Found problems: 1342
Let $S=\{(i,j) \vert i,j=1,2,\ldots ,100\}$ be a set consisting of points on the coordinate plane. Each element of $S$ is colored one of four given colors. A subset $T$ of $S$ is called [i]colorful[/i] if $T$ consists of exactly $4$ points with distinct colors, which are the vertices of a rectangle whose sides are parallel to the coordinate axes. Find the maximum possible number of colorful subsets $S$ can have, among all legitimate coloring patters.
In the rectangle $ABCD, M, N, P$ and $Q$ are the midpoints of the sides. If the area of the shaded triangle is $1$, calculate the area of the rectangle $ABCD$.
[img]https://2.bp.blogspot.com/-9iyKT7WP5fc/XNYuXirLXSI/AAAAAAAAKK4/10nQuSAYypoFBWGS0cZ5j4vn_hkYr8rcwCK4BGAYYCw/s400/may3.gif[/img]
Let $ABCD$ be a rectangle with diagonals of length $10.$ Let $P$ be the midpoint of $\overline{AD},$ $S$ be the midpoint of $\overline{BC},$ and $T$ be the midpoint of $\overline{CD}.$ Points $Q$ and $R$ are chosen on $\overline{AB}$ such that $AP=AQ$ and $BR=BS,$ and minor arcs $\widehat{PQ}$ and $\widehat{RS}$ centered at $A$ and $B,$ respectively, are drawn. Circle $\omega$ is tangent to $\overline{CD}$ at $T$ and externally tangent to $\widehat{PQ}$ and $\widehat{RS}.$ Suppose that the radius of $\omega$ is $\tfrac{43}{18}.$ Then the sum of all possible values of the area of $ABCD$ can be written in the form $\tfrac{a+b\sqrt{c}}{d},$ where $a,\ b,\ c,$ and $d$ are positive integers, $b$ and $d$ are relatively prime, and $c$ is prime. Find $a+b+c+d.$
A $ 9 \times 12$ rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres $ C_1,C_2,\ldots ,C_{96}$ in such way that the following to conditions are both fulfilled
i) the distances $C_1C_2,\ldots ,C_{95}C_{96}, C_{96}C_{1}$ are all equal to $ \sqrt {13}$,
ii) the closed broken line $ C_1C_2\ldots C_{96}C_1$ has a centre of symmetry?
[i]Bulgaria[/i]
Prove that there exists a positive real number $N$ such that for arbitrary real numbers $a,b>N$ it is possible to cover the perimeter of a rectangle with side lengths $a$ and $b$ using non-overlapping unit disks (the unit disks can be tangent to each other).
[i]Submitted by Benedek Váli, Budapest[/i]
A given rectangle $ R$ is divided into $mn$ small rectangles by straight lines parallel to its sides. (The distances between the parallel lines may not be equal.) What is the minimum number of appropriately selected rectangles’ areas that should be known in order to determine the area of $ R$?
The base of a new rectangle equals the sum of the diagonal and the greater side of a given rectangle, while the altitude of the new rectangle equals the difference of the diagonal and the greater side of the given rectangle. The area of the new rectangle is:
$ \textbf{(A)}$ greater than the area of the given rectangle
$ \textbf{(B)}$ equal to the area of the given rectangle
$ \textbf{(C)}$ equal to the area of a square with its side equal to the smaller side of the given rectangle
$ \textbf{(D)}$ equal to the area of a square with its side equal to the greater side of the given rectangle
$ \textbf{(E)}$ equal to the area of a rectangle whose dimensions are the diagonal and the shorter side of the given rectangle
Let $n$ and $m$ be positive integers in the range $[1, 10^{10}]$. Let $R$ be the rectangle with corners at $(0, 0), (n, 0), (n, m), (0, m)$ in the coordinate plane. A simple non-self-intersecting quadrilateral with vertices at integer coordinates is called [i]far-reaching[/i] if each of its vertices lie on or inside $R$, but each side of $R$ contains at least one vertex of the quadrilateral. Show that there is a far-reaching quadrilateral with area at most $10^6$.
Let $ABC$ be an equilateral triangle. Let $P$ and $S$ be points on $AB$ and $AC$, respectively, and let $Q$ and $R$ be points on $BC$ such that $PQRS$ is a rectangle. If $PQ = \sqrt3 PS$ and the area of $PQRS$ is $28\sqrt3$, what is the length of $PC$?
