Art of Problem Solving
AoPS Online AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy AoPS Academy
Small live classes for advanced math
and language arts learners in grades 1-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Stay ahead of learning milestones! Enroll in a class over the summer!
2015 AMC 12/AHSME
3
AMC 12/AHSME 2015
A
February 3rd
1
What is the value of $(2^0-1+5^2+0)^{-1}\times 5$?

$\textbf{(A) }-125\qquad\textbf{(B) }-120\qquad\textbf{(C) }\dfrac15\qquad\textbf{(D) }\dfrac5{24}\qquad\textbf{(E) }25$
djmathman
view topic
2
Two of the three sides of a triangle are $20$ and $15$. Which of the following numbers is not a possible perimeter of the triangle?

$\textbf{(A) }52\qquad\textbf{(B) }57\qquad\textbf{(C) }62\qquad\textbf{(D) }67\qquad\textbf{(E) }72$
djmathman
view topic
3
Mr. Patrick teaches math to $15$ students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was $80$. After he graded Payton's test, the class average became $81$. What was Payton's score on the test?

$\textbf{(A) }81\qquad\textbf{(B) }85\qquad\textbf{(C) }91\qquad\textbf{(D) }94\qquad\textbf{(E) }95$
djmathman
view topic
4
The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller?

$\textbf{(A) }\dfrac54\qquad\textbf{(B) }\dfrac32\qquad\textbf{(C) }\dfrac95\qquad\textbf{(D) }2\qquad\textbf{(E) }\dfrac52$
djmathman
view topic
5
Amelia needs to estimate the quantity $\tfrac ab-c$, where $a$, $b$, and $c$ are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of $\tfrac ab-c$?

$\textbf{(A) }\text{She rounds all three numbers up.}$

$\textbf{(B) }\text{She rounds }a\text{ and }b\text{ up, and she rounds }c\text{ down.}$

$\textbf{(C) }\text{She rounds }a\text{ and }c\text{ up, and she rounds }b\text{ down.}$

$\textbf{(D) }\text{She rounds }a\text{ up, and she rounds }b\text{ and }c\text{ down.}$

$\textbf{(E) }\text{She rounds }c\text{ up, and she rounds }a\text{ and }b\text{ down.}$
djmathman
view topic
6
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2:1$?

$\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8$
djmathman
view topic
7
Two right circular cylinders have the same volume. The radius of the second cylinder is $10\%$ more than the radius of the first. What is the relationship between the heights of the two cylinders?

$\textbf{(A) }\text{The second height is 10\% less than the first.}$

$\textbf{(B) }\text{The first height is 10\% more than the second.}$

$\textbf{(C) }\text{The second height is 21\% less than the first.}$

$\textbf{(D) }\text{The first height is 21\% more than the second.}$

$\textbf{(E) }\text{The second height is 80\% of the first.}$
djmathman
view topic
8
The ratio of the length to the width of a rectangle is $4:3$. If the rectangle has diagonal of length $d$, then the area may be expressed as $kd^2$ for some constant $k$. What is $k$?

$\textbf{(A) }\dfrac27\qquad\textbf{(B) }\dfrac37\qquad\textbf{(C) }\dfrac{12}{25}\qquad\textbf{(D) }\dfrac{16}{25}\qquad\textbf{(E) }\dfrac34$
djmathman
view topic
9
A box contains $2$ red marbles, $2$ green marbles, and $2$ yellow marbles. Carol takes $2$ marbles from the box at random; then Claudia takes $2$ of the remaining marbles at random; and then Cheryl takes the last two marbles. What is the probability that Cheryl gets $2$ marbles of the same color?

$\textbf{(A) }\dfrac1{10}\qquad\textbf{(B) }\dfrac16\qquad\textbf{(C) }\dfrac15\qquad\textbf{(D) }\dfrac13\qquad\textbf{(E) }\dfrac12$
djmathman
view topic
10
Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=80$. What is $x$?

$\textbf{(A) }8\qquad\textbf{(B) }10\qquad\textbf{(C) }15\qquad\textbf{(D) }18\qquad\textbf{(E) }26$
djmathman
view topic
11
On a sheet of paper, Isabella draws a circle of radius $2$, a circle of radius $3$, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly $k\geq 0$ lines. How many different values of $k$ are possible?

$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$
djmathman
view topic
12
The parabolas $y=ax^2-2$ and $y=4-bx^2$ intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area $12$. What is $a+b$?

