question string | answer string | difficulty int64 | problem_id string |
|---|---|---|---|
What is the difference between the sum of the first $2003$ even counting numbers and the sum of the first $2003$ odd counting numbers?
$\mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 2\qquad \mathrm{(D) \ } 2003\qquad \mathrm{(E) \ } 4006$ | D | 2 | 2003A-P1 |
Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $2366, how many members are in th... | B | 2 | 2003A-P2 |
A solid box is $15$ cm by $10$ cm by $8$ cm. A new solid is formed by removing a cube $3$ cm on a side from each corner of this box. What percent of the original volume is removed?
$\mathrm{(A) \ } 4.5\qquad \mathrm{(B) \ } 9\qquad \mathrm{(C) \ } 12\qquad \mathrm{(D) \ } 18\qquad \mathrm{(E) \ } 24$ | D | 2 | 2003A-P3 |
It takes Mary $30$ minutes to walk uphill $1$ km from her home to school, but it takes her only $10$ minutes to walk from school to her home along the same route. What is her average speed, in km/hr, for the round trip?
$\mathrm{(A) \ } 3\qquad \mathrm{(B) \ } 3.125\qquad \mathrm{(C) \ } 3.5\qquad \mathrm{(D) \ } 4\qq... | A | 2 | 2003A-P4 |
The sum of the two 5-digit numbers $AMC10$ and $AMC12$ is $123422$. What is $A+M+C$?
$\mathrm{(A) \ } 10\qquad \mathrm{(B) \ } 11\qquad \mathrm{(C) \ } 12\qquad \mathrm{(D) \ } 13\qquad \mathrm{(E) \ } 14$ | E | 2 | 2003A-P5 |
Define $x \heartsuit y$ to be $|x-y|$ for all real numbers $x$ and $y$. Which of the following statements is not true?
$\mathrm{(A) \ } x \heartsuit y = y \heartsuit x$ for all $x$ and $y$
$\mathrm{(B) \ } 2(x \heartsuit y) = (2x) \heartsuit (2y)$ for all $x$ and $y$
$\mathrm{(C) \ } x \heartsuit 0 = x$ for all $x$
... | C | 2 | 2003A-P6 |
How many non-congruent triangles with perimeter $7$ have integer side lengths?
$\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 3\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 5$ | B | 2 | 2003A-P7 |
What is the probability that a randomly drawn positive factor of $60$ is less than $7$?
$\mathrm{(A) \ } \frac{1}{10}\qquad \mathrm{(B) \ } \frac{1}{6}\qquad \mathrm{(C) \ } \frac{1}{4}\qquad \mathrm{(D) \ } \frac{1}{3}\qquad \mathrm{(E) \ } \frac{1}{2}$ | E | 2 | 2003A-P8 |
A set $S$ of points in the $xy$-plane is symmetric about the origin, both coordinate axes, and the line $y=x$. If $(2,3)$ is in $S$, what is the smallest number of points in $S$?
$\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 8\qquad \mathrm{(E) \ } 16$ | D | 2 | 2003A-P9 |
Al, Bert, and Carl are the winners of a school drawing for a pile of Halloween candy, which they are to divide in a ratio of $3:2:1$, respectively. Due to some confusion they come at different times to claim their prizes, and each assumes he is the first to arrive. If each takes what he believes to be the correct share... | D | 2 | 2003A-P10 |
A square and an equilateral triangle have the same perimeter. Let $A$ be the area of the circle circumscribed about the square and $B$ the area of the circle circumscribed around the triangle. Find $A/B$.
$\mathrm{(A) \ } \frac{9}{16}\qquad \mathrm{(B) \ } \frac{3}{4}\qquad \mathrm{(C) \ } \frac{27}{32}\qquad \mathrm{... | C | 3 | 2003A-P11 |
Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards?
$\mathrm{(A) ... | E | 3 | 2003A-P12 |
The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing?
