Found problems: 1342
Fold the next seven corners into a rectangle.
[img]https://cdn.artofproblemsolving.com/attachments/b/b/2b8b9d6d4b72024996a66d41f865afb91bb9b7.png[/img]
Do there exist a rectangle that can be partitioned into a regular hexagon with side length $1$, and several right triangles with side lengths $1, \sqrt3 , 2$?
A circle that passes through the vertex $A$ of a rectangle $ABCD$ intersects the side $AB$ at a second point $E$ different from $B.$ A line passing through $B$ is tangent to this circle at a point $T,$ and the circle with center $B$ and passing through $T$ intersects the side $BC$ at the point $F.$ Show that if $\angle CDF= \angle BFE,$ then $\angle EDF=\angle CDF.$
A square and a rectangle of the same perimeter have a common corner. Prove that the intersection point of the diagonals of the rectangle lies on the diagonal of the square.
(Yu. Blinkov)
Let the lines $e$ and $f$ be perpendicular and intersect each other at $H$. Let $A$ and $B$ lie on $e$ and $C$ and $D$ lie on $f$, such that all five points $A,B,C,D$ and $H$ are distinct. Let the lines $b$ and $d$ pass through $B$ and $D$ respectively, perpendicularly to $AC$; let the lines $a$ and $c$ pass through $A$ and $C$ respectively, perpendicularly to $BD$. Let $a$ and $b$ intersect at $X$ and $c$ and $d$ intersect at $Y$. Prove that $XY$ passes through $H$.
The octagon $P_1P_2P_3P_4P_5P_6P_7P_8$ is inscribed in a circle with the vertices around the circumference in the given order. Given that the polygon $P_1P_3P_5P_7$ is a square of area $5$, and the polygon $P_2P_4P_6P_8$ is a rectangle of area $4$, find the maximum possible area of the octagon.
A $1 \times 5n$ rectangle is partitioned into tiles, each of the tile being either a separate $1 \times 1$ square or a broken domino consisting of two such squares separated by four squares (not belonging to the domino). Prove that the number of such partitions is a perfect fifth power.
[i](K. Kokhas)[/i]
The numbers $1,2,\ldots,100$ are written in the cells of a $10\times 10$ table, each number is written once. In one move, Nazar may interchange numbers in any two cells. Prove that he may get a table where the sum of the numbers in every two adjacent (by side) cells is composite after at most $35$ such moves.
[i]N. Agakhanov[/i]
A large rectangle is tiled by some $1\times1$ tiles. In the center there is a small rectangle tiled by some white tiles. The small rectangle is surrounded by a red border which is fi ve tiles wide. That red border is surrounded by a white border which is fi ve tiles wide. Finally, the white border is surrounded by a red border which is five tiles wide. The resulting pattern is pictured below. In all, $2900$ red tiles are used to tile the large rectangle. Find the perimeter of the large rectangle.
[asy]
import graph;
size(5cm);
fill((-5,-5)--(0,-5)--(0,35)--(-5,35)--cycle^^(50,-5)--(55,-5)--(55,35)--(50,35)--cycle,red);
fill((0,30)--(0,35)--(50,35)--(50,30)--cycle^^(0,-5)--(0,0)--(50,0)--(50,-5)--cycle,red);
fill((-15,-15)--(-10,-15)--(-10,45)--(-15,45)--cycle^^(60,-15)--(65,-15)--(65,45)--(60,45)--cycle,red);
fill((-10,40)--(-10,45)--(60,45)--(60,40)--cycle^^(-10,-15)--(-10,-10)--(60,-10)--(60,-15)--cycle,red);
fill((-10,-10)--(-5,-10)--(-5,40)--(-10,40)--cycle^^(55,-10)--(60,-10)--(60,40)--(55,40)--cycle,white);
fill((-5,35)--(-5,40)--(55,40)--(55,35)--cycle^^(-5,-10)--(-5,-5)--(55,-5)--(55,-10)--cycle,white);
for(int i=0;i<16;++i){
draw((-i,-i)--(50+i,-i)--(50+i,30+i)--(-i,30+i)--cycle,linewidth(.5));
}
[/asy]
Let $ABCD$ be a tetrahedron such that $AD \perp BD$, $BD \perp CD$, $CD \perp AD$ and $AD=10$, $BD=15$, $CD=20$. Let $E$ and $F$ be points such that $E$ lies on $BC$, $DE \perp BC$, and $ADEF$ is a rectangle. If $S$ is the solid consisting of all points inside $ABCD$ but outside $FBCD$, compute the volume of $S$.