$\textbf{(A) }1\qquad\textbf{(B) }1.5\qquad\textbf{(C) }2\qquad\textbf{(D) }2.5\qquad\textbf{(E) }3$
djmathman
view topic
13
A league with $12$ teams holds a round-robin tournament, with each team playing every other team once. Games either end with one team victorious or else end in a draw. A team scores $2$ points for every game it wins and $1$ point for every game it draws. Which of the following is $\textbf{not}$ a true statement about the list of $12$ scores?

$\textbf{(A) }\text{There must be an even number of odd scores.}$

$\textbf{(B) }\text{There must be an even number of even scores.}$

$\textbf{(C) }\text{There cannot be two scores of 0.}$

$\textbf{(D) }\text{The sum of the scores must be at least 100.}$

$\textbf{(E) }\text{The highest score must be at least 12.}$
djmathman
view topic
14
What is the value of $a$ for which $\frac1{\log_2a}+\frac1{\log_3a}+\frac1{\log_4a}=1$?

$\textbf{(A) }9\qquad\textbf{(B) }12\qquad\textbf{(C) }18\qquad\textbf{(D) }24\qquad\textbf{(E) }36$
djmathman
view topic
15
What is the minimum number of digits to the right of the decimal point needed to express the fraction $\dfrac{123\,456\,789}{2^{26}\cdot 5^4}$ as a decimal?

$\textbf{(A) }4\qquad\textbf{(B) }22\qquad\textbf{(C) }26\qquad\textbf{(D) }30\qquad\textbf{(E) }104$
djmathman
view topic
16
Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\tfrac{12}5\sqrt2$. What is the volume of the tetrahedron?

$\textbf{(A) }3\sqrt2\qquad\textbf{(B) }2\sqrt5\qquad\textbf{(C) }\dfrac{24}5\qquad\textbf{(D) }3\sqrt3\qquad\textbf{(E) }\dfrac{24}5\sqrt2$
djmathman
view topic
17
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?

$\textbf{(A) }\dfrac{47}{256}\qquad\textbf{(B) }\dfrac{3}{16}\qquad\textbf{(C) }\dfrac{49}{256}\qquad\textbf{(D) }\dfrac{25}{128}\qquad\textbf{(E) }\dfrac{51}{256}$
djmathman
view topic
18
The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$?

$\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$
djmathman
view topic
19
For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. How many different values of $p<2015$ are possible?

$\textbf{(A) }30\qquad\textbf{(B) }31\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$
djmathman
view topic
20
Isosceles triangles $T$ and $T'$ are not congruent but have the same area and the same perimeter. The sides of $T$ have lengths $5$, $5$, and $8$, while those of $T'$ have lengths $a$, $a$, and $b$. Which of the following numbers is closest to $b$?

$\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8$
djmathman
view topic
21
A circle of radius $r$ passes through both foci of, and exactly four points on, the ellipse with equation $x^2+16y^2=16$. The set of all possible values of $r$ is an interval $[a,b)$. What is $a+b$?

$\textbf{(A) }5\sqrt2+4\qquad\textbf{(B) }\sqrt{17}+7\qquad\textbf{(C) }6\sqrt2+3\qquad\textbf{(D) }\sqrt{15}+8\qquad\textbf{(E) }12$
djmathman
view topic
22
For each positive integer $n$, let $S(n)$ be the number of sequences of length $n$ consisting solely of the letters $A$ and $B$, with no more than three $A$s in a row and no more than three $B$s in a row. What is the remainder when $S(2015)$ is divided by $12$?

$\textbf{(A) }0\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }10$
djmathman
view topic
23
Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\tfrac12$ is $\tfrac{a-b\pi}c$, where $a$, $b$, and $c$ are positive integers and $\gcd(a,b,c)=1$. What is $a+b+c$?

$\textbf{(A) }59\qquad\textbf{(B) }60\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$
djmathman
view topic
24
Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\tfrac nd$ where $n$ and $d$ are integers with $1\leq d\leq 5$. What is the probability that \[(\cos(a\pi)+i\sin(b\pi))^4\] is a real number?

