Found problems: 1342
In rectangle $ABCD$, $AD=1$, $P$ is on $AB$ and $DB$ and $DP$ trisect $\angle ADC$. What is the perimeter $\triangle BDP$
[list=1]
[*] $3+\frac{\sqrt{3}}{3}$
[*] $2+\frac{4\sqrt{3}}{3}$
[*] $2+2\sqrt{2}$
[*] $\frac{3+3\sqrt{5}}{2}$
[/list]
Let $ABCD$ be a convex quadrilateral and points $E$ and $F$ on sides $AB,CD$ such that
\[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\]
If $S$ is the area of $AEFD$ show that ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$
A rectangular grid whose side lengths are integers greater than $1$ is given. Smaller rectangles with area equal to an odd integer and length of each side equal to an integer greater than $1$ are cut out one by one. Finally one single unit is left. Find the least possible area of the initial grid before the cuttings.
Ps. Collected [url=https://artofproblemsolving.com/community/c949611_2019_nigerian_senior_mo_round_3]here[/url]
Given a positive integer $n$, determine the largest real number $\mu$ satisfying the following condition: for every set $C$ of $4n$ points in the interior of the unit square $U$, there exists a rectangle $T$ contained in $U$ such that
$\bullet$ the sides of $T$ are parallel to the sides of $U$;
$\bullet$ the interior of $T$ contains exactly one point of $C$;
$\bullet$ the area of $T$ is at least $\mu$.
Let $n$ be a positive integer. A 4-by-$n$ rectangle is divided into $4n$ unit squares in the usual way. Each unit square is colored black or white. Suppose that every white unit square shares an edge with at least one black unit square. Prove that there are at least $n$ black unit squares.
Let $ABCD$ be a rectangle and the circle of center $D$ and radius $DA$, which cuts the extension of the side $AD$ at point $P$. Line $PC$ cuts the circle at point $Q$ and the extension of the side $AB$ at point $R$. Show that $QB = BR$.
Carl has a rectangle whose side lengths are positive integers. This rectangle has the property that when he increases the width by 1 unit and decreases the length by 1 unit, the area increases by $x$ square units. What is the smallest possible positive value of $x$?
[i]Proposed by Ray Li[/i]
A line $l$ intersects the segment $AB$ perpendicular to $C$. Three circles are drawn successively with $AB, AC$ and $BC$ as the diameter. The largest circle intersects $l$ in $D$. The segments $DA$ and $DB$ still intersect the two smaller circles in $E$ and $F$.
a. Prove that quadrilateral $CFDE$ is a rectangle.
b. Prove that the line through $E$ and $F$ touches the circles with diameters $AC$ and $BC$ in $E$ and $F$.
[asy]
unitsize (2.5 cm);
pair A, B, C, D, E, F, O;
O = (0,0);
A = (-1,0);
B = (1,0);
C = (-0.3,0);
D = intersectionpoint(C--(C + (0,1)), Circle(O,1));
E = (C + reflect(A,D)*(C))/2;
F = (C + reflect(B,D)*(C))/2;
draw(Circle(O,1));
draw(Circle((A + C)/2, abs(A - C)/2));
draw(Circle((B + C)/2, abs(B - C)/2));
draw(A--B);
draw(interp(C,D,-0.4)--D);
draw(A--D--B);
dot("$A$", A, W);
dot("$B$", B, dir(0));
dot("$C$", C, SE);
dot("$D$", D, NW);
dot("$E$", E, SE);
dot("$F$", F, SW);
[/asy]
The lengths of the sides of a rectangle are given to be odd integers. Prove that there does not exist a point within that rectangle that has integer distances to each of its four vertices.
A rectangle \( m \times n \), where \( m \) and \( n \) are natural numbers strictly greater than 1, is partitioned into \( mn \) unit squares, each of which can be colored either black or white. An operation consists of changing the color of all the squares in a row or in a column to the opposite color. Is it possible that, although initially exactly one square is colored black and all the others are white, after a finite number of moves all squares have the same color?
An $8\times 8$ chessboard is made of unit squares. We put a rectangular piece of paper with sides of length 1 and 2. We say that the paper and a single square overlap if they share an inner point. Determine the maximum number of black squares that can overlap the paper.
