Found problems: 63
2014-2015 SDML (High School), 4
Two regular square pyramids have all edges $12$ cm in length. The pyramids have parallel bases and those bases have parallel edges, and each pyramid has its apex at the center of the other pyramid's base. What is the total number of cubic centimeters in the volume of the solid of intersection of the two pyramids?
2016-2017 SDML (Middle School), 7
Point $P$ is selected at random from the interior of the pentagon with vertices $A = (0, 2), B = (4, 0), C = (2\pi + 1, 0), D = (2\pi + 1, 4),$ and $E = (0, 4)$. What is the probability that $\angle ABP$ is obtuse? Express your answer as a common fraction.
2014-2015 SDML (Middle School), 4
If you pick a random $3$-digit number, what is the probability that its hundreds digit is triple the ones digit?
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?
$\mathrm{(A) \ } 3000000 \qquad \mathrm{(B) \ } 3600000 \qquad \mathrm {(C) \ } 3999960 \qquad \mathrm{(D) \ } 3999990 \qquad \mathrm{(E) \ } 5999940$
2016-2017 SDML (Middle School), 2
On a Cartesian coordinate plane, points $(1, 2)$ and $(7, 4)$ are opposite vertices of a square. What is the area of the square?
2012-2013 SDML (High School), 4
For what digit $A$ is the numeral $1AA$ a perfect square in base-$5$ and a perfect cube in base-$6$?
$\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }2\qquad\text{(D) }3\qquad\text{(E) }4$
2013-2014 SDML (Middle School), 3
Simplify $\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}$.
2014-2015 SDML (High School), 3
At summer camp, there are $20$ campers in each of the swimming class, the archery class, and the rock climbing class. Each camper is in at least one of these classes. If $4$ campers are in all three classes, and $24$ campers are in exactly one of the classes, how many campers are in exactly two classes?
$\text{(A) }10\qquad\text{(B) }11\qquad\text{(C) }12\qquad\text{(D) }13\qquad\text{(E) }14$
2014-2015 SDML (High School), 1
If you pick a random $3$-digit number, what is the probability that its hundreds digit is triple the ones digit?
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?
$\mathrm{(A) \ } -\frac{11}{7} \qquad \mathrm{(B) \ } -2 \qquad \mathrm {(C) \ } -1 \qquad \mathrm{(D) \ } -\frac{4}{7} \qquad \mathrm{(E) \ } \frac{1}{2}$
2016-2017 SDML (Middle School), 4
What is the sum of the last two digits of $7^{42} + 7^{43}$ in base $10$.
$\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }8\qquad\text{(D) }9\qquad\text{(E) }11$
2016-2017 SDML (Middle School), 11
Emily has an infinite number of balls and empty boxes available to her. The empty boxes, each capable of holding four balls, are arranged in a row from left to right. At the first step, she places a ball in the first box of the row. At each subsequent step, she places a ball in the first box of the row that still has room for a ball and empties any previous boxes. How many balls in total are in the boxes as a result of Emily's $2017$th step?
$\text{(A) }9\qquad\text{(B) }11\qquad\text{(C) }13\qquad\text{(D) }15\qquad\text{(E) }17$
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?
$\mathrm{(A) \ } 9 \qquad \mathrm{(B) \ } 10 \qquad \mathrm {(C) \ } 11 \qquad \mathrm{(D) \ } 12 \qquad \mathrm{(E) \ } 13$
2014-2015 SDML (Middle School), 12
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 $f\left(g\left(0\right)\right)=103$.
$\text{(A) }2\qquad\text{(B) }5\qquad\text{(C) }7\qquad\text{(D) }10\qquad\text{(E) }17$
2014-2015 SDML (Middle School), 8
Two regular square pyramids have all edges $12$ cm in length. The pyramids have parallel bases and those bases have parallel edges, and each pyramid has its apex at the center of the other pyramid's base. What is the total number of cubic centimeters in the volume of the solid of intersection of the two pyramids?
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]
$\mathrm{(A) \ } 40 \qquad \mathrm{(B) \ } 90 \qquad \mathrm {(C) \ } \frac{85}{2} \qquad \mathrm{(D) \ } 50 \qquad \mathrm{(E) \ } 45$
2016-2017 SDML (Middle School), 2
Each term of the sequence $5, 12, 19, 26, \cdots$ is $7$ more than the term that precedes it. What is the first term of the sequence that is greater than $2017$?
$\text{(A) }2018\qquad\text{(B) }2019\qquad\text{(C) }2020\qquad\text{(D) }2021\qquad\text{(E) }2022$
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 $x$?
$\mathrm{(A) \ } 80 \qquad \mathrm{(B) \ } 81 \qquad \mathrm {(C) \ } 82 \qquad \mathrm{(D) \ } 83 \qquad \mathrm{(E) \ } 84$
2016-2017 SDML (Middle School), 15
A regular hexagon $ABCDEF$ has area $36$. Find the area of the region which lies in the overlap of the triangles $ACE$ and $BDF$.
$\text{(A) }3\qquad\text{(B) }9\qquad\text{(C) }12\qquad\text{(D) }18\qquad\text{(E) }24$
2014-2015 SDML (Middle School), 15
How many triangles formed by three vertices of a regular $17$-gon are obtuse?
$\text{(A) }156\qquad\text{(B) }204\qquad\text{(C) }357\qquad\text{(D) }476\qquad\text{(E) }524$
2012-2013 SDML (Middle School), 11
Six different-sized cubes are glued together, one on top of the other. The bottom cube has edge length $8$. Each of the other cubes has four vertices at the midpoints of the edges of the cube below it as shown. The entire solid is then dipped in red paint. What is the total area of the red-painted surface on the solid?
(will insert image here later)
$\text{(A) }630\qquad\text{(B) }632\qquad\text{(C) }648\qquad\text{(D) }694\qquad\text{(E) }756$