Found problems: 1342
Let $ABC$ be a triangle right in $A$. Construct a point $P$ on the hypotenuse $BC$ such that if $Q$ is the foot of the perpendicular drawn from $P$ to side $AC$, then the area of the square of side $PQ$ is equal to the area of the rectangle of sides $PB$ and $PC$. Show construction steps.
Find the length of the longer side of the rectangle on the picture, if the shorter side has length $1$ and the circles touch each other and the sides of the rectangle as shown.
[img]https://cdn.artofproblemsolving.com/attachments/b/8/3986683247293bd089d8e83911309308ce0c3a.png[/img]
Show that amongst any $ 8$ points in the interior of a $7 \times 12$ rectangle, there exists a pair whose distance is less than $5$.
Note: The interior of a rectangle excludes points lying on the sides of the rectangle.
Points $E$ and $F$ lie inside rectangle $ABCD$ with $AE=DE=BF=CF=EF$. If $AB=11$ and $BC=8$, find the area of the quadrilateral $AEFB$.
In the plane are given two mutually perpendicular lines and $n$ rectangles with sides parallel to the two lines. Show that if every two rectangles have a common point, then all the rectangles have a common point.
Let $P_1, P_2, P_3, P_4$ be four points on a circle, let $I_1$ be incenter of the triangle of vertices $P_2P_3P_4$, $I_2$ the incenter of the triangle $P_1P_3P_4$, $I_3$ the incenter of the triangle $P_1P_2P_4$, $I_4$ the incenter of the triangle $P_2P_3P_1$. Prove that $I_1I_2I_3I_4$ is a rectangle.
The length of a rectangle is increased by $ 10\%$ and the width is decreased by $ 10\%$. What percent of the old area is the new area?
$ \textbf{(A)}\ 90 \qquad
\textbf{(B)}\ 99 \qquad
\textbf{(C)}\ 100 \qquad
\textbf{(D)}\ 101 \qquad
\textbf{(E)}\ 110$
Let's call $1 \times 2$ rectangle, which can be a rotated, a domino. Prove that there exists polygon, who can be covered by dominoes in exactly $2021$ different ways.
Diagonal $ DB$ of rectangle $ ABCD$ is divided into $ 3$ segments of length $ 1$ by parallel lines $ L$ and $ L'$ that pass through $ A$ and $ C$ and are perpendicular to $ DB$. The area of $ ABCD$, rounded to the nearest tenth, is
[asy]size(200);
defaultpen(linewidth(0.7)+fontsize(10));
real x=sqrt(6), y=sqrt(3), a=0.4;
pair D=origin, A=(0,y), B=(x,y), C=(x,0), E=foot(C,B,D), F=foot(A,B,D);
real r=degrees(B);
pair M1=F+3*dir(r)*dir(90), M2=F+3*dir(r)*dir(-90), N1=E+3*dir(r)*dir(90), N2=E+3*dir(r)*dir(-90);
markscalefactor=0.02;
draw(B--C--D--A--B--D^^M1--M2^^N1--N2^^rightanglemark(A,F,B,6)^^rightanglemark(N1,E,B,6));
pair W=A+a*dir(135), X=B+a*dir(45), Y=C+a*dir(-45), Z=D+a*dir(-135);
label("A", A, NE);
label("B", B, NE);
label("C", C, dir(0));
label("D", D, dir(180));
label("$L$", (x/2,0), SW);
label("$L^\prime$", C, SW);
label("1", D--F, NW);
label("1", F--E, SE);
label("1", E--B, SE);
clip(W--X--Y--Z--cycle);
[/asy]
$ \textbf{(A)}\ 4.1 \qquad \textbf{(B)}\ 4.2 \qquad \textbf{(C)}\ 4.3 \qquad \textbf{(D)}\ 4.4 \qquad \textbf{(E)}\ 4.5$
The rectangle is cut into 6 squares, as shown on the figure below. The gray square in the middle has a side equal to 1.
What is the area of the rectangle?
[img]https://i.ibb.co/gg1tBTN/Kyiv-MO-2023-7-1.png[/img]
There are several equal (possibly overlapping) square-shaped napkins on a rectangular table, with sides parallel to the sides of the table. Prove that it is possible to nail some of them to the table in such a way that every napkin is nailed exactly once.
Find all rectangles that can be cut into $13$ equal squares.
Given the equilateral triangle $ABC$. It is known that the radius of the inscribed circle is in this triangle is equal to $1$. The rectangle $ABDE$ is such that point $C$ belongs to its side $DE$. Find the radius of the circle circumscribed around the rectangle $ABDE$.