Problem 4. A chessboard with $2n \times 2n$ tiles is coloured such that every tile is coloured with one out of $n$ colours. Prove that there exists 2 tiles in either the same column or row such that if the colours of both tiles are swapped, then there exists a rectangle where all its four corner tiles have the same colour.
Given triangle $ABC$. Point $O_1$ is the center of the $BCDE$ rectangle, constructed so that the side $DE$ of the rectangle contains the vertex $A$ of the triangle. Points $O_2$ and $O_3$ are the centers of rectangles constructed in the same way on the sides $AC$ and $AB$, respectively. Prove that lines $AO_1, BO_2$ and $CO_3$ meet at one point.
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]
Two congruent rectangles are located as shown in the figure. Find the area of the shaded part.
[img]https://cdn.artofproblemsolving.com/attachments/2/e/10b164535ab5b3a3b98ce1a0b84892cd11d76f.png[/img]
Two $5\times1$ rectangles have 2 vertices in common as on the picture.
(a) Determine the area of overlap
(b) Determine the length of the segment between the other 2 points of intersection, $A$ and $B$.
[img]https://cdn.artofproblemsolving.com/attachments/9/0/4f1721c7ccdecdfe4d9cc05a17a553a0e9f670.png[/img]
Pinocchio claims that he can take some non-right-angled triangles , all of which are similar to one another and some of which may be congruent to one another, and put them together to form a rectangle. Is Pinocchio lying?
(A Fedotov)
A figure is an equiangular parallelogram if and only if it is a
$ \textbf{(A)}\ \text{rectangle}\qquad\textbf{(B)}\ \text{regular polygon}\qquad\textbf{(C)}\ \text{rhombus}\qquad\textbf{(D)}\ \text{square}\qquad\textbf{(E)}\ \text{trapezoid} $
Three acute triangles are inscribed in the same circle with their vertices being nine distinct points. Show that one can choose a vertex from each triangle so that the three chosen points determine a triangle each of whose angles is at most $90^\circ$.
Point $K$ is marked on the diagonal $AC$ in rectangle $ABCD$ so that $CK = BC$. On the side $BC$, point $M$ is marked so that $KM = CM$. Prove that $AK + BM = CM$.
$ABCD$ is a rectangle with side lengths $AB = CD = 1$ and $BC = DA = 2$. Let $ M$ be the midpoint of $AD$. Point $P$ lies on the opposite side of line $MB$ to $A$, such that triangle $MBP$ is equilateral. Find the value of $\angle PCB$.
How many rectangles can be formed by the vertices of a cube? (Note: square is also a special rectangle).
A. $6$ B. $8$ C. $12$ D. $18$ E. $16$
In the rectangle $ABCD$, point $M$ is the midpoint of the side $BC$. The points $P$ and $Q$ lie on the diagonal $AC$ such that $\angle DPC = \angle DQM = 90^o$. Prove that $Q$ is the midpoint of the segment $AP$.
Some squares of an $n \times n$ chessboard have been marked ($n \in N^*$). Prove that if the number of marked squares is at least $n\left(\sqrt{n} + \frac12\right)$, then there exists a rectangle whose vertices are centers of marked squares.
In the diagram below, square $ABCD$ with side length 23 is cut into nine rectangles by two lines parallel to $\overline{AB}$ and two lines parallel to $\overline{BC}$. The areas of four of these rectangles are indicated in the diagram. Compute the largest possible value for the area of the central rectangle.
[asy]
size(250);
defaultpen (linewidth (0.7) + fontsize (10));
draw ((0,0)--(23,0)--(23,23)--(0,23)--cycle);
label("$A$", (0,23), NW);
label("$B$", (23, 23), NE);
label("$C$", (23,0), SE);
label("$D$", (0,0), SW);
draw((0,6)--(23,6));
draw((0,19)--(23,19));
draw((5,0)--(5,23));
draw((12,0)--(12,23));
label("13", (17/2, 21));
label("111",(35/2,25/2));
label("37",(17/2,3));
label("123",(2.5,12.5));[/asy]
[i]Proposed by Lewis Chen[/i]