Found problems: 1342
Let $k$ and $n$ be positive integers. Consider an array of $2\left(2^n-1\right)$ rows by $k$ columns. A $2$-coloring of the elements of the array is said to be [i]acceptable[/i] if any two columns agree on less than $2^n-1$ entries on the same row.
Given $n$, determine the maximum value of $k$ for an acceptable $2$-coloring to exist.
Given a rectangle $ABCD$, points $M$ -- the midpoint of $[AD]$ side, $N$ -- the midpoint of $[BC]$ side. Let us take a point $P$ on the extension of the $[DC]$ segment over the point $D$. Let us denote the intersection point of lines $(PM)$ and $(AC)$ as $Q$. Prove that the $\angle QNM= \angle MNP$
Label the squares of a $1991 \times 1992$ rectangle $(m, n)$ with $1 \leq m \leq 1991$ and $1 \leq n \leq 1992$. We wish to color all the squares red. The first move is to color red the squares $(m, n), (m+1, n+1), (m+2, n+1)$for some $m < 1990, n < 1992$. Subsequent moves are to color any three (uncolored) squares in the same row, or to color any three (uncolored) squares in the same column. Can we color all the squares in this way?
Let $ I(A)$ and $ I(B)$ be the centers of the excircles of a triangle $ ABC,$ which touches the sides $ BC$ and $ CA$ in its interior. Furthermore let $ P$ a point on the circumcircle $ \omega$ of the triangle $ ABC.$ Show that the center of the segment which connects the circumcenters of the triangles $ I(A)CP$ and $ I(B)CP$ coincides with the center of the circle $ \omega.$
Chubby makes nonstandard checkerboards that have $31$ squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard?
$ \textbf{(A)}\ 480
\qquad\textbf{(B)}\ 481
\qquad\textbf{(C)}\ 482
\qquad\textbf{(D)}\ 483
\qquad\textbf{(E)}\ 484
$
Let points $A=(0,0,0)$, $B=(1,0,0)$, $C=(0,2,0)$, and $D=(0,0,3)$. Points $E,F,G$, and $H$ are midpoints of line segments $\overline{BD},\overline{AB},\overline{AC}$, and $\overline{DC}$ respectively. What is the area of $EFGH$?
$ \textbf{(A)}\ \sqrt2
\qquad\textbf{(B)}\ \frac{2\sqrt5}{3}
\qquad\textbf{(C)}\ \frac{3\sqrt5}{4}
\qquad\textbf{(D)}\ \sqrt3
\qquad\textbf{(E)}\ \frac{2\sqrt7}{3}
$
In a rectangle with sides of length 20 and 25 there are 120 squares of side length 1. Prove that there is a circle with a diameter of 1 contained in this rectangle and having no points in common with any of these squares.
Matías has a rectangular sheet of paper $ABCD$, with $AB<AD$.Initially, he folds the sheet along a straight line $AE$, where $E$ is a point on the side $DC$ , so that vertex $D$ is located on side $BC$, as shown in the figure. Then folds the sheet again along a straight line $AF$, where $F$ is a point on side $BC$, so that vertex $B$ lies on the line $AE$; and finally folds the sheet along the line $EF$. Matías observed that the vertices $B$ and $C$ were located on the same point of segment $AE$ after making the folds. Calculate the measure of the angle $\angle DAE$.
[img]https://cdn.artofproblemsolving.com/attachments/0/9/b9ab717e1806c6503a9310ee923f20109da31a.png[/img]
Let $EFGH$, $EFDC$, and $EHBC$ be three adjacent square faces of a cube, for which $EC=8$, and let $A$ be the eighth vertex of the cube. Let $I$, $J$, and $K$, be the points on $\overline{EF}$, $\overline{EH}$, and $\overline{EC}$, respectively, so that $EI=EJ=EK=2$. A solid $S$ is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to $\overline{AE}$, and containing the edges, $\overline{IJ}$, $\overline{JK}$, and $\overline{KI}$. The surface area of $S$, including the walls of the tunnel, is $m+n\sqrt{p}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find $m+n+p$.
Let $ABCD$ be the rectangle. Points $E$, $F$ lies on the sides $BC$ and $CD$ respectively, such that $\sphericalangle EAF = 45^{\circ}$ and $BE = DF$. Prove that area of the triangle $AEF$ is equal to the sum of the areas of the triangles $ABE$ and $ADF$.
A figure $\Phi$ composed of unit squares has the following property: if the squares of an $m \times n$ rectangle ($m,n$ are fixed) are filled with numbers whose sum is positive, the figure $\Phi$ can be placed within the rectangle (possibly after being rotated) so that the sum of the covered numbers is also positive. Prove that a number of such figures can be put on the $m\times n$ rectangle so that each square is covered by the same number of figures.
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]
How many ways are there to divide a $2\times n$ rectangle into rectangles having integral sides, where $n$ is a positive integer?
On the figure, the quadrilateral $ ABCD$ is a rectangle, $ P$ lies on $ AD$ and $ Q$ on $ AB.$ The triangles $ PAQ, QBC,$ and $ PCD$ all have the same areas, and $ BQ \equal{} 2.$ How long is $ AQ$?
