Found problems: 63
2016-2017 SDML (Middle School), 10
For how many positive integer values of $a$ is it true that $x = 2$ is the only positive integer solution of the system of inequalities $$\begin{cases} 2x > 3x - 3 \\ 3x - a > -6 \end{cases}$$
$\text{(A) }1\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }4\qquad\text{(E) }5$
2016-2017 SDML (Middle School), 1
What is the integer value of $\left(\sqrt{3}^{\sqrt{2}}\right)^{\sqrt{8}}$?
2016-2017 SDML (Middle School), 7
If $f(1) = 1$ and $f(n+1) = \frac{2f(n) + 1}{2}$, then find $f(237)$.
$\text{(A) }117\qquad\text{(B) }118\qquad\text{(C) }119\qquad\text{(D) }120\qquad\text{(E) }121$
2016-2017 SDML (Middle School), 6
There are $4$ pairs of men and women, and all $8$ people are arranged in a row so that in each pair the woman is somewhere to the left of the man. How many such arrangements are there?
2016-2017 SDML (Middle School), 1
A "domino" is made up of two small squares:
[asy]
unitsize(10);
draw((0,0) -- (2,0) -- (2,1) -- (0,1) -- cycle);
fill((0,0) -- (1,0) -- (1,1) -- (0,1) -- cycle);
[/asy]
Which of the "checkerboards" illustrated below CANNOT be covered exactly and completely by a whole number of non-overlapping dominoes?
[diagram requires in-line asy]
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 (Middle School), 2
Suppose $a=1332$ and $b=-222$. Find $c$ such that $\left(\frac{a}{c}\right)^3=\sqrt{b^6}$.
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 $f\left(g\left(0\right)\right)=103$.
$\text{(A) }2\qquad\text{(B) }5\qquad\text{(C) }7\qquad\text{(D) }10\qquad\text{(E) }17$
2017-2018 SDML (Middle School), 7
Nathan has a collection of weights each weighing either $1, 2, 3,$ or $5$ pounds (and he has an infinite number of each weight). In how many ways can he measure out eight pounds?
$\mathrm{(A) \ } 11 \qquad \mathrm{(B) \ } 12 \qquad \mathrm {(C) \ } 13 \qquad \mathrm{(D) \ } 14 \qquad \mathrm{(E) \ } 15$
2016-2017 SDML (Middle School), 12
What is the area of the region enclosed by the graph of the equations $x^2 - 14x + 3y + 70 = 21 + 11y - y^2$ that lies below the line $y = x-3$?
$\text{(A) }6\pi\qquad\text{(B) }7\pi\qquad\text{(C) }8\pi\qquad\text{(D) }9\pi\qquad\text{(E) }10\pi$
2016-2017 SDML (Middle School), 3
The five tires of a car (four road tires and a full-sized spare) were rotated so that each tire was used the same number of miles during the first $30,000$ miles the car traveled. For how many miles was each tire used?
$\text{(A) }6000\qquad\text{(B) }7500\qquad\text{(C) }24,000\qquad\text{(D) }30,000\qquad\text{(E) }37,500$
2016-2017 SDML (Middle School), 8
Find the coefficient of $x^7$ in the polynomial expansion of $(1 + 2x - x^2)^4$.
2017-2018 SDML (Middle School), 13
In the diagram, two circles, each with center D, have radii of $1$ and $2$. The total area of the shaded region is $\frac{5}{12}$ of the area of the larger circle. How many degrees are in the measure of $\angle ADC$?
[asy]
int angle = 100;
path A = arc(0, 1, 0, angle);
path B = arc(0, 1, angle, 360);
path C = arc(0, 2, 0, angle);
path D = arc(0, 2, angle, 360);
filldraw(C -- origin -- cycle, gray);
filldraw(D -- origin -- cycle, white);
filldraw(A -- origin -- cycle, white);
filldraw(B -- origin -- cycle, gray);
label("$D$", origin, NE);
label("$C$", (2, 0), E);
label("$A$", (2, 0) * dir(angle), N);
[/asy]
$\mathrm{(A) \ } 100 \qquad \mathrm{(B) \ } 105 \qquad \mathrm {(C) \ } 110 \qquad \mathrm{(D) \ } 115 \qquad \mathrm{(E) \ } 120$
2017-2018 SDML (Middle School), 6
Lori makes a list of all the numbers between $1$ and $999$ inclusive. She first colors all the multiples of $5$ red. Then she colors blue every number which is adjacent to a red number. How many numbers in her list are left uncolored?
