Found problems: 1342
A circle can be inscribed in the convex quadrilateral $ ABCD$. The incircle touches the sides $ AB, BC, CD, DA$ at $ A_1, B_1, C_1, D_1$ respectively. The points $ E, F, G, H$ are the midpoints of $ A_1B_1, B_1C_1, C_1D_1, D_1A_1$ respectively. Prove that the quadrilateral $ EFGH$ is a rectangle if and only if $ A, B, C, D$ are concyclic.
In rectangle $PQRS$, $PQ=8$ and $QR=6$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, points $E$ and $F$ lie on $\overline{RS}$, and points $G$ and $H$ lie on $\overline{SP}$ so that $AP=BQ<4$ and the convex octagon $ABCDEFGH$ is equilateral. The length of a side of this octagon can be expressed in the form $k+m\sqrt{n}$, where $k$, $m$, and $n$ are integers and $n$ is not divisible by the square of any prime. What is $k+m+n$?
$\textbf{(A) } 1 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 21 \qquad \textbf{(D) } 92 \qquad \textbf{(E) } 106$
Two players play a game involving an $n \times n$ grid of chocolate. Each turn, a player may either eat a piece of chocolate (of any size), or split an existing piece of chocolate into two rectangles along a grid-line. The player who moves last loses. For how many positive integers $n$ less than $1000$ does the second player win?
(Splitting a piece of chocolate refers to taking an $a \times b$ piece, and breaking it into an $(a-c) \times b$ and a $c \times b$ piece, or an $a \times (b-d)$ and an $a \times d$ piece.)
[i]Proposed by Lewis Chen[/i]
Two players, Red and Blue, play in alternating turns on a 10x10 board. Blue goes first. In his turn, a player picks a row or column (not chosen by any player yet) and color all its squares with his own color. If any of these squares was already colored, the new color substitutes the old one.
The game ends after 20 turns, when all rows and column were chosen. Red wins if the number of red squares in the board exceeds at least by 10 the number of blue squares; otherwise Blue wins.
Determine which player has a winning strategy and describe this strategy.
Given the true statement: If a quadrilateral is a square, then it is a rectangle. It follows that, of the converse and the inverse of this true statement is:
$ \textbf{(A)}\ \text{only the converse is true} \qquad\textbf{(B)}\ \text{only the inverse is true }\qquad \textbf{(C)}\ \text{both are true} \qquad$ $\textbf{(D)}\ \text{neither is true} \qquad\textbf{(E)}\ \text{the inverse is true, but the converse is sometimes true} $
From each vertex of triangle $ABC$ we draw 3 arbitary parrallell lines, and from each vertex we draw a perpendicular to these lines. There are 3 rectangles that one of their diagnals is triangle's side. We draw their other diagnals and call them $\ell_1$, $\ell_2$ and $\ell_3$.
a) Prove that $\ell_1$, $\ell_2$ and $\ell_3$ are concurrent at a point $P$.
b) Find the locus of $P$ as we move the 3 arbitary lines.
Given quadrilateral $ABCD$ and the directions of its sides. Inscribe a rectangle in $ABCD$.
Given a rectangle $ABCD$ with a point $P$ inside it. It is known that $PA =
17, PB = 15,$ and $PC = 6.$
What is the length of $PD$?
A non-square rectangle is cut into $N$ rectangles of various shapes and sizes. Prove that one can always cut each of these rectangles into two rectangles so that one can construct a square and rectangle, each figure consisting of $N$ pieces.