[img]http://i250.photobucket.com/albums/gg265/geometry101/NielsHenrikAbel1995Number2.jpg[/img]
A. 7/2
B. $ \sqrt{7}$
C. $ 2 \sqrt{3}$
D. $ 1 \plus{} \sqrt{5}$
E. Not uniquely determined
Let $ ABC$ be a triangle. Squares $ AB_{c}B_{a}C$, $ CA_{b}A_{c}B$ and $ BC_{a}C_{b}A$ are outside the triangle. Square $ B_{c}B_{c}'B_{a}'B_{a}$ with center $ P$ is outside square $ AB_{c}B_{a}C$. Prove that $ BP,C_{a}B_{a}$ and $ A_{c}B_{c}$ are concurrent.
The sides of rectangle $ABCD$ have lengths 10 and 11. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD.$ The maximum possible area of such a triangle can be written in the form $p\sqrt{q}-r,$ where $p, q,$ and $r$ are positive integers, and $q$ is not divisible by the square of any prime number. Find $p+q+r.$
Let $ABC$ be a triangle . Let point $X$ be in the triangle and $AX$ intersects $BC$ in $Y$ . Draw the perpendiculars $YP,YQ,YR,YS$ to lines $CA,CX,BX,BA$ respectively. Find the necessary and sufficient condition for $X$ such that $PQRS$ be cyclic .
$200 \times 200$ square is colored in chess order. In one move we can take every $2 \times 3$ rectangle and change color of all its cells. Can we make all cells of square in same color ?
Let $I$ and $J$ be the centers of the incircle and the excircle in the angle $BAC$ of the triangle $ABC$. For any point $M$ in the plane of the triangle, not on the line $BC$, denote by $I_M$ and $J_M$ the centers of the incircle and the excircle (touching $BC$) of the triangle $BCM$. Find the locus of points $M$ for which $II_MJJ_M$ is a rectangle.
Let's take a $ABCD$ rectangle of paper; the side $AB$ measures $5$ cm and the side $BC$ measures $9$ cm.
We do three folds:
1.We take the $AB$ side on the $BC$ side and call $P$ the point on the $BC$ side that coincides with $A$.
A right trapezoid $BCDQ$ is then formed.
2. We fold so that $B$ and $Q$ coincide. A $5$-sided polygon $RPCDQ$ is formed.
3. We fold again by matching $D$ with $C$ and $Q$ with $P$. A new right trapezoid $RPCS$.
After making these folds, we make a cut perpendicular to $SC$ by its midpoint $T$, obtaining the right trapezoid $RUTS$.
Calculate the area of the figure that appears as we unfold the last trapezoid $RUTS$.
Let $ABCD$ be a rectangle with: $AB=a$, $BC=b$. Inside the rectangle we have to exteriorly tangents circles such that one is tangent to the sides $AB$ and $AD$,the other is tangent to the sides $CB$ and $CD$.
1. Find the distance between the centers of the circles(using $a$ and $b$).
2. When the radiums of both circles change the tangency point between both of them changes, and describes a locus. Find that locus.
Given a rectangle $ABCD$, determine two points $K$ and $L$ on the sides $BC$ and $CD$ such that the triangles $ABK, AKL$ and $ADL$ have same area.
Circle $\omega_1$ with radius $6$ centered at point $A$ is internally tangent at point $B$ to circle $\omega_2$ with radius $15$. Points $C$ and $D$ lie on $\omega_2$ such that $\overline{BC}$ is a diameter of $\omega_2$ and $\overline{BC} \perp \overline{AD}$. The rectangle $EFGH$ is inscribed in $\omega_1$ such that $\overline{EF} \perp \overline{BC}$, $C$ is closer to $\overline{GH}$ than to $\overline{EF}$, and $D$ is closer to $\overline{FG}$ than to $\overline{EH}$, as shown. Triangles $\triangle DGF$ and $\triangle CHG$ have equal areas. The area of rectangle $EFGH$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[asy]
size(5cm);
defaultpen(fontsize(10pt));
pair A = (9, 0), B = (15, 0), C = (-15, 0), D = (9, 12), E = (9+12/sqrt(5), -6/sqrt(5)), F = (9+12/sqrt(5), 6/sqrt(5)), G = (9-12/sqrt(5), 6/sqrt(5)), H = (9-12/sqrt(5), -6/sqrt(5));
filldraw(G--H--C--cycle, lightgray);
filldraw(D--G--F--cycle, lightgray);
draw(B--C);
draw(A--D);
draw(E--F--G--H--cycle);
draw(circle(origin, 15));
draw(circle(A, 6));
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(H);
label("$A$", A, (.8, -.8));
label("$B$", B, (.8, 0));
label("$C$", C, (-.8, 0));
label("$D$", D, (.4, .8));
label("$E$", E, (.8, -.8));
label("$F$", F, (.8, .8));
label("$G$", G, (-.8, .8));
label("$H$", H, (-.8, -.8));
label("$\omega_1$", (9, -5));
label("$\omega_2$", (-1, -13.5));
[/asy]
We have a $10 \times 10$ table. $T$ is a set of rectangles with vertices from the table and sides parallel to the sides of the table such that no rectangle from the set is a subrectangle of another rectangle from the set. $t$ is the maximum number of elements of $T$.
(a) Prove that $t>300$.
(b) Prove that $t<600$.
[i]Proposed by Mir Omid Haji Mirsadeghi and Kasra Alishahi[/i]
A floor has $2 \times 11$ dimension, and this floor covering with $1 \times 2$ rectangles. (No two rectangles overlap). How many cases we done this job?