Found problems: 22
2014-2015 SDML (High School), 5
The squares in a $7\times7$ grid are colored one of two colors: green and purple. The coloring has the property that no green square is directly above or to the right of a purple square. Find the total number of ways this can be done.
2014-2015 SDML (High School), 8
Triangles $ABC$ and $BDC$ are such that $\angle{ABC}=\angle{BDC}=90^{\circ}$ and $\angle{DBC}=\angle{CAB}$. Let $Q$ be a point on $\overline{BD}$ such that $\overline{QC}\perp\overline{AD}$. Suppose that $BD=15$. Then $DQ$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2014-2015 SDML (High School), 5
Beth adds the consecutive positive integers $a$, $b$, $c$, $d$, and $e$, and finds that the sum is a perfect square. She then adds $b$, $c$, and $d$ and finds that this sum is a perfect cube. What is the smallest possible value of $e$?
$\text{(A) }47\qquad\text{(B) }137\qquad\text{(C) }227\qquad\text{(D) }677\qquad\text{(E) }1127$
2014-2015 SDML (High School), 15
Find the sum of all $\left\lfloor x\right\rfloor$ such that $x^2-15\left\lfloor x\right\rfloor+36=0$.
$\text{(A) }15\qquad\text{(B) }26\qquad\text{(C) }45\qquad\text{(D) }49\qquad\text{(E) }75$
2014-2015 SDML (High School), 3
A light flashes in one of three different colors: red, green, and blue. Every $3$ seconds, it flashes green. Every $5$ seconds, it flashes red. Every $7$ seconds, it flashes blue. If it is supposed to flash in two colors at once, it flashes the more infrequent color. How many times has the light flashed green after $671$ seconds?
$\text{(A) }148\qquad\text{(B) }154\qquad\text{(C) }167\qquad\text{(D) }217\qquad\text{(E) }223$
2014-2015 SDML (High School), 4
Evaluate $$1+\frac{1+\frac{1+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}{2+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}}{2+\frac{1+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}{2+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}}.$$
$\text{(A) }\frac{\sqrt{3}}{2}\qquad\text{(B) }\frac{1+\sqrt{5}}{2}\qquad\text{(C) }\frac{2+\sqrt{3}}{2}\qquad\text{(D) }\frac{3+\sqrt{5}}{2}\qquad\text{(E) }\frac{3+\sqrt{13}}{2}$
2014-2015 SDML (High School), 9
The quadrilateral $ABCD$ can be inscribed in a circle and $\angle{ABD}$ is a right angle. $M$ is the midpoint of $BD$, where $CM$ is an altitude of $\triangle{BCD}$. If $AB=14$ and $CD=6\sqrt{11}$, what [is] the length of $AD$?
$\text{(A) }36\qquad\text{(B) }38\qquad\text{(C) }41\qquad\text{(D) }42\qquad\text{(E) }44$
2014-2015 SDML (High School), 7
Find the sum of all positive integers $n$ such that $$\frac{n^3+8n^2+8n+80}{n+7}$$ is an integer.
$\text{(A) }31\qquad\text{(B) }57\qquad\text{(C) }66\qquad\text{(D) }87\qquad\text{(E) }112$
2014-2015 SDML (High School), 14
Dave's Amazing Hotel has $3$ floors. If you press the up button on the elevator from the $3$rd floor, you are immediately transported to the $1$st floor. Similarly, if you press the down button from the $1$st floor, you are immediately transported to the $3$rd floor. Dave gets in the elevator at the $1$st floor and randomly presses up or down at each floor. After doing this $482$ times, the probability that Dave is on the first floor can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is the remainder when $m+n$ is divided by $1000$?
$\text{(A) }136\qquad\text{(B) }294\qquad\text{(C) }508\qquad\text{(D) }692\qquad\text{(E) }803$
2014-2015 SDML (High School), 4
A rubber band is wrapped around two pipes as shown. One has radius $3$ inches and the other has radius $9$ inches. The length of the band can be expressed as $a\pi+b\sqrt{c}$ where $a$, $b$, $c$ are integers and $c$ is square free. What is $a+b+c$?
[asy]
size(4cm);
draw(circle((0,0),3));
draw(circle((12,0),9));
draw(3*dir(120)--(12,0)+9*dir(120));
draw(3*dir(240)--(12,0)+9*dir(240));
[/asy]
2014-2015 SDML (High School), 11
The numbers $1,2,\ldots,9$ are arranged so that the $1$st term is not $1$ and the $9$th term is not $9$. What is the probability that the third term is $3$?
