Found problems: 1342
On the side $ \overline{DC} $ of the rectangle $ ABCD $ in which $ \frac{AB}{AD} = \sqrt{2} $ a semicircle is built externally. Any point $ M $ of the semicircle is connected by segments with $ A $ and $ B $ to obtain points $ K $ and $ L $ on $ \overline{DC} $, respectively. Prove that $ DL^2 + KC^2 = AB^2 $.
$(BUL 3)$ One hundred convex polygons are placed on a square with edge of length $38 cm.$ The area of each of the polygons is smaller than $\pi cm^2,$ and the perimeter of each of the polygons is smaller than $2\pi cm.$ Prove that there exists a disk with radius $1$ in the square that does not intersect any of the polygons.
The convex pentagon $ABCDE$ has $\angle A=\angle B=120^{\circ}$, $EA=AB=BC=2$ and $CD=DE=4$. What is the area of $ABCDE$?
[asy]
draw((0,0)--(1,0)--(1.5,sqrt(3)/2)--(0.5,3sqrt(3)/2)--(-0.5,sqrt(3)/2)--cycle);
dot((0,0));
dot((1,0));
dot((1.5,sqrt(3)/2));
dot((0.5,3sqrt(3)/2));
dot((-0.5,sqrt(3)/2));
label("A", (0,0), SW);
label("B", (1,0), SE);
label("C", (1.5,sqrt(3)/2), E);
label("D", (0.5,3sqrt(3)/2), N);
label("E", (-.5, sqrt(3)/2), W);
[/asy]
$ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 7\sqrt{3} \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 9\sqrt{3} \qquad\textbf{(E)}\ 12\sqrt{5} $
Squares of an $n \times n$ table ($n \ge 3$) are painted black and white as in a chessboard. A move allows one to choose any $2 \times 2$ square and change all of its squares to the opposite color. Find all such n that there is a finite number of the moves described after which all squares are the same color.
The city is a rectangle divided onto squares by $m$ streets coming from the West to the East and $n$ streets coming from the North to the South. There are militioners (policemen) on the streets but not on the crossroads. They watch the certain automobile, moving along the closed route, marking the time and the direction of its movement. Its trace is not known in advance, but they know, that it will not pass over the same segment of the way twice. What is the minimal number of the militioners providing the unique determination of the route according to their reports?
What is the probability that an integer in the set $ \{1,2,3,\ldots,100\}$ is divisible by $ 2$ and not divisible by $ 3$?
$ \textbf{(A)}\ \frac{1}{6} \qquad
\textbf{(B)}\ \frac{33}{100} \qquad
\textbf{(C)}\ \frac{17}{50} \qquad
\textbf{(D)}\ \frac{1}{2} \qquad
\textbf{(E)}\ \frac{18}{25}$
In rectangle $ABCD$, $AB=6$, $AD=30$, and $G$ is the midpoint of $\overline{AD}$. Segment $AB$ is extended $2$ units beyond $B$ to point $E$, and $F$ is the intersection of $\overline{ED}$ and $\overline{BC}$. What is the area of $BFDG$?
$ \textbf{(A)}\ \frac{133}{2}\qquad\textbf{(B)}\ 67\qquad\textbf{(C)}\ \frac{135}{2}\qquad\textbf{(D)}\ 68\qquad\textbf{(E)}\ \frac{137}{2}$
Let $p$ be a prime number. Prove that from a $p^2\times p^2$ array of squares, we can select $p^3$ of the squares such that the centers of any four of the selected squares are not the vertices of a rectangle with sides parallel to the edges of the array.
The rectangle $ABCD$ below has dimensions $AB = 12 \sqrt{3}$ and $BC = 13 \sqrt{3}$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P$. If triangle $ABP$ is cut out and removed, edges $\overline{AP}$ and $\overline{BP}$ are joined, and the figure is then creased along segments $\overline{CP}$ and $\overline{DP}$, we obtain a triangular pyramid, all four of whose faces are isosceles triangles. Find the volume of this pyramid.
[asy]
pair D=origin, A=(13,0), B=(13,12), C=(0,12), P=(6.5, 6);
draw(B--C--P--D--C^^D--A);
filldraw(A--P--B--cycle, gray, black);
label("$A$", A, SE);
label("$B$", B, NE);
label("$C$", C, NW);
label("$D$", D, SW);
label("$P$", P, N);
label("$13\sqrt{3}$", A--D, S);
label("$12\sqrt{3}$", A--B, E);[/asy]
Inside an $11\times 11$ square, Pablo drew a rectangle and extending its sides divided the square into $5$ rectangles, as shown in the figure.