$\textbf{(A) }\dfrac3{50}\qquad\textbf{(B) }\dfrac4{25}\qquad\textbf{(C) }\dfrac{41}{200}\qquad\textbf{(D) }\dfrac6{25}\qquad\textbf{(E) }\dfrac{13}{50}$
djmathman
view topic
25
A collection of circles in the upper half-plane, all tangent to the $x$-axis, is constructed in layers as follows. Layer $L_0$ consists of two circles of radii $70^2$ and $73^2$ that are externally tangent. For $k\geq 1$, the circles in $\textstyle\bigcup_{j=0}^{k-1} L_j$ are ordered according to their points of tangency with the $x$-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer $L_k$ consists of the $2^{k-1}$ circles constructed in this way. Let $S=\textstyle\bigcup_{j=0}^6 L_j$, and for every circle $C$ denote by $r(C)$ its radius. What is \[\sum_{C\in S}\dfrac1{\sqrt{r(C)}}?\]

[asy] import olympiad; size(350); defaultpen(linewidth(0.7)); // define a bunch of arrays and starting points pair[] coord = new pair[65]; int[] trav = {32,16,8,4,2,1}; coord[0] = (0,73^2); coord[64] = (2*73*70,70^2); // draw the big circles and the bottom line path arc1 = arc(coord[0],coord[0].y,260,360); path arc2 = arc(coord[64],coord[64].y,175,280); fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.78)); fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.78)); draw(arc1^^arc2); draw((-930,0)--(70^2+73^2+850,0)); // We now apply the findCenter function 63 times to get // the location of the centers of all 63 constructed circles. // The complicated array setup ensures that all the circles // will be taken in the right order for(int i = 0;i<=5;i=i+1) { int skip = trav[i]; for(int k=skip;k<=64 - skip; k = k + 2*skip) { pair cent1 = coord[k-skip], cent2 = coord[k+skip]; real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2); real shiftx = cent1.x + sqrt(4*r1*rn); coord[k] = (shiftx,rn); } // Draw the remaining 63 circles } for(int i=1;i<=63;i=i+1) { filldraw(circle(coord[i],coord[i].y),gray(0.78)); }[/asy]

$\textbf{(A) }\dfrac{286}{35}\qquad\textbf{(B) }\dfrac{583}{70}\qquad\textbf{(C) }\dfrac{715}{73}\qquad\textbf{(D) }\dfrac{143}{14}\qquad\textbf{(E) }\dfrac{1573}{146}$
djmathman
view topic
B
February 25th
1
What is the value of $2-(-2)^{-2}$?

$ \textbf{(A) } -2 \qquad\textbf{(B) } \dfrac{1}{16} \qquad\textbf{(C) } \dfrac{7}{4} \qquad\textbf{(D) } \dfrac{9}{4} \qquad\textbf{(E) } 6 $
chezbgone
view topic
2
Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?

$ \textbf{(A) }\text{3:10 PM}\qquad\textbf{(B) }\text{3:30 PM}\qquad\textbf{(C) }\text{4:00 PM}\qquad\textbf{(D) }\text{4:10 PM}\qquad\textbf{(E) }\text{4:30 PM} $
ABCDE
view topic
3
Isaac has written down one integer two times and another integer three times. The sum of the five numbers is $100$, and one of the numbers is $28$. What is the other number?

$\textbf{(A) }8\qquad\textbf{(B) }11\qquad\textbf{(C) }14\qquad\textbf{(D) }15\qquad\textbf{(E) }18$
EulerMacaroni
view topic
4
David, Hikmet, Jack, Marta, Rand, and Todd were in a $12$-person race with $6$ other people. Rand finished $6$ places ahead of Hikmet. Marta finished $1$ place behind Jack. David finished $2$ places behind Hikmet. Jack finished $2$ places behind Todd. Todd finished $1$ place behind Rand. Marta finished in $6$th place. Who finished in $8$th place?

$\textbf{(A) } \text{David} \qquad\textbf{(B) } \text{Hikmet} \qquad\textbf{(C) } \text{Jack} \qquad\textbf{(D) } \text{Rand} \qquad\textbf{(E) } \text{Todd} $
chezbgone
view topic
5
The Tigers beat the Sharks $2$ out of the first $3$ times they played. They then played $N$ more times, and the Sharks ended up winning at least $95\%$ of all the games played. What is the minimum possible value for $N$?

$\textbf{(A) }35\qquad\textbf{(B) }37\qquad\textbf{(C) }39\qquad\textbf{(D) }41\qquad\textbf{(E) }43$
EulerMacaroni
view topic
6
Back in 1930, Tillie had to memorize her multiplication tables from $0\times 0$ through $12\times 12$. The multiplication table she was given had rows and columns labeled with the factors, and the products formed the body of the table. To the nearest hundredth, what fraction of the numbers in the body of the table are odd?