$\text{(A) }\frac{17}{75}\qquad\text{(B) }\frac{43}{399}\qquad\text{(C) }\frac{127}{401}\qquad\text{(D) }\frac{16}{19}\qquad\text{(E) }\frac{6}{7}$
2014-2015 SDML (High School), 2
The number $15$ is written on a blackboard. A move consists of erasing the number $x$ and replacing it with $x+y$ where $y$ is a randomly chosen number between $1$ and $5$ (inclusive). The game ends when the number on the blackboard exceeds $51$. Which number is most likely to be on the blackboard at the end of the game?
$\text{(A) }52\qquad\text{(B) }53\qquad\text{(C) }54\qquad\text{(D) }55\qquad\text{(E) }56$
2014-2015 SDML (High School), 10
What is the sum of all $k\leq25$ such that one can completely cover a $k\times k$ square with $T$ tetrominos (shown in the diagram below) without any overlap?
[asy]
size(2cm);
draw((0,0)--(3,0));
draw((0,1)--(3,1));
draw((1,2)--(2,2));
draw((0,0)--(0,1));
draw((1,0)--(1,2));
draw((2,0)--(2,2));
draw((3,0)--(3,1));
[/asy]
$\text{(A) }20\qquad\text{(B) }24\qquad\text{(C) }84\qquad\text{(D) }108\qquad\text{(E) }154$
2014-2015 SDML (High School), 7
Let $S$ be a finite set of real numbers such that given any three distinct elements $x,y,z\in\mathbb{S}$, at least one of $x+y$, $x+z$, or $y+z$ is also contained in $S$. Find the largest possible number of elements that $S$ could have.
2014-2015 SDML (High School), 13
Six points are chosen on the unit circle such that the product of the distances from any other point on the unit circle is at most $2$. Find the area of the hexagon with these six points as vertices.
$\text{(A) }\frac{1}{2}\qquad\text{(B) }\frac{3}{2}\qquad\text{(C) }\frac{\sqrt{3}}{2}\qquad\text{(D) }\frac{3\sqrt{3}}{2}\qquad\text{(E) }\frac{3+\sqrt{3}}{2}$
2014-2015 SDML (High School), 3
Let $a$ and $b$ be the roots of the equation $x^2-47x+289=0$. Compute $\sqrt{a}+\sqrt{b}$.
2014-2015 SDML (High School), 2
Sally is thinking of a positive four-digit integer. When she divides it by any one-digit integer greater than $1$, the remainder is $1$. How many possible values are there for Sally's four-digit number?
2014-2015 SDML (High School), 6
Let $a$ and $b$ be positive reals such that $$a=1+\frac{a}{b}$$$$b=3+\frac{4+a}{b-2}$$ What is $a$?
$\text{(A) }\sqrt{2}\qquad\text{(B) }2+\sqrt{2}\qquad\text{(C) }2+\sqrt{2}+\sqrt[3]{2}\qquad\text{(D) }\sqrt{2}+\sqrt[3]{2}\qquad\text{(E) }\sqrt[3]{2}$
2014-2015 SDML (High School), 6
Find the largest integer $k$ such that $$k\leq\sqrt{2}+\sqrt[3]{\frac{3}{2}}+\sqrt[4]{\frac{4}{3}}+\sqrt[5]{\frac{5}{4}}+\cdots+\sqrt[2015]{\frac{2015}{2014}}.$$
2014-2015 SDML (High School), 12
An ant starts at the bottom left corner of a $5\times5$ grid of dots and walks to the top right corner. It can walk from one dot to any dot that is horizontally or vertically adjacent to it. If it never walks between the same pair of dots twice, what is the length of the longest path the ant can take?
$\text{(A) }30\qquad\text{(B) }31\qquad\text{(C) }32\qquad\text{(D) }33\qquad\text{(E) }34$
2014-2015 SDML (High School), 1
How many ways are there to color the vertices of a square green, red, or blue so that no two adjacent vertices have the same color? (Two colorings are considered different even if one coloring can be rotated to product the other coloring.)
2014-2015 SDML (High School), 8
A penny is placed in the coordinate plane $\left(0,0\right)$. The penny can be moved $1$ unit to the right, $1$ unit up, or diagonally $1$ unit to the right and $1$ unit up. How many different ways are there for the penny to get to the point $\left(5,5\right)$?
$\text{(A) }8\qquad\text{(B) }25\qquad\text{(C) }99\qquad\text{(D) }260\qquad\text{(E) }351$