Image:2003amc10a10.gif
$\mathr... | E | 3 | 2003A-P13 |
Points $K, L, M,$ and $N$ lie in the plane of the square $ABCD$ such that $AKB$, $BLC$, $CMD$, and $DNA$ are equilateral triangles. If $ABCD$ has an area of 16, find the area of $KLMN$.
```asy
unitsize(2cm);
defaultpen(fontsize(8)+linewidth(0.8));
pair A=(-0.5,0.5), B=(0.5,0.5), C=(0.5,-0.5), D=(-0.5,-0.5);
pair K=(0,... | D | 3 | 2003A-P14 |
A semicircle of diameter $1$ sits at the top of a semicircle of diameter $2$, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a ''lune''. Determine the area of this lune.
Image:2003amc10a19.gif
$\mathrm{(A) \ } \frac{1}{6}\pi-\frac{\sqrt{3}}{4}\qquad \mathrm{(B) \ }... | C | 3 | 2003A-P15 |
A point P is chosen at random in the interior of equilateral triangle $ABC$. What is the probability that $\triangle ABP$ has a greater area than each of $\triangle ACP$ and $\triangle BCP$?
\textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{4}\qquad\textbf{(C)}\ \frac{1}{3}\qquad\textbf{(D)}\ \frac{1}{2}\qquad\tex... | C | 3 | 2003A-P16 |
Square $ABCD$ has sides of length $4$, and $M$ is the midpoint of $\overline{CD}$. A circle with radius $2$ and center $M$ intersects a circle with radius $4$ and center $A$ at points $P$ and $D$. What is the distance from $P$ to $\overline{AD}$?
```asy
pair A,B,C,D,M,P;
D=(0,0);
C=(10,0);
B=(10,10);
A=(0,10);
M=(5,0)... | B | 3 | 2003A-P17 |
Let $n$ be a $5$-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by $100$. For how many values of $n$ is $q+r$ divisible by $11$?
$\mathrm{(A) \ } 8180\qquad \mathrm{(B) \ } 8181\qquad \mathrm{(C) \ } 8182\qquad \mathrm{(D) \ } 9000\qquad \mathrm{(E) \ } 9090$ | B | 3 | 2003A-P18 |
A parabola with equation $y=ax^2+bx+c$ is reflected about the $x$-axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of $y=f(x)$ and $y=g(x)$, respectively. Which of the following describes the graph of $y=(f+g)(x)$?
\textbf{(A)}\ \text{a parabola t... | D | 3 | 2003A-P19 |
How many $15$-letter arrangements of $5$ A's, $5$ B's, and $5$ C's have no A's in the first $5$ letters, no B's in the next $5$ letters, and no C's in the last $5$ letters?
$\textrm{(A)}\ \sum_{k=0}^{5}\binom{5}{k}^{3}\qquad\textrm{(B)}\ 3^{5}\cdot 2^{5}\qquad\textrm{(C)}\ 2^{15}\qquad\textrm{(D)}\ \frac{15!}{(5!)^{3}... | A | 3 | 2003A-P20 |
The graph of the polynomial
$P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e$
has five distinct $x$-intercepts, one of which is at $(0,0)$. Which of the following coefficients cannot be zero?
\textbf{(A)}\ a \qquad \textbf{(B)}\ b \qquad \textbf{(C)}\ c \qquad \textbf{(D)}\ d \qquad \textbf{(E)}\ e | D | 4 | 2003A-P21 |
Objects $A$ and $B$ move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object $A$ starts at $(0,0)$ and each of its steps is either right or up, both equally likely. Object $B$ starts at $(5,7)$ and each of its steps is either to the left or down, both equally likely. Which of the ... | C | 4 | 2003A-P22 |
How many perfect squares are divisors of the product $1! \cdot 2! \cdot 3! \cdot \hdots \cdot 9!$?