[i]2015 CCA Math Bonanza Individual Round #13[/i]
In rectangle $ABCD$, $AB=1$, $BC=2$, and points $E$, $F$, and $G$ are midpoints of $\overline{BC}$, $\overline{CD}$, and $\overline{AD}$, respectively. Point $H$ is the midpoint of $\overline{GE}$. What is the area of the shaded region?
[asy]
import graph;
size(9cm);
pen dps = fontsize(10); defaultpen(dps);
pair D = (0,0);
pair F = (1/2,0);
pair C = (1,0);
pair G = (0,1);
pair E = (1,1);
pair A = (0,2);
pair B = (1,2);
pair H = (1/2,1);
// do not look
pair X = (1/3,2/3);
pair Y = (2/3,2/3);
draw(A--B--C--D--cycle);
draw(G--E);
draw(A--F--B);
draw(D--H--C);
filldraw(H--X--F--Y--cycle,grey);
label("$A$",A,NW);
label("$B$",B,NE);
label("$C$",C,SE);
label("$D$",D,SW);
label("$E$",E,E);
label("$F$",F,S);
label("$G$",G,W);
label("$H$",H,N);
label("$\displaystyle\frac12$",(0.25,0),S);
label("$\displaystyle\frac12$",(0.75,0),S);
label("$1$",(1,0.5),E);
label("$1$",(1,1.5),E);
[/asy]
$ \textbf{(A)}\ \dfrac1{12}\qquad\textbf{(B)}\ \dfrac{\sqrt3}{18}\qquad\textbf{(C)}\ \dfrac{\sqrt2}{12}\qquad\textbf{(D)}\ \dfrac{\sqrt3}{12}\qquad\textbf{(E)}\ \dfrac16 $
Let $ABCD$ be a rectangle of centre $O$, such that $\angle DAC=60^{\circ}$. The angle bisector of $\angle DAC$ meets $DC$ at $S$. Lines $OS$ and $AD$ meet at $L$, and lines $BL$ and $AC$ meet at $M$. Prove that lines $SM$ and $CL$ are parallel.
A large square is divided into a small square surrounded by four congruent rectangles as shown. The perimeter of each of the congruent rectangles is 14. What is the area of the large square?
[asy]unitsize(3mm);
defaultpen(linewidth(.8pt));
draw((0,0)--(7,0)--(7,7)--(0,7)--cycle);
draw((1,0)--(1,6));
draw((7,1)--(1,1));
draw((6,7)--(6,1));
draw((0,6)--(6,6));[/asy]$ \textbf{(A)}\ \ 49 \qquad \textbf{(B)}\ \ 64 \qquad \textbf{(C)}\ \ 100 \qquad \textbf{(D)}\ \ 121 \qquad \textbf{(E)}\ \ 196$
A $3000\times 3000$ square is tiled by dominoes (i. e. $1\times 2$ rectangles) in an arbitrary way. Show that one can color the dominoes in three colors such that the number of the dominoes of each color is the same, and each dominoe $d$ has at most two neighbours of the same color as $d$. (Two dominoes are said to be [i]neighbours[/i] if a cell of one domino has a common edge with a cell of the other one.)
A circle of radius 1 is randomly placed in a 15-by-36 rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
In this diagram the center of the circle is $O$, the radius is $a$ inches, chord $EF$ is parallel to chord $CD, O, G, H, J$ are collinear, and $G$ is the midpoint of $CD$. Let $K$ (sq. in.) represent the area of trapezoid $CDFE$ and let $R$ (sq. in.) represent the area of rectangle $ELMF$. Then, as $CD$ and $EF$ are translated upward so that $OG$ increases toward the value $a$, while $JH$ always equals $HG$, the ratio $K:R$ become arbitrarily close to:
[asy]
size((270));
draw((0,0)--(10,0)..(5,5)..(0,0));
draw((5,0)--(5,5));
draw((9,3)--(1,3)--(1,1)--(9,1)--cycle);
draw((9.9,1)--(.1,1));
label("O", (5,0), S);
label("a", (7.5,0), S);
label("G", (5,1), SE);
label("J", (5,5), N);
label("H", (5,3), NE);
label("E", (1,3), NW);
label("L", (1,1), S);
label("C", (.1,1), W);
label("F", (9,3), NE);
label("M", (9,1), S);
label("D", (9.9,1), E);
[/asy]
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{1}{\sqrt{2}}+\frac{1}{2} \qquad\textbf{(E)}\ \frac{1}{\sqrt{2}}+1$
Cut the $10$ cm $\times 25$ cm rectangle into two pieces with one straight cut so that they can fit inside the $22.1 $ cm circle without crossing.