[img]https://cdn.artofproblemsolving.com/attachments/5/a/7774da7085f283b3aae74fb5ff472572571827.gif[/img]
Sofía did the same, but she also managed to make the lengths of the sides of the $5$ rectangles be whole numbers between $1$ and $10$, all different. Show a figure like the one Sofia made.
The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$. Suppose that the radius of the circle is 5, that $BC = 6$, and that $AD$ is bisected by $BC$. Suppose further that $AD$ is the only chord starting at $A$ which is bisected by $BC$. It follows that the sine of the minor arc $AB$ is a rational number. If this fraction is expressed as a fraction $m/n$ in lowest terms, what is the product $mn$?
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair A=dir(200), D=dir(95), M=midpoint(A--D), C=dir(30), BB=C+2*dir(C--M), B=intersectionpoint(M--BB, Circle(origin, 1));
draw(Circle(origin, 1)^^A--D^^B--C);
real r=0.05;
pair M1=midpoint(M--D), M2=midpoint(M--A);
draw((M1+0.1*dir(90)*dir(A--D))--(M1+0.1*dir(-90)*dir(A--D)));
draw((M2+0.1*dir(90)*dir(A--D))--(M2+0.1*dir(-90)*dir(A--D)));
pair point=origin;
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$D$", D, dir(point--D));[/asy]
A certain number of cubes are painted in six colours, each cube having six faces of different colours (the colours in different cubes may be arranged differently) . The cubes are placed on a table so as to form a rectangle. We are allowed to take out any column of cubes, rotate it (as a whole) along its long axis and replace it in the rectangle. A similar operation with rows is also allowed. Can we always make the rectangle monochromatic (i.e. such that the
top faces of all the cubes are the same colour) by means of such operations?
( D. Fomin , Leningrad)
Let $f:[0,1]\to\mathbb R$ be a function satisfying
$$|f(x)-f(y)|\le|x-y|$$for every $x,y\in[0,1]$. Show that for every $\varepsilon>0$ there exists a countable family of rectangles $(R_i)$ of dimensions $a_i\times b_i$, $a_i\le b_i$ in the plane such that
$$\{(x,f(x)):x\in[0,1]\}\subset\bigcup_iR_i\text{ and }\sum_ia_i<\varepsilon.$$(The edges of the rectangles are not necessarily parallel to the coordinate axes.)
A square is divided into rectangles.
A "chain" is a subset $K$ of the set of these rectangles such that there exists a side of the square which is covered by projections of rectangles of $K$ and such that no point of this side is a projection of two inner points of two inner points of two different rectangles of $K$.
(a) Prove that every two rectangles in such a division are members of a certain "chain".
(b) Solve the similar problem for a cube, divided into rectangular parallelopipeds (in the definition of chain , replace "side" by"edge") .
(A.I . Golberg, V.A. Gurevich)
Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by $P$, $Q$, $R$, $S$ the feet of the perpendiculars from $Y$ onto lines $AX$, $BX$, $AZ$, $BZ$, respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$, where $O$ is the midpoint of segment $AB$.
$D$ is an arbitrary point lying on side $BC$ of $\triangle{ABC}$. Circle $\omega_1$ is tangent to segments $AD$ , $BD$ and the circumcircle of $\triangle{ABC}$ and circle $\omega_2$ is tangent to segments $AD$ , $CD$ and the circumcircle of $\triangle{ABC}$. Let $X$ and $Y$ be the intersection points of $\omega_1$ and $\omega_2$ with $BC$ respectively and take $M$ as the midpoint of $XY$. Let $T$ be the midpoint of arc $BC$ which does not contain $A$. If $I$ is the incenter of $\triangle{ABC}$, prove that $TM$ goes through the midpoint of $ID$.
Consider a $m\times n$ rectangular board consisting of $mn$ unit squares. Two of its unit squares are called [i]adjacent[/i] if they have a common edge, and a [i]path[/i] is a sequence of unit squares in which any two consecutive squares are adjacent. Two parths are called [i]non-intersecting[/i] if they don't share any common squares.