\textbf{(A)}\ 504\qquad\textbf{(B)}\ 672\qquad\textbf{(C)}\ 864\qquad\textbf{(D)}\ 936\qquad\textbf{(E)}\ 1008 | B | 4 | 2003A-P23 |
If $a\geq b > 1,$ what is the largest possible value of $\log_{a}(a/b) + \log_{b}(b/a)?$
$\mathrm{(A)}\ -2 \qquad
\mathrm{(B)}\ 0 \qquad
\mathrm{(C)}\ 2 \qquad
\mathrm{(D)}\ 3 \qquad
\mathrm{(E)}\ 4$ | B | 4 | 2003A-P24 |
Let $f(x)= \sqrt{ax^2+bx}$. For how many real values of $a$ is there at least one positive value of $b$ for which the domain of $f$ and the range of $f$ are the same set?
$\mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ \mathrm{infinitely \ many} }$ | C | 4 | 2003A-P25 |
Which of the following is the same as
$$
\frac{2-4+6-8+10-12+14}{3-6+9-12+15-18+21}?
$$
$\text {(A) } -1 \qquad \text {(B) } -\frac{2}{3} \qquad \text {(C) } \frac{2}{3} \qquad \text {(D) } 1 \qquad \text {(E) } \frac{14}{3}$ | C | 2 | 2003B-P1 |
Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs 1 dollar more than a pink pill, and Al's pills cost a total of 546 dollars for the two weeks. How much does one green pill cost?
$\text {(A) } 7 \qquad \text {(B) } 14 \qquad \text {(C) } 19 \qquad ... | D | 2 | 2003B-P2 |
Rose fills each of the rectangular regions of her rectangular flower bed with a
different type of flower. The lengths, in feet, of the rectangular regions in her
flower bed are as shown in the figure. She plants one flower per square foot in
each region. Asters cost \$1 each, begonias \$1.50 each, cannas \$2 each, dahl... | A | 2 | 2003B-P3 |
Moe uses a mower to cut his rectangular 90-foot by 150-foot lawn. The swath he cuts is 28 inches wide, but he overlaps each cut by 4 inches to make sure that no grass is missed. he walks at the rate of 5000 feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe ... | C | 2 | 2003B-P4 |
Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is 4 : 3. The horizontal length of a "27-inch" television screen is closest, in inches, to which of the following?
File:Problem_5.PNG
$\text {(A) ... | D | 2 | 2003B-P5 |
The second and fourth terms of a geometric sequence are 2 and 6. Which of the following is a possible first term?
$\text {(A) } -\sqrt{3} \qquad \text {(B) } \frac{-2\sqrt{3}}{3} \qquad \text {(C) } \frac{-\sqrt{3}}{3} \qquad \text {(D) } \sqrt{3} \qquad \text {(E) } 3$ | B | 2 | 2003B-P6 |
Penniless Pete's piggy bank has no pennies in it, but it has 100 coins, all nickels, dimes, and quarters, whose total value is \$8.35. It does not necessarily contain coins of all three types. What is the difference between the largest and smallest number of dimes that could be in the bank?
$\text {(A) } 0 \qquad \tex... | D | 2 | 2003B-P7 |
Let $\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For example, $\clubsuit(8) = 8$ and $\clubsuit(123) = 1 + 2 + 3 = 6.$ For how many two-digit values of $x$ is $\clubsuit(\clubsuit(x)) = 3?$
$\text{(A) }3\qquad\text{(B) }4\qquad\text{(C) }6\qquad\text{(D) }9\qquad\text{(E) }10$ | E | 2 | 2003B-P8 |
Let $f$ be a linear function for which $f(6) - f(2) = 12.$ What is $f(12) - f(2)?$
$\text {(A) } 12 \qquad \text {(B) } 18 \qquad \text {(C) } 24 \qquad \text {(D) } 30 \qquad \text {(E) } 36$ | D | 2 | 2003B-P9 |
Several figures can be made by attaching two equilateral triangles to the regular pentagon ABCDE in two of the five positions shown. How many non-congruent figures can be constructed in this way?