A point $P$ lies on the internal angle bisector of $\angle BAC$ of a triangle $\triangle ABC$. Point $D$ is the midpoint of $BC$ and $PD$ meets the external angle bisector of $\angle BAC$ at point $E$. If $F$ is the point such that $PAEF$ is a rectangle then prove that $PF$ bisects $\angle BFC$ internally or externally.
Instead of walking along two adjacent sides of a rectangular field, a boy took a shortcut along the diagonal of the field and saved a distance equal to $ \frac{1}{2}$ the longer side. The ratio of the shorter side of the rectangle to the longer side was:
$ \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{2}{3} \qquad\textbf{(C)}\ \frac{1}{4} \qquad\textbf{(D)}\ \frac{3}{4} \qquad\textbf{(E)}\ \frac{2}{5}$
Let $a, b, c, d$ be the lengths of the sides of a quadrilateral circumscribed about a circle and let $S$ be its area. Prove that $S \leq \sqrt{abcd}$ and find conditions for equality.
In the rectangle $ABCD$ with $AB> BC$, the perpendicular bisecotr of $AC$ intersects the side $CD$ at point $E$. The circle with the center at point $E$ and the radius $AE$ intersects again the side $AB$ at point $F$. If point $O$ is the orthogonal projection of point $C$ on the line $EF$, prove that points $B, O$ and $D$ are collinear.
Given an oriented line $\Delta$ and a fixed point $A$ on it, consider all trapezoids $ABCD$ one of whose bases $AB$ lies on $\Delta$, in the positive direction. Let $E,F$ be the midpoints of $AB$ and $CD$ respectively. Find the loci of vertices $B,C,D$ of trapezoids that satisfy the following:
[i](i) [/i] $|AB| \leq a$ ($a$ fixed);
[i](ii) [/i] $|EF| = l$ ($l$ fixed);
[i](iii)[/i] the sum of squares of the nonparallel sides of the trapezoid is constant.
[hide="Remark"]
[b]Remark.[/b] The constants are chosen so that such trapezoids exist.[/hide]
Let $ABCD$ be a rectangle with $AB=a >CD =b$. Given circles $(K_1,r_1) , (K_2,r_2)$ with $r_1<r_2$ tangent externally at point $K$ and also tangent to the sides of the rectangle, circle $(K_1,r_1)$ tangent to both $AD$ and $AB$, circle $(K_2,r_2)$ tangent to both $AB$ and $BC$. Let also the internal common tangent of those circles pass through point $D$.
(i) Express sidelengths $a$ and $b$ in terms of $r_1$ and $r_2$.
(ii) Calculate the ratios $\frac{r_1}{r_2}$ and $\frac{a}{b}$ .
(iii) Find the length of $DK$ in terms of $r_1$ and $r_2$.
In the plane the points with integer coordinates are the vertices of unit squares. The squares are coloured alternately black and white (as on a chessboard). For any pair of positive integers $ m$ and $ n$, consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths $ m$ and $ n$, lie along edges of the squares. Let $ S_1$ be the total area of the black part of the triangle and $ S_2$ be the total area of the white part. Let $ f(m,n) \equal{} | S_1 \minus{} S_2 |$.
a) Calculate $ f(m,n)$ for all positive integers $ m$ and $ n$ which are either both even or both odd.
b) Prove that $ f(m,n) \leq \frac 12 \max \{m,n \}$ for all $ m$ and $ n$.
c) Show that there is no constant $ C\in\mathbb{R}$ such that $ f(m,n) < C$ for all $ m$ and $ n$.
In rectangular coordinate system, the equation $\frac{|x+y|}{2a}+\frac{|x-y|}{2b}=1$ ($a,b$ are different positive numbers) refers to
$\text{(A)}$ a triangle
$\text{(B)}$ a square
$\text{(C)}$ rectangle, not square
$\text{(D)}$ rhombus, not square