Let $ABCD$ be a rectangle and $P$ be a point on the side$ AD$ such that $\angle BPC = 90^o$. The perpendicular from $A$ on $BP$ cuts $BP$ at $M$ and the perpendicular from $D$ on $CP$ cuts $CP$ in $N$. Show that the center of the rectangle lies in the $MN$ segment.
Find the locus of the center of a rectangle, whose four vertices lies on the sides of a given triangle.
Determine all natural numbers $n$ for which it is possible to construct a rectangle of sides $15$ and $n$, with pieces congruent to:
[asy]
unitsize(0.6 cm);
draw((0,0)--(3,0));
draw((0,1)--(3,1));
draw((0,2)--(1,2));
draw((2,2)--(3,2));
draw((0,0)--(0,2));
draw((1,0)--(1,2));
draw((2,0)--(2,2));
draw((3,0)--(3,2));
draw((5,-0.5)--(6,-0.5));
draw((4,0.5)--(7,0.5));
draw((4,1.5)--(7,1.5));
draw((5,2.5)--(6,2.5));
draw((4,0.5)--(4,1.5));
draw((5,-0.5)--(5,2.5));
draw((6,-0.5)--(6,2.5));
draw((7,0.5)--(7,1.5));
[/asy]
The squares of the pieces have side $1$ and the pieces cannot overlap or leave free spaces
On the cartesian plane are drawn several rectangles with the sides parallel to the coordinate axes. Assume that any two rectangles can be cut by a vertical or a horizontal line. Show that it's possible to draw one horizontal and one vertical line such that each rectangle is cut by at least one of these two lines.
Rectangle $ABCD$ is inscribed in circle $O$. Let the projections of a point $P$ on minor arc $CD$ onto $AB,AC,BD$ be $K,L,M$, respectively. Prove that $\angle LKM=45$if and only if $ABCD$ is a square.
More than three quarters of the circumference of a circle is colored black. Prove that there exists a rectangle such that all of its vertices are black.
Actually the result holds if "three quarters" is replaced by "one half"...
The points $S(i, j)$ with integer Cartesian coordinates $0 < i \leq n, 0 < j \leq m, m \leq n$, form a lattice. Find the number of:
[b](a)[/b] rectangles with vertices on the lattice and sides parallel to the coordinate axes;
[b](b)[/b] squares with vertices on the lattice and sides parallel to the coordinate axes;
[b](c)[/b] squares in total, with vertices on the lattice.
In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a [i]box[/i]. Two boxes [i]intersect[/i] if they have a common point in their interior or on their boundary. Find the largest $ n$ for which there exist $ n$ boxes $ B_1$, $ \ldots$, $ B_n$ such that $ B_i$ and $ B_j$ intersect if and only if $ i\not\equiv j\pm 1\pmod n$.
[i]Proposed by Gerhard Woeginger, Netherlands[/i]
The area of the rectangular region is
[asy]
draw((0,0)--(4,0)--(4,2.2)--(0,2.2)--cycle,linewidth(.5 mm));
label(".22 m",(4,1.1),E);
label(".4 m",(2,0),S);
[/asy]
$\text{(A)}\ \text{.088 m}^2 \qquad \text{(B)}\ \text{.62 m}^2 \qquad \text{(C)}\ \text{.88 m}^2 \qquad \text{(D)}\ \text{1.24 m}^2 \qquad \text{(E)}\ \text{4.22 m}^2$
In the cartesian plane consider rectangles with sides parallel to the coordinate axes. We say that one rectangle is [i]below[/i] another rectangle if there is a line $g$ parallel to the $x$-axis such that the first rectangle is below $g$, the second one above $g$ and both rectangles do not touch $g$.
Similarly, we say that one rectangle is [i]to the right of[/i] another rectangle if there is a line $h$ parallel to the $y$-axis such that the first rectangle is to the right of $h$, the second one to the left of $h$ and both rectangles do not touch $h$.
Show that any finite set of $n$ pairwise disjoint rectangles with sides parallel to the coordinate axes can be enumerated as a sequence $(R_1,\dots,R_n)$ so that for all indices $i,j$ with $1 \le i<j \le n$ the rectangle $R_i$ is to the right of or below the rectangle $R_j$
For every positive integer $m$, a $m\times 2018$ rectangle consists of unit squares (called "cell") is called [i]complete[/i] if the following conditions are met:
i. In each cell is written either a "$0$", a "$1$" or nothing;
ii. For any binary string $S$ with length $2018$, one may choose a row and complete the empty cells so that the numbers in that row, if read from left to right, produce $S$ (In particular, if a row is already full and it produces $S$ in the same manner then this condition ii. is satisfied).