[i](6 points)[/i]
$ABCD$ is a quadrilateral with right angles at $A$ and $C$. Points $E$ and $F$ are on $AC$, and $DE$ and $BF$ are perpendicular to $AC$. If $AE=3$, $DE=5$, and $CE=7$, then $BF=$
[asy]
draw((0,0)--(10,0)--(3,-5)--(0,0)--(6.5,3)--(10,0));
draw((6.5,0)--(6.5,3));
draw((3,0)--(3,-5));
dot((0,0));
dot((10,0));
dot((3,0));
dot((3,-5));
dot((6.5,0));
dot((6.5,3));
label("A", (0,0), W);
label("B", (6.5,3), N);
label("C", (10,0), E);
label("D", (3,-5), S);
label("E", (3,0), N);
label("F", (6.5,0), S);[/asy]
$\text{(A)} \ 3.6 \qquad \text{(B)} \ 4 \qquad \text{(C)} \ 4.2 \qquad \text{(D)} \ 4.5 \qquad \text{(E)} \ 5$
Suppose $ABCD$ and $AEFG$ are rectangles such that the points $B,E,D,G$ are collinear (in this order). Let the lines $BC$ and $GF$ intersect at point $T$ and let the lines $DC$ and $EF$ intersect at point $H$. Prove that points $A, H$ and $T$ are collinear.
$ABCD$ is a cyclic quadrilateral. The lines $AD$ and $BC$ meet at X, and the lines $AB$ and $CD$ meet at $Y$ . The line joining the midpoints $M$ and $N$ of the diagonals $AC$ and $BD$, respectively, meets the internal bisector of angle $AXB$ at $P$ and the external bisector of angle $BYC$ at $Q$. Prove that $PXQY$ is a rectangle
The length of rectangle $ABCD$ is $5$ inches and its width is $3$ inches. Diagonal $AC$ is dibided into three equal segments by points $E$ and $F$. The area of triangle $BEF$, expressed in square inches, is:
$\text{(A)} \ \frac 32 \qquad \text{(B)} \ \frac 53 \qquad \text{(C)} \ \frac 52 \qquad \text{(D)} \ \frac13\sqrt{34} \qquad \text{(E)} \ \frac13\sqrt{68}$
Given three congruent rectangles in the space. Their centers coincide, but the planes they lie in are mutually perpendicular. For any two of the three rectangles, the line of intersection of the planes of these two rectangles contains one midparallel of one rectangle and one midparallel of the other rectangle, and these two midparallels have different lengths. Consider the convex polyhedron whose vertices are the vertices of the rectangles.
[b]a.)[/b] What is the volume of this polyhedron ?
[b]b.)[/b] Can this polyhedron turn out to be a regular polyhedron ? If yes, what is the condition for this polyhedron to be regular ?
Let $ABC$ be acute triangle. Inscribe a rectangle $DEFG$ in this triangle such that $D\in AB,E\in AC,F\in BC,G\in BC$. Describe the locus of (i.e., the curve occupied by) the intersections of the diagonals of all possible rectangles $DEFG$.
Every point in the plane is coloured one of seven distinct colours. Is there an inscribed trapezoid whose vertices are all of the same colour?
In a $2^n \times 2^n$ square with $n$ positive integer is covered with at least two non-overlapping rectangle pieces with integer dimensions and a power of two as surface. Prove that two rectangles of the covering have the same dimensions (Two rectangles have the same dimensions as they have the same width and the same height, wherein they, not allowed to be rotated.)
Let $E$ be the point of intersection of the diagonals of convex quadrilateral $ABCD$, and let $P,Q,R,$ and $S$ be the centers of the circles circumscribing triangles $ABE,$ $BCE$, $CDE$, and $ADE$, respectively. Then
$\textbf{(A) }PQRS\text{ is a parallelogram}$
$\textbf{(B) }PQRS\text{ is a parallelogram if an only if }ABCD\text{ is a rhombus}$
$\textbf{(C) }PQRS\text{ is a parallelogram if an only if }ABCD\text{ is a rectangle}$
$\textbf{(D) }PQRS\text{ is a parallelogram if an only if }ABCD\text{ is a parallelogram}$
$\textbf{(E) }\text{none of the above are true}$
The set $ G$ is defined by the points $ (x,y)$ with integer coordinates, $ 3\le|x|\le7$, $ 3\le|y|\le7$. How many squares of side at least $ 6$ have their four vertices in $ G$?