<center>
```asy
size(200);
defaultpen(0.9);
real r = 5/dir(54).x, h = 5 tan(54*pi/180);
pair A = (5,0), B = A+10*dir(72), C... | B | 2 | 2003B-P10 |
Cassandra sets her watch to the correct time at noon. At the actual time of 1:00 PM, she notices that her watch reads 12:57 and 36 seconds. Assuming that her watch loses time at a constant rate, what will be the actual time when her
watch first reads 10:00 PM?
$\text {(A) 10:22 PM and 24 seconds} \qquad \text {(B) 10:... | C | 3 | 2003B-P11 |
What is the largest integer that is a divisor of $(n+1)(n+3)(n+5)(n+7)(n+9)$ for all positive even integers $n$?
$\text {(A) } 3 \qquad \text {(B) } 5 \qquad \text {(C) } 11 \qquad \text {(D) } 15 \qquad \text {(E) } 165$ | D | 3 | 2003B-P12 |
An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies $75\%$ of the volume of the frozen ice cream. What is the ratio of the cone’s height to its rad... | B | 3 | 2003B-P13 |
In rectangle $ABCD, AB=5$ and $BC=3$. Points $F$ and $G$ are on $\overline{CD}$ so that $DF=1$ and $GC=2$. Lines $AF$ and $BG$ intersect at $E$. Find the area of $\triangle AEB$.
File:Problem_14.png
$\text {(A) } 10 \qquad \text {(B) } \frac{21}{2} \qquad \text {(C) } 12 \qquad \text {(D) } \frac{25}{2} \qquad \text ... | D | 3 | 2003B-P14 |
A regular octagon $ABCDEFGH$ has an area of one square unit. What is the area of the rectangle $ABEF$?
File:Problem_15.PNG
$\text {(A) } 1-\frac{\sqrt{2}}{2} \qquad \text {(B) } \frac{\sqrt{2}}{4} \qquad \text {(C) } \sqrt{2}-1 \qquad \text {(D) } \frac{1}{2} \qquad \text {(E) } \frac{1+\sqrt{2}}{4}$ | D | 3 | 2003B-P15 |
Three semicircles of radius 1 are constructed on diameter AB of a semicircle of
radius 2. The centers of the small semicircles divide AB into four line segments
of equal length, as shown. What is the area of the shaded region that lies within
the large semicircle but outside the smaller semicircles?
```asy
import grap... | E | 3 | 2003B-P16 |
If $\log (xy^3) = 1$ and $\log (x^2y) = 1$, what is $\log (xy)$?
$\mathrm{(A)}\ -\frac 12
\qquad\mathrm{(B)}\ 0
\qquad\mathrm{(C)}\ \frac 12
\qquad\mathrm{(D)}\ \frac 35
\qquad\mathrm{(E)}\ 1$ | D | 3 | 2003B-P17 |
Let $x$ and $y$ be positive integers such that $7x^5 = 11y^{13}.$ The minimum possible value of $x$ has a prime factorization $a^cb^d.$ What is $a + b + c + d$?
\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34 | B | 3 | 2003B-P18 |
Let $S$ be the set of permutations of the sequence $1,2,3,4,5$ for which the first term is not $1$. A permutation is chosen randomly from $S$. The probability that the second term is $2$, in lowest terms, is $a/b$. What is $a+b$?
$\mathrm{(A)}\ 5
\qquad\mathrm{(B)}\ 6
\qquad\mathrm{(C)}\ 11
\qquad\mathrm{(D)}\ 16
\qqu... | E | 3 | 2003B-P19 |
Part of the graph of $f(x) = ax^3 + bx^2 + cx + d$ is shown. What is $b$?