Let $ABCD$ be a convex circumscribed quadrilateral such that $\angle ABC+\angle ADC<180^{\circ}$ and $\angle ABD+\angle ACB=\angle ACD+\angle ADB$. Prove that one of the diagonals of quadrilateral $ABCD$ passes through the other diagonals midpoint.
There is an integer in each cell of a $2m\times 2n$ table. We define the following operation: choose three cells forming an L-tromino (namely, a cell $C$ and two other cells sharing a side with $C$, one being horizontal and the other being vertical) and sum $1$ to each integer in the three chosen cells. Find a necessary and sufficient condition, in terms of $m$, $n$ and the initial numbers on the table, for which there exists a sequence of operations that makes all the numbers on the table equal.
$ABCD$ is a rectangle. Segment $BA$ is extended through $A$ to a point $E$. Let the intersection of $EC$ and $AD$ be point $F$. Suppose that [the] measure of $\angle{ACD}$ is $60$ degrees, and that the length of segment $EF$ is twice the length of diagonal $AC$. What is the measure of $\angle{ECD}$?
The midpoints of the four sides of a rectangle are $(-3, 0), (2, 0), (5, 4)$ and $(0, 4)$. What is the area of the rectangle?
$\textbf{(A)} ~20\qquad\textbf{(B)} ~25\qquad\textbf{(C)} ~40\qquad\textbf{(D)} ~50\qquad\textbf{(E)} ~80\qquad$
Given an integer $n\geq 3$. For each $3\times3$ squares on the grid, call this $3\times3$ square isolated if the center unit square is white and other 8 squares are black, or the center unit square is black and other 8 squares are white.
Now suppose one can paint an infinite grid by white or black, so that one can select an $a\times b$ rectangle which contains at least $n^2-n$ isolated $3\times 3$ square. Find the minimum of $a+b$ that such thing can happen.
(Note that $a,b$ are positive reals, and selected $a\times b$ rectangle may have sides not parallel to grid line of the infinite grid.)
Define a figure which is constructed by unit squares "cross star" if it satisfies the following conditions:
$(1)$Square bar $AB$ is bisected by square bar $CD$
$(2)$At least one square of $AB$ lay on both sides of $CD$
$(3)$At least one square of $CD$ lay on both sides of $AB$
There is a rectangular grid sheet composed of $38\times 53=2014$ squares,find the number of such cross star in this rectangle sheet
Define an integer $n \ge 0$ to be \textit{two-far} if there exist integers $a$ and $b$ such that $a$, $b$, and $n + a + b$ are all powers of two. If $N$ is the number of two-far integers less than 2048, find the remainder when $N$ is divided by 100.
Let $T = TNYWR$. Let $CMU$ be a triangle with $CM=13$, $MU=14$, and $UC=15$. Rectangle $WEAN$ is inscribed in $\triangle CMU$ with points $W$ and $E$ on $\overline{MU}$, point $A$ on $\overline{CU}$, and point $N$ on $\overline{CM}$. If the area of $WEAN$ is $T$, what is its perimeter?
Suppose we are given a $19\times19$ chessboard (a table with $19^2$ squares) and remove the central square. Is it possible to tile the remaining $19^2-1=360$ squares with $4\times1$ and $1\times4$ rectangles? (So that each of the $360$ squares is covered by exactly one rectangle.) Justify your answer.
Let $ABCD$ be a rectangle. A line $r$ moves parallel to $AB$ and intersects diagonal $AC$ , forming two triangles opposite the vertex, inside the rectangle. Prove that the sum of the areas of these triangles is minimal when $r$ passes through the midpoint of segment $AD$ .
Treasure was buried in a single cell of an $M\times N$ ($2\le M$, $N$) grid. Detectors were brought to find the cell with the treasure. For each detector, you can set it up to scan a specific subgrid $[a,b]\times[c,d]$ with $1\le a\le b\le M$ and $1\le c\le d\le N$. Running the detector will tell you whether the treasure is in the region or not, though it cannot say where in the region the treasure was detected. You plan on setting up $Q$ detectors, which may only be run simultaneously after all $Q$ detectors are ready.
In terms of $M$ and $N$, what is the minimum $Q$ required to gaurantee to determine the location of the treasure?
(a) Suppose that each square of a 4 x 7 chessboard is colored either black or white. Prove that with [i]any[/i] such coloring, the board must contain a rectangle (formed by the horizontal and vertical lines of the board) whose four distinct unit corner squares are all of the same color.
(b) Exhibit a black-white coloring of a 4 x6 board in which the four corner squares of every rectangle, as described above, are not all of the same color.
Find the volume of the solid generated by a rotation of the region enclosed by the curve $y=x^3-x$ and the line $y=x$ about the line $y=x$ as the axis of rotation.