[asy]defaultpen(black+0.75bp+fontsize(8pt));
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draw((-8,0)--(8.5,0),Arrow(HookHead,1mm));
draw((0,-8)--(0,8.5),Arrow(HookHead,1mm));
int i,j;
for (i=-7;i<8;++i) {
for (j=-7;j<8;++j) {
if (((-7 <= i) && (i <= -3)) || ((3 <= i) && (i<= 7))) { if (((-7 <= j) && (j <= -3)) || ((3 <= j) && (j<= 7))) { fill(shift(i,j)*p,black); }}}} draw((-7,-.2)--(-7,.2),black+0.5bp);
draw((-3,-.2)--(-3,.2),black+0.5bp);
draw((3,-.2)--(3,.2),black+0.5bp);
draw((7,-.2)--(7,.2),black+0.5bp);
draw((-.2,-7)--(.2,-7),black+0.5bp);
draw((-.2,-3)--(.2,-3),black+0.5bp);
draw((-.2,3)--(.2,3),black+0.5bp);
draw((-.2,7)--(.2,7),black+0.5bp);
label("$-7$",(-7,0),S);
label("$-3$",(-3,0),S);
label("$3$",(3,0),S);
label("$7$",(7,0),S);
label("$-7$",(0,-7),W);
label("$-3$",(0,-3),W);
label("$3$",(0,3),W);
label("$7$",(0,7),W);[/asy]$ \textbf{(A)}\ 125\qquad \textbf{(B)}\ 150\qquad \textbf{(C)}\ 175\qquad \textbf{(D)}\ 200\qquad \textbf{(E)}\ 225$
If the diagonals of a quadrilateral are perpendicular to each other, the figure would always be included under the general classification:
$ \textbf{(A)}\ \text{rhombus} \qquad\textbf{(B)}\ \text{rectangles} \qquad\textbf{(C)}\ \text{square} \qquad\textbf{(D)}\ \text{isosceles trapezoid}\qquad\textbf{(E)}\ \text{none of these} $
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.
[i]Proposed by Jeck Lim, Singapore[/i]
Andy has 2010 square tiles, each of which has a side length of one unit. He plans to arrange
the tiles in an m x n rectangle, where mn = 2010. Compute the sum of the perimeters of all
of the different possible rectangles he can make. Two rectangles are considered to be the same
if one can be rotated to become the other, so, for instance, a 1 x 2010 rectangle is considered
to be the same as a 2010 x 1 rectangle.
In the diagram of rectangles below, with lengths as labeled, let $A$ be the area of the rectangle labeled $A$, and so on. Find $36A+6B+C+6D$.
[asy]
size(3cm);
real[] A = {0,8,13};
real[] B = {0,7,12};
for (int i = 0; i < 3; ++i) {
draw((A[i],0)--(A[i],-B[2]));
draw((0,-B[i])--(A[2],-B[i]));
}
label("8", (4,0), N);
label("5", (10.5,0),N);
label("7", (0,-3.5),W);
label("5", (0,-9.5),W);
label("$A$", (4,-3.5));
label("$B$", (10.5,-3.5));
label("$C$", (10.5,- 9.5));
label("$D$", (4, -9.5));
[/asy]
[i]2018 CCA Math Bonanza Team Round #1[/i]
Circles $ C_1$ and $ C_2$ with centers at $ O_1$ and $ O_2$ respectively meet at points $ A$ and $ B$. The radii $ O_1B$ and $ O_2B$ meet $ C_1$ and $ C_2$ at $ F$ and$ E$. The line through $ B$ parallel to $ EF$ intersects $ C_1$ again at $ M$ and $ C_2$ again at $ N$. Prove that $ MN \equal{} AE \plus{} AF$.
In an $n \times n$ square array of $1\times1$ cells, at least one cell is colored pink. Show that you can always divide the square into rectangles along cell borders such that each rectangle contains exactly one pink cell.