$\textbf{(A) }0.21\qquad\textbf{(B) }0.25\qquad\textbf{(C) }0.46\qquad\textbf{(D) }0.50\qquad\textbf{(E) }0.75$
EulerMacaroni
view topic
7
A regular $15$-gon has $L$ lines of symmetry, and the smallest positive angle for which it has rotational symmetry is $R$ degrees. What is $L+R$?

$\textbf{(A) }24\qquad\textbf{(B) }27\qquad\textbf{(C) }32\qquad\textbf{(D) }39\qquad\textbf{(E) }54$
EulerMacaroni
view topic
8
What is the value of $(625^{\log_{5}{2015}})^{\frac{1}{4}}$?

$\textbf{(A) }5\qquad\textbf{(B) }\sqrt[4]{2015}\qquad\textbf{(C) }625\qquad\textbf{(D) }2015\qquad\textbf{(E) }\sqrt[4]{5^{2015}}$
EulerMacaroni
view topic
9
Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is $\frac{1}{2}$, independently of what has happened before. What is the probability that Larry wins the game?

$\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{3}{5}\qquad\textbf{(C) }\frac{2}{3}\qquad\textbf{(D) }\frac{3}{4}\qquad\textbf{(E) }\frac{4}{5}$
EulerMacaroni
view topic
10
How many noncongruent integer-sided triangles with positive area and perimeter less than $15$ are neither equilateral, isosceles, nor right triangles?

$\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7$
EulerMacaroni
view topic
11
The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?

$\textbf{(A) } 20 \qquad\textbf{(B) } \dfrac{360}{17} \qquad\textbf{(C) } \dfrac{107}{5} \qquad\textbf{(D) } \dfrac{43}{2} \qquad\textbf{(E) } \dfrac{281}{13} $
chezbgone
view topic
12
Let $a$, $b$, and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $(x-a)(x-b)+(x-b)(x-c)=0$?

$ \textbf {(A) } 15 \qquad \textbf {(B) } 15.5 \qquad \textbf {(C) } 16 \qquad \textbf {(D) } 16.5 \qquad \textbf {(E) } 17 $
ahaanomegas
view topic
13
Quadrilateral $ABCD$ is inscribed inside a circle with $\angle BAC= 70^{\circ}, \angle ADB= 40^{\circ}, AD=4$, and $BC=6$. What is $AC$?

$\textbf{(A) }3+\sqrt{5}\qquad\textbf{(B) }6\qquad\textbf{(C) }\frac{9}{2}\sqrt{2}\qquad\textbf{(D) }8-\sqrt{2}\qquad\textbf{(E) }7$
EulerMacaroni
view topic
14
A circle of radius $2$ is centered at $A$. An equilateral triangle with side $4$ has a vertex at $A$. What is the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle?

$ \textbf {(A) } 8-\pi \qquad \textbf {(B) } \pi + 2 \qquad \textbf {(C) } 2\pi - \frac {\sqrt{2}}{2} \qquad \textbf {(D) } 4(\pi - \sqrt{3}) \qquad \textbf {(E) } 2\pi + \frac {\sqrt{3}}{2} $
ahaanomegas
view topic
15
At Rachelle's school an A counts 4 points, a B 3 points, a C 2 points, and a D 1 point. Her GPA on the four classes she is taking is computed as the total sum of points divided by $4$. She is certain that she will get As in both Mathematics and Science, and at least a C in each of English and History. She think she has a $\frac{1}{6}$ chance of getting an A in English, and a $\frac{1}{4}$ chance of getting a B. In History, she has a $\frac{1}{4}$ chance of getting an A, and a $\frac{1}{3}$ chance of getting a B, independently of what she gets in English. What is the probability that Rachelle will get a GPA of at least 3.5?

$\textbf{(A) }\frac{11}{72}\qquad\textbf{(B) }\frac{1}{6}\qquad\textbf{(C) }\frac{3}{16}\qquad\textbf{(D) }\frac{11}{24}\qquad\textbf{(E) }\frac{1}{2}$
EulerMacaroni
view topic
16
A regular hexagon with sides of length $6$ has an isosceles triangle attached to each side. Each of these triangles has two sides of length $8$. The isosceles triangles are folded to make a pyramid with the hexagon as the base of the pyramid. What is the volume of the pyramid?

