Found problems: 22
2017-2018 SDML (Middle School), 14
Amy made a list of every possible distinct five-digit positive integer that can be formed using each of the digits $1, 2, 3, 4,$ and $5$ exactly once in each integer. What is the sum of the integers on Amy's list?
$\mathrm{(A) \ } 3000000 \qquad \mathrm{(B) \ } 3600000 \qquad \mathrm {(C) \ } 3999960 \qquad \mathrm{(D) \ } 3999990 \qquad \mathrm{(E) \ } 5999940$
2017-2018 SDML (Middle School), 1
Evaluate $\frac{3^4 + 3^7}{84}$.
$ \mathrm{(A) \ } 27 \qquad \mathrm{(B) \ } 29 \qquad \mathrm {(C) \ } 33 \qquad \mathrm{(D) \ } 37 \qquad \mathrm{(E) \ } 39$
2017-2018 SDML (Middle School), 6
In the figure, a circle is located inside a trapezoid with two right angles so that a point of tangency of the circle is the midpoint of the side perpendicular to the two bases. The circle also has points of tangency on each base of the trapezoid. The diameter of the circle is $\frac{2}{3}$ the length of $EF$. If the area of the circle is $9\pi$ square units, what is the area of the trapezoid?
[asy]
draw((0,0) -- (11, 0) -- (7,6) -- (0,6) -- cycle);
draw((0,3) -- (9,3));
draw(circle((3,3), 3));
draw(rightanglemark((1,0),(0,0),(0,1),12));
draw(rightanglemark((0,0),(0,6),(6,6), 12));
label("E", (0,3), W);
label("F", (9,3), E);
[/asy]
2017-2018 SDML (Middle School), 8
Albert and Bob and Charlie are each thinking of a number. Albert's number is one more than twice Bob's. Bob's number is one more than twice Charlie's, and Charlie's number is two more than twice Albert's. What number is Albert thinking of?
$\mathrm{(A) \ } -\frac{11}{7} \qquad \mathrm{(B) \ } -2 \qquad \mathrm {(C) \ } -1 \qquad \mathrm{(D) \ } -\frac{4}{7} \qquad \mathrm{(E) \ } \frac{1}{2}$
2017-2018 SDML (Middle School), 10
Mrs. Krabappel gives a five-question pop quiz one Monday. Nobody is ready, so everyone guesses and gets exactly three questions correct. The students later discover that they each answered a different set of three questions correctly. What is the largest possible number of students in the class?
$\mathrm{(A) \ } 9 \qquad \mathrm{(B) \ } 10 \qquad \mathrm {(C) \ } 11 \qquad \mathrm{(D) \ } 12 \qquad \mathrm{(E) \ } 13$
2017-2018 SDML (Middle School), 11
How many three-digit numbers leave remainder $2$ when divided by $5$ and leave remainder $7$ when divided by $9$?
$\mathrm{(A) \ } 20 \qquad \mathrm{(B) \ } 21 \qquad \mathrm {(C) \ } 22 \qquad \mathrm{(D) \ } 23 \qquad \mathrm{(E) \ } 24$
2017-2018 SDML (Middle School), 4
Two congruent squares are packed into an isoceles right triangle as shown below. Each of the squares has area 10. What is the area of the triangle?
[asy]
draw((0,0) -- (3*sqrt(10), 0) -- (0, 3*sqrt(10)) -- cycle);
draw((0,0) -- (2*sqrt(10), 0) -- (2*sqrt(10), sqrt(10)) -- (0, sqrt(10)));
draw((sqrt(10), sqrt(10)) -- (sqrt(10), 0));
[/asy]
$\mathrm{(A) \ } 40 \qquad \mathrm{(B) \ } 90 \qquad \mathrm {(C) \ } \frac{85}{2} \qquad \mathrm{(D) \ } 50 \qquad \mathrm{(E) \ } 45$
2017-2018 SDML (Middle School), 2
A circle and a square are drawn on the plane so that they overlap. Together, the two shapes cover an area of $329$ square units. The area common to both shapes is $101$ square units. The area of the circle is $234$ square units. What is the perimeter of the square in units?