Each unit square of the rectangular board can be colored black or white. We speak of a [i]coloring[/i] of the board if all its $mn$ unit squares are colored.
Let $N$ be the number of colorings of the board such that there exists at least one black path from the left edge of the board to its right edge. Let $M$ be the number of colorings of the board for which there exist at least two non-intersecting black paths from the left edge of the board to its right edge.
Prove that $N^{2}\geq M\cdot 2^{mn}$.
A convex quadrilateral is determined by the points of intersection of the curves $x^4+y^4=100$ and $xy=4$; determine its area.
A square board is dissected into $n^2$ rectangular cells by $n-1$ horizontal and $n-1$ vertical lines. The cells are painted alternately black and white in a chessboard pattern. One diagonal consists of $n$ black cells which are squares. Prove that the total area of all black cells is not less than the total area of all white cells.
In a multiplication table, the entry in the $i$-th row and the $j$-th column is the product $ij$ From an $m\times n$ subtable with both $m$ and $n$ odd, the interior $(m-2) (n-2)$ rectangle is removed, leaving behind a frame of width $1$. The squares of the frame are painted alternately black and white. Prove that the sum of the numbers in the black squares is equal to the sum of the numbers in the white squares.
Let $ n $ and $ r $ be positive integers and $ A $ be a set of lattice points in the plane such that any open disc of radius $ r $ contains a point of $ A $. Show that
for any coloring of the points of $ A $ in $ n $ colors there exists four points of the same color which are the vertices of a rectangle.
Six points are marked inside the $3\times 4$ rectangle. Prove that there is a pair of marked points with the distance between them not greater than $\sqrt5$.
Jorge's teacher asks him to plot all the ordered pairs $ (w, l)$ of positive integers for which $ w$ is the width and $ l$ is the length of a rectangle with area 12. What should his graph look like?
$ \textbf{(A)}$[asy]size(75);
draw((0,-1)--(0,13));
draw((-1,0)--(13,0));
dot((1,12));
dot((2,6));
dot((3,4));
dot((4,3));
dot((6,2));
dot((12,1));
label("$l$", (0,6), W);
label("$w$", (6,0), S);[/asy]
$ \textbf{(B)}$[asy]size(75);
draw((0,-1)--(0,13));
draw((-1,0)--(13,0));
dot((1,1));
dot((3,3));
dot((5,5));
dot((7,7));
dot((9,9));
dot((11,11));
label("$l$", (0,6), W);
label("$w$", (6,0), S);[/asy]
$ \textbf{(C)}$[asy]size(75);
draw((0,-1)--(0,13));
draw((-1,0)--(13,0));
dot((1,11));
dot((3,9));
dot((5,7));
dot((7,5));
dot((9,3));
dot((11,1));
label("$l$", (0,6), W);
label("$w$", (6,0), S);[/asy]
$ \textbf{(D)}$[asy]size(75);
draw((0,-1)--(0,13));
draw((-1,0)--(13,0));
dot((1,6));
dot((3,6));
dot((5,6));
dot((7,6));
dot((9,6));
dot((11,6));
label("$l$", (0,6), W);
label("$w$", (6,0), S);[/asy]
$ \textbf{(E)}$[asy]size(75);
draw((0,-1)--(0,13));
draw((-1,0)--(13,0));
dot((6,1));
dot((6,3));
dot((6,5));
dot((6,7));
dot((6,9));
dot((6,11));
label("$l$", (0,6), W);
label("$w$", (6,0), S);[/asy]
The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle?
$\textbf{(A)} \text{ } \frac{\sqrt{3}-1}{2} \qquad \textbf{(B)} \text{ } \frac{1}{2} \qquad \textbf{(C)} \text{ } \frac{\sqrt{5}-1}{2} \qquad \textbf{(D)} \text{ } \frac{\sqrt{2}}{2} \qquad \textbf{(E)} \text{ } \frac{\sqrt{6}-1}{2}$
Rectangle on a checked paper with length of a unit square side being $1$ Is divided into domino figures( two unit square sharing a common edge). Prove that you colour all corners of squares on the edge of rectangle and inside rectangle with $3$ colours such that for any two corners with distance $1$ the following conditions hold: they are coloured in different colour if the line connecting the two corners is on the border of two domino figures and coloured in same colour if the line connecting the two corners is inside a domino figure.