Found problems: 120
2018-2019 SDML (High School), 11
For the system of equations $x^2 + x^2y^2 + x^2y^4 = 525$ and $x + xy + xy^2 = 35$, the sum of the real $y$ values that satisfy the equations is
$ \mathrm{(A) \ } 2 \qquad \mathrm{(B) \ } \frac{5}{2} \qquad \mathrm {(C) \ } 5 \qquad \mathrm{(D) \ } 20 \qquad \mathrm{(E) \ } \frac{55}{2}$
2012-2013 SDML (High School), 1
Let $\bullet$ be the operation such that $a\bullet b=10a-b$. What is the value of $\left(\left(\left(2\bullet0\right)\bullet1\right)\bullet3\right)$?
$\text{(A) }1969\qquad\text{(B) }1987\qquad\text{(C) }1993\qquad\text{(D) }2007\qquad\text{(E) }2013$
2012-2013 SDML (High School), 3
Let $b=\log_53$. What is $\log_b\left(\log_35\right)$?
$\text{(A) }-1\qquad\text{(B) }-\frac{3}{5}\qquad\text{(C) }0\qquad\text{(D) }\frac{3}{5}\qquad\text{(E) }1$
2011-2012 SDML (High School), 3
The $42$ points $P_1,P_2,\ldots,P_{42}$ lie on a straight line, in that order, so that the distance between $P_n$ and $P_{n+1}$ is $\frac{1}{n}$ for all $1\leq n\leq41$. What is the sum of the distances between every pair of these points? (Each pair of points is counted only once.)
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), 8
What is the maximum area of a triangle that can be inscribed in an ellipse with semi-axes $a$ and $b$?
$\text{(A) }ab\frac{3\sqrt{3}}{4}\qquad\text{(B) }ab\qquad\text{(C) }ab\sqrt{2}\qquad\text{(D) }\left(a+b\right)\frac{3\sqrt{3}}{4}\qquad\text{(E) }\left(a+b\right)\sqrt{2}$
2014-2015 SDML (High School), 4
What is the maximum number of points that can be placed in the interior of an equilateral triangle of side length $2$ such that the distance between any two points is greater than one?
$\text{(A) }3\qquad\text{(B) }4\qquad\text{(C) }5\qquad\text{(D) }6\qquad\text{(E) }7$
2014-2015 SDML (High School), 6
Let $f\left(x\right)=x^2-14x+52$ and $g\left(x\right)=ax+b$, where $a$ and $b$ are positive. Find $a$, given that $f\left(g\left(-5\right)\right)=3$ and $f\left(g\left(0\right)\right)=103$.
$\text{(A) }2\qquad\text{(B) }5\qquad\text{(C) }7\qquad\text{(D) }10\qquad\text{(E) }17$
2018-2019 SDML (High School), 12
How many ordered pairs $(s, d)$ of positive integers with $4 \leq s \leq d \leq 2019$ are there such that when $s$ silver balls and $d$ diamond balls are randomly arranged in a row, the probability that the balls on each end have the same color is $\frac{1}{2}$?
$ \mathrm{(A) \ } 58 \qquad \mathrm{(B) \ } 59 \qquad \mathrm {(C) \ } 60 \qquad \mathrm{(D) \ } 61 \qquad \mathrm{(E) \ } 62$
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.
$\text{(A) }31\qquad\text{(B) }57\qquad\text{(C) }66\qquad\text{(D) }87\qquad\text{(E) }112$
2011-2012 SDML (High School), 7
Let $x$ and $y$ be nonnegative real numbers such that $x+y=1$. Find the maximum value of $x^4y+xy^4$.
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 $1000$?
$\text{(A) }136\qquad\text{(B) }294\qquad\text{(C) }508\qquad\text{(D) }692\qquad\text{(E) }803$
2012-2013 SDML (High School), 11
Suppose that $\cos\left(3x\right)+3\cos\left(x\right)=-2$. What is the value of $\cos\left(2x\right)$?
$\text{(A) }-\frac{1}{2}\qquad\text{(B) }-\frac{1}{\sqrt[3]{2}}\qquad\text{(C) }\frac{1}{\sqrt[3]{2}}\qquad\text{(D) }\sqrt[3]{2}-1\qquad\text{(E) }\frac{1}{2}$
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]
2012-2013 SDML (High School), 2
If five boys and three girls are randomly divided into two four-person teams, what is the probability that all three girls will end up on the same team?
$\text{(A) }\frac{1}{7}\qquad\text{(B) }\frac{2}{7}\qquad\text{(C) }\frac{1}{10}\qquad\text{(D) }\frac{1}{14}\qquad\text{(E) }\frac{1}{28}$
2018-2019 SDML (High School), 2
Given that $\frac{x}{\sqrt{x} + \sqrt{y}} = 18$ and $\frac{y}{\sqrt{x} + \sqrt{y}} = 2$, find $\sqrt{x} - \sqrt{y}$.
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 $3$?
$\text{(A) }\frac{17}{75}\qquad\text{(B) }\frac{43}{399}\qquad\text{(C) }\frac{127}{401}\qquad\text{(D) }\frac{16}{19}\qquad\text{(E) }\frac{6}{7}$
2011-2012 SDML (High School), 3
Two standard six-sided dice are tossed. What is the probability that the sum of the numbers is greater than $7$?
$\text{(A) }1\qquad\text{(B) }\frac{5}{12}\qquad\text{(C) }\frac{2}{3}\qquad\text{(D) }\frac{4}{9}\qquad\text{(E) }\frac{7}{36}$
2012-2013 SDML (High School), 10
Pentagon $ABCDE$ is inscribed in a circle such that $ACDE$ is a square with area $12$. What is the largest possible area of pentagon $ABCDE$?
$\text{(A) }9+3\sqrt{2}\qquad\text{(B) }13\qquad\text{(C) }12+\sqrt{2}\qquad\text{(D) }14\qquad\text{(E) }12+\sqrt{6}-\sqrt{3}$
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 (Middle School), 5
A circle of radius $5$ is inscribed in an isosceles right triangle, $ABC$. The length of the hypotenuse of $ABC$ can be expressed as $a+a\sqrt{2}$ for some $a$. What is $a$?
2018-2019 SDML (High School), 1
Seven children, each with the same birthday, were born in seven consecutive years. The sum of the ages of the youngest three children in $42$. What is the sum of the ages of the oldest three?
$ \mathrm{(A) \ } 51 \qquad \mathrm{(B) \ } 54 \qquad \mathrm {(C) \ } 57 \qquad \mathrm{(D) \ } 60 \qquad \mathrm{(E) \ } 63$
2018-2019 SDML (High School), 5
The graph of the equation $y = ax^2 + bx + c$ is shown in the diagram. Which of the following must be positive?
[DIAGRAM NEEDED]
$ \mathrm{(A) \ } a \qquad \mathrm{(B) \ } ab^2 \qquad \mathrm {(C) \ } b - c \qquad \mathrm{(D) \ } bc \qquad \mathrm{(E) \ } c - a$