Image:2003_12B_AMC-20.png
$\mathrm{(A)}\ -4
\qquad\mathrm{(B)}\ -2
\qquad\mathrm{(C)}\ 0
\qquad\mathrm{(D)}\ 2
\qquad\mathrm{(E)}\ 4$ | B | 3 | 2003B-P20 |
An object moves $8$ cm in a straight line from $A$ to $B$, turns at an angle $\alpha$, measured in radians and chosen at random from the interval $(0,\pi)$, and moves $5$ cm in a straight line to $C$. What is the probability that $AC < 7$?
$\mathrm{(A)}\ \frac{1}{6}
\qquad\mathrm{(B)}\ \frac{1}{5}
\qquad\mathrm{(C)}\ ... | D | 4 | 2003B-P21 |
Let $ABCD$ be a rhombus with $AC = 16$ and $BD = 30$. Let $N$ be a point on $\overline{AB}$, and let $P$ and $Q$ be the feet of the perpendiculars from $N$ to $\overline{AC}$ and $\overline{BD}$, respectively. Which of the following is closest to the minimum possible value of $PQ$?
<center>
```asy
size(200);
defaultp... | C | 4 | 2003B-P22 |
The number of $x$-intercepts on the graph of $y=\sin(1/x)$ in the interval $(0.0001,0.001)$ is closest to
$\mathrm{(A)}\ 2900
\qquad\mathrm{(B)}\ 3000
\qquad\mathrm{(C)}\ 3100
\qquad\mathrm{(D)}\ 3200
\qquad\mathrm{(E)}\ 3300$ | A | 4 | 2003B-P23 |
Positive integers $a,b,$ and $c$ are chosen so that $a<b<c$, and the system of equations
<center>$2x + y = 2003 \quad$ and $\quad y = |x-a| + |x-b| + |x-c|$</center>
has exactly one solution. What is the minimum value of $c$?
$\mathrm{(A)}\ 668
\qquad\mathrm{(B)}\ 669
\qquad\mathrm{(C)}\ 1002
\qquad\mathrm{(D)}\ 2003
... | C | 4 | 2003B-P24 |
Three points are chosen randomly and independently on a circle. What is the probability that all three pairwise distances between the points are less than the radius of the circle?
$\mathrm{(A)}\ \dfrac{1}{36}
\qquad\mathrm{(B)}\ \dfrac{1}{24}
\qquad\mathrm{(C)}\ \dfrac{1}{18}
\qquad\mathrm{(D)}\ \dfrac{1}{12}
\qquad\... | D | 4 | 2003B-P25 |
Alicia earns $20$ dollars per hour, of which $1.45\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?
$\text{(A) } 0.0029 \qquad \text{(B) } 0.029 \qquad \text{(C) } 0.29 \qquad \text{(D) } 2.9 \qquad \text{(E) } 29$ | E | 2 | 2004A-P1 |
On the AMC 12, each correct answer is worth $6$ points, each incorrect answer is worth $0$ points, and each problem left unanswered is worth $2.5$ points. If Charlyn leaves $8$ of the $25$ problems unanswered, how many of the remaining problems must she answer correctly in order to score at least $100$?
$\text{(A) } 1... | C | 2 | 2004A-P2 |
For how many ordered pairs of positive integers $(x,y)$ is $x+2y=100$?
$\text{(A) } 33 \qquad \text{(B) } 49 \qquad \text{(C) } 50 \qquad \text{(D) } 99 \qquad \text{(E) } 100$ | B | 2 | 2004A-P3 |
Bertha has $6$ daughters and no sons. Some of her daughters have $6$ daughters, and the rest have none. Bertha has a total of $30$ daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters?
$\text{(A) } 22 \qquad \text{(B) } 23 \qquad \text{(C) } 24... | E | 2 | 2004A-P4 |
The graph of the line $y=mx+b$ is shown. Which of the following is true?