$\mathrm{(A) \ } 400 \qquad \mathrm{(B) \ } 402 \qquad \mathrm {(C) \ } 597 \qquad \mathrm{(D) \ } 600 \qquad \mathrm{(E) \ } 602$
2017-2018 SDML (Middle School), 3
Charlie plans to sell bananas for forty cents and apples for fifty cents at his fruit stand, but Dave accidentally reverses the prices. After selling all their fruit they earn a dollar more than they would have with the original prices. How many more bananas than apples did they sell?
$\mathrm{(A) \ } 2 \qquad \mathrm{(B) \ } 4 \qquad \mathrm {(C) \ } 5 \qquad \mathrm{(D) \ } 10 \qquad \mathrm{(E) \ } 20$
2014-2015 SDML (Middle School), 3
Layna wants to paint a rectangular wall green, but she only has blue and yellow paint. She finds that a $2:1$ mix of blue paint to yellow paint produces the color green she wants, and she knows that one gallon of paint will cover $80$ square feet of wall. If the wall is $8$ feet tall and $21$ feet long, how many gallons of blue paint does Layna need? Express your answer as a fraction in simplest form.
2016-2017 SDML (Middle School), 8
An ice cream cone has radius $1$ and height $4$ inches. What is the number of inches in the radius of a sphere of ice cream which has the same volume of the cone?
$\text{(A) }\frac{1}{2}\qquad\text{(B) }1\qquad\text{(C) }\frac{3}{2}\qquad\text{(D) }2\qquad\text{(E) }\frac{5}{2}$
2017-2018 SDML (Middle School), 4
The diagram below shows an equilateral triangle and a square of side length $2$ joined along an edge. What is the area of the shaded triangle?
[asy]
fill((2,0) -- (2,2) -- (1, 2 + sqrt(3)) -- cycle, gray);
draw((0,0) -- (2,0) -- (2,2) -- (1, 2 + sqrt(3)) -- (0,2) -- (0,0));
draw((0,2) -- (2,2));
[/asy]
2016-2017 SDML (Middle School), 6
What is the probability that a random arrangement of the letters in the word 'ARROW' will have both R's next to each other?
$\text{(A) }\frac{1}{10}\qquad\text{(B) }\frac{2}{15}\qquad\text{(C) }\frac{1}{5}\qquad\text{(D) }\frac{3}{10}\qquad\text{(E) }\frac{2}{5}$
2017-2018 SDML (Middle School), 15
For all positive integers $n$ the function $f$ satisfies $f(1) = 1, f(2n + 1) = 2f(n),$ and $f(2n) = 3f(n) + 2$. For how many positive integers $x \leq 100$ is the value of $f(x)$ odd?
$\mathrm{(A) \ } 4 \qquad \mathrm{(B) \ } 5 \qquad \mathrm {(C) \ } 6 \qquad \mathrm{(D) \ } 7 \qquad \mathrm{(E) \ } 10$
2014-2015 SDML (High School), 13
How many triangles formed by three vertices of a regular $17$-gon are obtuse?
$\text{(A) }156\qquad\text{(B) }204\qquad\text{(C) }357\qquad\text{(D) }476\qquad\text{(E) }524$
2017-2018 SDML (Middle School), 9
Jerry has a four-sided die, a six-sided die, and an eight-sided die. Each die is numbered starting at one. Jerry rolls the three dice simultaneously. What is the probability that they all show different numbers?
$\mathrm{(A) \ } \frac{35}{48} \qquad \mathrm{(B) \ } \frac{35}{64} \qquad \mathrm {(C) \ } \frac{3}{8} \qquad \mathrm{(D) \ } \frac{5}{12} \qquad \mathrm{(E) \ } \frac{5}{8}$
2016-2017 SDML (Middle School), 14
Evaluate the sum $$\frac{1}{3^1} + \frac{2}{3^2} + \frac{3}{3^3} + \cdots + \frac{k}{3^k} + \cdots$$
$\text{(A) }\frac{5}{9}\qquad\text{(B) }\frac{5}{8}\qquad\text{(C) }\frac{2}{3}\qquad\text{(D) }\frac{3}{4}\qquad\text{(E) }\frac{7}{9}$
2016-2017 SDML (Middle School), 13
If Scott rolls four fair six-sided dice, what is the probability that he rolls more 2's than 1's?
$\text{(A) }\frac{8}{27}\qquad\text{(B) }\frac{25}{81}\qquad\text{(C) }\frac{103}{324}\qquad\text{(D) }\frac{421}{1296}\qquad\text{(E) }\frac{65}{162}$
2016-2017 SDML (Middle School), 5
A group of $25$ friends were discussing a large positive integer. "It can be divided by $1$," said the first friend. "It can be divided by $2$," said the second friend. "And by $3$," said the third friend. "And by $4$," added the fourth friend. This continued until everyone had made such a comment. If exactly $2$ friends were incorrect, and those two friends said consecutive numbers, what was the least possible integer they were discussing?