$\mathrm{(A) \ } 14 \qquad \mathrm{(B) \ } 48 \qquad \mathrm {(C) \ } 56 \qquad \mathrm{(D) \ } 64 \qquad \mathrm{(E) \ } 196$
2017-2018 SDML (Middle School), 5
If $(x + 1) + (x + 2) + ... + (x + 20) = 174 + 176 + 178 + ... + 192$, then what is the value of $x$?
$\mathrm{(A) \ } 80 \qquad \mathrm{(B) \ } 81 \qquad \mathrm {(C) \ } 82 \qquad \mathrm{(D) \ } 83 \qquad \mathrm{(E) \ } 84$
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?
$\mathrm{(A) \ } 11 \qquad \mathrm{(B) \ } 12 \qquad \mathrm {(C) \ } 13 \qquad \mathrm{(D) \ } 14 \qquad \mathrm{(E) \ } 15$
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?
$\mathrm{(A) \ } 2 \qquad \mathrm{(B) \ } 4 \qquad \mathrm {(C) \ } 5 \qquad \mathrm{(D) \ } 10 \qquad \mathrm{(E) \ } 20$
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]
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?
$\mathrm{(A) \ } 4 \qquad \mathrm{(B) \ } 5 \qquad \mathrm {(C) \ } 6 \qquad \mathrm{(D) \ } 7 \qquad \mathrm{(E) \ } 10$
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?
$\mathrm{(A) \ } \frac{35}{48} \qquad \mathrm{(B) \ } \frac{35}{64} \qquad \mathrm {(C) \ } \frac{3}{8} \qquad \mathrm{(D) \ } \frac{5}{12} \qquad \mathrm{(E) \ } \frac{5}{8}$
2017-2018 SDML (Middle School), 1
Let $N = \frac{1}{3} + \frac{3}{5} + \frac{5}{7} + \frac{7}{9} + \frac{9}{11}$. What is the greatest integer which is less than $N$?
2017-2018 SDML (Middle School), 3
Evaluate the following expression: $$0 - 1 -2 + 3 - 4 + 5 + 6 + 7 - 8 + ... + 2000$$ The terms with minus signs are exactly the powers of two.
2017-2018 SDML (Middle School), 5
If the sum of the slope and the $y$-intercept of a line is $3$, then through which point is the line guaranteed to pass?
2017-2018 SDML (Middle School), 12
If $n$ is an integer such that $2 \leq n \leq 2017$, for how many values of $n$ is $\left(1 + \frac{1}{2}\right)\left(1 + \frac{1}{3}\right)\cdots\left(1 + \frac{1}{n}\right)$ equal to a positive integer?
$\mathrm{(A) \ } 0 \qquad \mathrm{(B) \ } 1 \qquad \mathrm {(C) \ } 1007 \qquad \mathrm{(D) \ } 1008 \qquad \mathrm{(E) \ } 2016$
2017-2018 SDML (Middle School), 8
Gorf the frog is standing on the first lily pad in a row of lily pads numbered from $1$ to $20$ from left to right. On a single jump, Gorf is able to jump either $1,2,$ or $3$ lily pads to the right. Unfortunately all the prime-numbered lily pads are contaminated with a deadly poison. How many sequences of jumps are there that allow Gorf to jump to the twentieth lily pad, while avoiding the poison?
2017-2018 SDML (Middle School), 2
How many ways are there to cover this region with dominoes?
[asy]
unitsize(20);
int[][] a = {
{999, 999, 000, 000, 000, 999, 999, 999},
{999, 999, 000, 888, 000, 999, 999, 999},
{999, 999, 000, 000, 000, 000, 000, 000},
{000, 000, 000, 888, 888, 000, 888, 000},
{000, 888, 000, 888, 888, 000, 000, 000},
{000, 000, 000, 000, 000, 000, 999, 999},
{999, 999, 999, 000, 888, 000, 999, 999},
{999, 999, 999, 000, 000, 000, 999, 999}};
for (int i = 0; i < 8; ++i) {
for (int j = 0; j < 8; ++j) {
if (a[j][i] != 999) draw((i, -j)--(i+1, -j)--(i+1, -j-1)--(i, -j-1)--cycle);
if (a[j][i] == 888) fill((i, -j)--(i+1, -j)--(i+1, -j-1)--(i, -j-1)--cycle);
}
}
[/asy]