Image:2004 AMC 12A Problem 5.png
$\text{(A) } mb<-1 \qquad \text{(B) } -1<mb<0 \qquad \text{(C) } mb=0 \qquad \text{(D) } 0<mb<1 \qquad \text{(E) } mb>1$ | B | 2 | 2004A-P5 |
Let $U=2\cdot 2004^{2005}$, $V=2004^{2005}$, $W=2003\cdot 2004^{2004}$, $X=2\cdot 2004^{2004}$, $Y=2004^{2004}$ and $Z=2004^{2003}$. Which of the following is the largest?
$\text{(A) } U-V \qquad \text{(B) } V-W \qquad \text{(C) } W-X \qquad \text{(D) } X-Y \qquad \text{(E) } Y-Z \qquad$ | A | 2 | 2004A-P6 |
A game is played with tokens according to the following rules. In each round, the player with the most tokens gives one token to each of the other players and also places one token into a discard pile. The game ends when some player runs out of tokens. Players $A$, $B$ and $C$ start with $15$, $14$ and $13$ tokens, res... | B | 2 | 2004A-P7 |
In the overlapping triangles $\triangle{ABC}$ and $\triangle{ABE}$ sharing common side $AB$, $\angle{EAB}$ and $\angle{ABC}$ are right angles, $AB=4$, $BC=6$, $AE=8$, and $\overline{AC}$ and $\overline{BE}$ intersect at $D$. What is the difference between the areas of $\triangle{ADE}$ and $\triangle{BDC}$?
$\text{(A) ... | B | 2 | 2004A-P8 |
A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars would increase sales. If the diameter of the jars is increased by $25\%$ without altering the volume, by what percent must the height be decreased?
$\text {(A) } 10\% \qquad \text {(B) } 25\% \qquad \text {(C) } 36\% \... | C | 2 | 2004A-P9 |
The sum of $49$ consecutive integers is $7^5$. What is their median?
$\text {(A) } 7 \qquad \text {(B) } 7^2\qquad \text {(C) } 7^3\qquad \text {(D) } 7^4\qquad \text {(E) }7^5$ | C | 2 | 2004A-P10 |
The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is $20$ cents. If she had one more quarter, the average value would be $21$ cents. How many dimes does she have in her purse?
$\text {(A) }0 \qquad \text {(B) } 1 \qquad \text {(C) } 2 \qquad \text {(D) } 3\qquad \text {(E) }4$ | A | 3 | 2004A-P11 |
Let $A = (0,9)$ and $B = (0,12)$. Points $A'$ and $B'$ are on the line $y = x$, and $\overline{AA'}$ and $\overline{BB'}$ intersect at $C = (2,8)$. What is the length of $\overline{A'B'}$?
$\text {(A) } 2 \qquad \text {(B) } 2\sqrt2 \qquad \text {(C) } 3 \qquad \text {(D) } 2 + \sqrt 2\qquad \text {(E) }3\sqrt 2$ | B | 3 | 2004A-P12 |
Let $S$ be the set of points $(a,b)$ in the coordinate plane, where each of $a$ and $b$ may be $- 1$, $0$, or $1$. How many distinct lines pass through at least two members of $S$?
$\text {(A) } 8 \qquad \text {(B) } 20 \qquad \text {(C) } 24 \qquad \text {(D) } 27\qquad \text {(E) }36$ | B | 3 | 2004A-P13 |
A sequence of three real numbers forms an arithmetic progression with a first term of $9$. If $2$ is added to the second term and $20$ is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term in the geometric progression?
$\text {(A) }... | A | 3 | 2004A-P14 |
Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run $100$ meters. They next meet after Sally has run $150$ meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters?
$... | C | 3 | 2004A-P15 |
The set of all real numbers $x$ for which
$$
\log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x})))
$$
is defined is $\{x\mid x > c\}$. What is the value of $c$?
$\textbf {(A) } 0\qquad \textbf {(B) }2001^{2002} \qquad \textbf {(C) }2002^{2003} \qquad \textbf {(D) }2003^{2004} \qquad \textbf {(E) }2001^{2002^{2003}}$ | B | 3 | 2004A-P16 |
Let $f$ be a function with the following properties:
(i) $f(1) = 1$, and
(ii) $f(2n) = n \cdot f(n)$ for any positive integer $n$.
