is a square. If Barack and Michelle each have perimeter $ 12$, find the area of the polygon with larger area.
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]
Prove that the number
\[
3 \underbrace{99\ldots9}_{2025} \underbrace{60\ldots01}_{2025}
\]
is a square of a positive integer.
* Cut out of a $3 \times 3$ square an unfolding of the cube with edge $1$.
In square $ABCD,$ points $E$ and $F$ are chosen in the interior of sides $BC$ and $CD$, respectively. The line drawn from $F$ perpendicular to $AE$ passes through the intersection point $G$ of $AE$ and diagonal $BD$. A point $K$ is chosen on $FG$ such that $|AK|= |EF|$. Find $\angle EKF.$
Consider a $N\times N$ square board where $N\geq 3$ is an odd integer. The caterpillar Carl sits at the center of the square; all other cells contain distinct positive integers. An integer $n$ weights $1\slash n$ kilograms. Carl wants to leave the board but can eat at most $2$ kilograms. Determine whether Carl can always find a way out when
a) $N=2003.$
b) $N$ is an arbitrary odd integer.
Given a square sheet of paper with a side of $2016$. Is it possible to bend its not more than ten times, construct a segment of length $1$?
Prove that the squares with sides $\frac{1}{1}, \frac{1}{2}, \frac{1}{3},\ldots$ may be put into the square with side $\frac{3}{2} $ in such a way that no two of them have any interior point in common.
Two squares are arranged as shown. Prove that the area of the black triangle equal to the sum of the gray areas.
[img]https://2.bp.blogspot.com/-byhWqNr1ras/XTq-NWusg2I/AAAAAAAAKZA/1sxEZ751v_Evx1ij7K_CGiuZYqCjhm-mQCK4BGAYYCw/s400/Oral%2BSharygin%2B2017%2B8.9%2Bp5.png[/img]
A square is cut into a finitely number of triangles in an arbitrary way. Show the sum of the diameters of the inscribed circles in these triangles is greater than the side length of the square.
A point $M$ is chosen inside the square $ABCD$ in such a way that $\angle MAC = \angle MCD = x$ . Find $\angle ABM$.
Let $P$ be a point inside the square $ABCD$ and $PA = 1$, $PB = \sqrt2$ and $PC =\sqrt3$.
a) Determine the length of segment $[PD]$.
b) Determine the angle $\angle APB$.
On the sides of a parallelogram, squares are constructed outwards. Prove that the centers of these squares are vertices of a square.
Someone wants to unscrew a square nut with side $a$, with a wrench whose hole has the form of a regular hexagon with side $b$. What condition should the lengths $a$ and $b$ meet to make this possible?
Is it possible to cut a square with side $\sqrt{2015}$ into no more than five pieces so that these pieces can be rearranged into a rectangle with sides of integer length? (The cuts should be made using straight lines, and flipping of the pieces is disallowed.)
Let $E$ be an arbitrary point different from $A$ and $B$ on the side $AB$ of a square $ABCD$, and let $F$ and $G$ be points on the segment $CE$ so that $BF$ and $DG$ are perpendicular to $CE$. Prove that $DF = AG$.
Let $ABC$ be an acute triangle. Squares $AA_1A_2A_3$, $BB_1B_2B_3$ and $CC_1C_2C_3$ are located such that the lines $A_1A_2$, $B_1B_2$, $C_1C_2$ pass through the points $B$, $C$ and $A$ respectively and the lines $A_2A_3$, $B_2B_3$, $C_2C_3$ pass through the points $C$, $A$ and $B$ respectively. Prove that
(a) the lines $AA_2$, $B_1B_2$ and $C_1C_3$ intersect at one point.
(b) the lines $AA_2$, $BB_2$ and $CC_2$ intersect at one point.
(Mykhailo Plotnikov)
[img]https://cdn.artofproblemsolving.com/attachments/3/d/ad2fe12ae2c82d04b48f5e683b7d54e0764baf.png[/img]
a) Do there exist four equilateral triangles in the plane such that each two have
exactly one vertex in common, and every point in the plane lies on the boundary of at most two
of them?
b) Do there exist four squares in the plane such that each two have exactly one vertex in common, and every point in the plane lies on the boundary of at most two of them?
(Note that in both parts, there is no assumption on the intersection of interior of polygons.)
[i]Proposed by Hesam Rajabzadeh[/i]
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.
There are $n \ge 2$ points in a square side $1$. Show that one can label the points $P_1, P_2, ... , P_n$ such that $\sum_{i=1}^n |P_{i-1} - P_i|^2 \le 4$, where we use cyclic subscripts, so that $P_0$ means $P_n$.
The squares $ABMN$, $BCKL$ and $ACPQ$ are constructed outside triangle $ABC$. The difference between the areas of $AB MN$ and $BCKL$ is $d$. Find the difference between the areas of the squares with sides $NQ$ and $PK$ respectively, if $\angle ABC$ is
(a) a right angle;
(b) not necessarily a right angle.
(A Gerko)
The figure shows a square and three circles of equal radius tangent to each other and square passes. Find the radius of the circles if the square length is $1$.
[img]http://3.bp.blogspot.com/-iIjwupkz7DQ/XnrIRhKIJnI/AAAAAAAALhA/clERrIDqEtcujzvZk_qu975wsTjKaxCLQCK4BGAYYCw/s400/97%2Bestonia%2Bopen%2Bs2.3.png[/img]
In a square $ABCD$ of diagonals $AC$ and $BD$, we call $O$ at the center of the square. A square $PQRS$ is constructed with sides parallel to those of $ABCD$ with $P$ in segment $AO, Q$ in segment $BO, R$ in segment $CO, S$ in segment $DO$. If area of $ABCD$ equals two times the area of $PQRS$, and $M$ is the midpoint of the $AB$ side, calculate the measure of the angle $\angle AMP$.