A [i]complete[/i] rectangle is called [i]minimal[/i], if we remove any of its rows and then making it no longer [i]complete[/i].
a. Prove that for any positive integer $k\le 2018$ there exists a [i]minimal[/i] $2^k\times 2018$ rectangle with exactly $k$ columns containing both $0$ and $1$.
b. A [i]minimal[/i] $m\times 2018$ rectangle has exactly $k$ columns containing at least some $0$ or $1$ and the rest of columns are empty. Prove that $m\le 2^k$.
Let $ ABC$ be an acute-angled triangle; $ AD$ be the bisector of $ \angle BAC$ with $ D$ on $ BC$; and $ BE$ be the altitude from $ B$ on $ AC$.
Show that $ \angle CED > 45^\circ .$
[b][weightage 17/100][/b]
Cells of a $n*n$ square are filled with positive integers in the way that in the intersection of the $i-$th column and $j-$th row, the number $i+j$ is written. In every step, we can choose two non-intersecting equal rectangles with one dimension equal to $n$ and swap all the numbers inside these two rectangles with one another. ( without reflection or rotation ) Find the minimum number of moves one should do to reach the position where the intersection of the $i-$th column and $j-$row is written $2n+2-i-j$.
In the $3\times5$ grid shown, fill in each empty box with a two-digit positive integer such that:
[list][*]no number appears in more than one box, and
[*] for each of the $9$ lines in the grid consisting of three boxes connected by line segments, the box in the middle of the line contains the least common multiple of the numbers in the two boxes on the line.[/list]
You do not need to prove that your answer is the only one possible; you merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)
[asy]
import graph; size(7cm);
real labelscalefactor = 0.5;
pen dps = linewidth(0.8) + fontsize(14);
defaultpen(dps);
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
draw((2,0)--(3,0)--(3,1)--(2,1)--cycle);
draw((4,0)--(5,0)--(5,1)--(4,1)--cycle);
draw((6,0)--(7,0)--(7,1)--(6,1)--cycle);
draw((8,0)--(9,0)--(9,1)--(8,1)--cycle);
draw((0,2)--(1,2)--(1,3)--(0,3)--cycle);
draw((0,4)--(1,4)--(1,5)--(0,5)--cycle);
draw((2,2)--(3,2)--(3,3)--(2,3)--cycle);
draw((2,4)--(3,4)--(3,5)--(2,5)--cycle);
draw((4,4)--(5,4)--(5,5)--(4,5)--cycle);
draw((4,2)--(5,2)--(5,3)--(4,3)--cycle);
draw((6,2)--(7,2)--(7,3)--(6,3)--cycle);
draw((6,4)--(7,4)--(7,5)--(6,5)--cycle);
draw((8,4)--(9,4)--(9,5)--(8,5)--cycle);
draw((8,2)--(9,2)--(9,3)--(8,3)--cycle);
draw((0.5,1)--(0.5,2));
draw((0.5,3)--(0.5,4));
draw((1,4)--(2,3));
draw((2.5,1)--(2.5,2));
draw((2.5,3)--(2.5,4));
draw((3,4)--(4,3));
draw((3,2)--(4,1));
draw((4.5,1)--(4.5,2));
draw((4.5,3)--(4.5,4));
draw((5,4.5)--(6,4.5));
draw((7,4.5)--(8,4.5));
draw((5,4)--(6,3));
draw((7,2)--(8,1));
draw((5,2)--(6,1));
draw((5,0.5)--(6,0.5));
draw((7,0.5)--(8,0.5));
draw((8.5,1)--(8.5,2));
draw((8.5,3)--(8.5,4));
label("$4$",(4.5, 0.5));
label("$9$",(8.5, 4.5));
[/asy]
Jeck and Lisa are playing a game on table dimensions $m \times n$ , where $m , n >2$. Lisa starts so that she puts knight figurine on arbitrary square of table.After that Jeck and Lisa put new figurine on table by the following rules:
[b]1.[/b] Jeck puts queen figurine on any empty square of a table which is two squares vertically and one square horizontally distant, or one square vertically and two squares horizontally distant from last knight figurine which Lisa put on the table
[b]2.[/b] Lisa puts knight figurine on any empty square of a table which is in the same row, column or diagonal as last queen figurine Jeck put on the table.
Player which cannot put his figurine loses. For which pairs of $(m,n)$ Lisa has winning strategy?
[i]
Proposed by Stijn Cambie[/i]
There are 5 points in a rectangle (including its boundary) with area 1, no three of them are in the same line. Find the minimum number of triangles with the area not more than $\frac 1{4}$, vertex of which are three of the five points.