What is the value of $f(2^{100})$?
$\text {(A)}\ 1 \qquad \text {(B)}\ 2^{99} \qquad \text {(C)}\ 2^{100} \qquad \text {(D)}\ 2^{4950} \qquad \text {(E)}\ 2^{9999}$ | D | 3 | 2004A-P17 |
Square $ABCD$ has side length $2$. A semicircle with diameter $\overline{AB}$ is constructed inside the square, and the tangent to the semicircle from $C$ intersects side $\overline{AD}$ at $E$. What is the length of $\overline{CE}$?
```asy
size(100);
defaultpen(fontsize(10));
pair A=(0,0), B=(2,0), C=(2,2), D=(0,2), ... | D | 3 | 2004A-P18 |
Circles $A, B$ and $C$ are externally tangent to each other, and internally tangent to circle $D$. Circles $B$ and $C$ are congruent. Circle $A$ has radius $1$ and passes through the center of $D$. What is the radius of circle $B$?
<center>
```asy
unitsize(15mm);
pair A=(-1,0),B=(2/3,8/9),C=(2/3,-8/9),D=(0,0);
draw(... | D | 3 | 2004A-P19 |
Select numbers $a$ and $b$ between $0$ and $1$ independently and at random, and let $c$ be their sum. Let $A, B$ and $C$ be the results when $a, b$ and $c$, respectively, are rounded to the nearest integer. What is the probability that $A + B = C$?
$\text {(A) } \frac14 \qquad \text {(B) } \frac13 \qquad \text {(C) } ... | E | 3 | 2004A-P20 |
If $\sum_{n = 0}^{\infty}\cos^{2n}\theta = 5$, what is the value of $\cos{2\theta}$?
$\text {(A) } \frac15 \qquad \text {(B) } \frac25 \qquad \text {(C) } \frac {\sqrt5}{5}\qquad \text {(D) } \frac35 \qquad \text {(E) }\frac45$ | D | 4 | 2004A-P21 |
Three mutually tangent spheres of radius $1$ rest on a horizontal plane. A sphere of radius $2$ rests on them. What is the distance from the plane to the top of the larger sphere?
$\text {(A) } 3 + \frac {\sqrt {30}}{2} \qquad \text {(B) } 3 + \frac {\sqrt {69}}{3} \qquad \text {(C) } 3 + \frac {\sqrt {123}}{4}\qquad ... | B | 4 | 2004A-P22 |
A polynomial
$$
P(x) = c_{2004}x^{2004} + c_{2003}x^{2003} + ... + c_1x + c_0
$$
has real coefficients with $c_{2004}\not = 0$ and $2004$ distinct complex zeroes $z_k = a_k + b_ki$, $1\leq k\leq 2004$ with $a_k$ and $b_k$ real, $a_1 = b_1 = 0$, and
$$
\sum_{k = 1}^{2004}{a_k} = \sum_{k = 1}^{2004}{b_k}.
$$
Which of... | E | 4 | 2004A-P23 |
A plane contains points $A$ and $B$ with $AB = 1$. Let $S$ be the union of all disks of radius $1$ in the plane that cover $\overline{AB}$. What is the area of $S$?
$\text {(A) } 2\pi + \sqrt3 \qquad \text {(B) } \frac {8\pi}{3} \qquad \text {(C) } 3\pi - \frac {\sqrt3}{2} \qquad \text {(D) } \frac {10\pi}{3} - \sqrt3... | C | 4 | 2004A-P24 |
For each integer $n\geq 4$, let $a_n$ denote the base-$n$ number $0.\overline{133}_n$. The product $a_4a_5...a_{99}$ can be expressed as $\frac {m}{n!}$, where $m$ and $n$ are positive integers and $n$ is as small as possible. What is the value of $m$?