$\textbf{(A) }18\qquad\textbf{(B) }162\qquad\textbf{(C) }36\sqrt{21}\qquad\textbf{(D) }18\sqrt{138}\qquad\textbf{(E) }54\sqrt{21}$
EulerMacaroni
view topic
17
An unfair coin lands on heads with a probability of $\tfrac{1}{4}$. When tossed $n$ times, the probability of exactly two heads is the same as the probability of exactly three heads. What is the value of $n$?

$ \textbf {(A) } 5 \qquad \textbf {(B) } 8 \qquad \textbf {(C) } 10 \qquad \textbf {(D) } 11 \qquad \textbf {(E) } 13 $
ahaanomegas
view topic
18
For every composite positive integer $n$, define $r(n)$ to be the sum of the factors in the prime factorization of $n$. For example, $r(50)=12$ because the prime factorization of $50$ is $ 2 \cdot 5^2 $, and $ 2 + 5 + 5 = 12 $. What is the range of the function $r$, $ \{ r(n) : n \ \text{is a composite positive integer} \} $?

(A) the set of positive integers
(B) the set of composite positive integers
(C) the set of even positive integers
(D) the set of integers greater than 3
(E) the set of integers greater than 4
ahaanomegas
view topic
19
In $\triangle{ABC}$, $\angle{C} = 90^{\circ}$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$, and $W$ lie on a circle. What is the perimeter of the triangle?

$ \textbf{(A)}\ 12+9\sqrt{3}\qquad\textbf{(B)}\ 18+6\sqrt{3}\qquad\textbf{(C)}\ 12+12\sqrt{2}\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 32 $
va2010
view topic
20
For every positive integer $n$, let $\operatorname{mod_5}(n)$ be the remainder obtained when $n$ is divided by $5$. Define a function $f : \{0, 1, 2, 3, \dots\} \times \{0, 1, 2, 3, 4\} \to \{0, 1, 2, 3, 4\}$ recursively as follows:
\[f(i, j) = \begin{cases} \operatorname{mod_5}(j+1) & \text{if }i=0\text{ and }0\leq j\leq 4 \\ f(i-1, 1) & \text{if }i\geq 1\text{ and }j=0 \text{, and}\\ f(i-1, f(i, j-1)) & \text{if }i\geq 1\text{ and }1\leq j\leq 4 \end{cases}\]
What is $f(2015, 2)$?

$\textbf{(A) }0 \qquad\textbf{(B) }1 \qquad\textbf{(C) }2 \qquad\textbf{(D) }3 \qquad\textbf{(E) }4$
numbersandnumbers
view topic
21
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose the Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$?

$\textbf{(A) } 9 \qquad\textbf{(B) } 11 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 13 \qquad\textbf{(E) } 15 $
droid347
view topic
22
Six chairs are evenly spaced around a circular table. One person is seated in each chair. Each person gets up and sits down in a chair that is not the same chair and is not adjacent to the chair he or she originally occupied, so that again one person is seated in each chair. In how many ways can this be done?
$ \textbf{(A) }14\qquad\textbf{(B) }16\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad\textbf{(E) }24 $
raxu
view topic
23
A rectangular box measures $a \times b \times c$, where $a,$ $b,$ and $c$ are integers and $1 \leq a \leq b \leq c$. The volume and surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible?

$ \textbf{(A) }4\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }21\qquad\textbf{(E) }26 $
zacchro
view topic
24
Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. What is $AR+BR+CR+DR$?

$ \textbf{(A)}\ 180 \qquad\textbf{(B)}\ 184 \qquad\textbf{(C)}\ 188 \qquad\textbf{(D)}\ 192\qquad\textbf{(E)}\ 196 $
brandbest1
view topic
25
A bee starts flying from point $P_0$. She flies 1 inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015}$ she is exactly $a\sqrt{b} + c\sqrt{d}$ inches away from $P_0$, where $a$, $b$, $c$ and $d$ are positive integers and $b$ and $d$ are not divisible by the square of any prime. What is $a+b+c+d$?

$ \textbf{(A)}\ 2016 \qquad\textbf{(B)}\ 2024 \qquad\textbf{(C)}\ 2032 \qquad\textbf{(D)}\ 2040 \qquad\textbf{(E)}\ 2048$
brandbest1
view topic
https://data.artofproblemsolving.com/images/maa_logo.png These problems are copyright $\copyright$ Mathematical Association of America.
rrusczyk
view topic
a