$\text {(A) } 98 \qquad \text {(B) } 101 \qquad \text {(C) } 132\q... | E | 4 | 2004A-P25 |
At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her fifth practice she made 48 free throws. How many free throws did she make at the first practice?
$(\mathrm {A}) 3\qquad (\mathrm {B}) 6 \qquad (\mathrm {C}) 9 \qquad (\mathrm {D}) 12 \qquad (\mathrm... | A | 2 | 2004B-P1 |
In the expression $c\cdot a^b-d$, the values of $a$, $b$, $c$, and $d$ are 0, 1, 2, and 3, although not necessarily in that order. What is the maximum possible value of the result?
$(\mathrm {A}) 5\qquad (\mathrm {B}) 6 \qquad (\mathrm {C}) 8 \qquad (\mathrm {D}) 9 \qquad (\mathrm {E}) 10$ | D | 2 | 2004B-P2 |
If $x$ and $y$ are positive integers for which $2^x3^y=1296$, what is the value of $x+y$?
$(\mathrm {A}) 8\qquad (\mathrm {B}) 9 \qquad (\mathrm {C}) 10 \qquad (\mathrm {D}) 11 \qquad (\mathrm {E}) 12$ | A | 2 | 2004B-P3 |
An integer $x$, with $10\leq x\leq 99$, is to be chosen. If all choices are equally likely, what is the probability that at least one digit of $x$ is a 7?
$(\mathrm {A}) \dfrac{1}{9} \qquad (\mathrm {B}) \dfrac{1}{5} \qquad (\mathrm {C}) \dfrac{19}{90} \qquad (\mathrm {D}) \dfrac{2}{9} \qquad (\mathrm {E}) \dfrac{1}{3... | B | 2 | 2004B-P4 |
On a trip from the United States to Canada, Isabella took $d$ U.S. dollars. At the border she exchanged them all, receiving 10 Canadian dollars for every 7 U.S. dollars. After spending 60 Canadian dollars, she had $d$ Canadian dollars left. What is the sum of the digits of $d$?
$(\mathrm {A}) 5\qquad (\mathrm {B}) 6 \... | A | 2 | 2004B-P5 |
Minneapolis-St. Paul International Airport is 8 miles southwest of downtown St. Paul and 10 miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis?
$(\mathrm {A}) 13\qquad (\mathrm {B}) 14 \qquad (\mathrm {C}) 15 \qquad (\mat... | A | 2 | 2004B-P6 |
A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the square and the circle?
$(\mathrm {A}) 200+25\pi \quad (\mathrm {B}) 100+75\pi \quad (\mathrm {C}) 75+100\pi \quad (\mathrm {D}) 100+100\pi \quad (\mathrm {E}) 100+12... | B | 2 | 2004B-P7 |
A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains 100 cans, how many rows does it contain?
$(\mathrm {A}) 5 \qquad (\mathrm {B}) 8 \qquad (\mathrm {C}) 9 \qquad (\mathrm {D}) 10 \qquad (\mathrm {E}) 11$ | D | 2 | 2004B-P8 |
The point $(-3,2)$ is rotated $90^\circ$ clockwise around the origin to point $B$. Point $B$ is then reflected over the line $x=y$ to point $C$. What are the coordinates of $C$?
$\mathrm{(A)}\ (-3,-2)
\qquad
\mathrm{(B)}\ (-2,-3)
\qquad
\mathrm{(C)}\ (2,-3)
\qquad
\mathrm{(D)}\ (2,3)
\qquad
\mathrm{(E)}\ (3,2)$ | E | 2 | 2004B-P9 |
An annulus is the region between two concentric circles. The concentric circles in the figure have radii $b$ and $c$, with $b>c$. Let $OX$ be a radius of the larger circle, let $XZ$ be tangent to the smaller circle at $Z$, and let $OY$ be the radius of the larger circle that contains $Z$. Let $a=XZ$, $d=YZ$, and $e=XY$.... | A | 2 | 2004B-P10 |
All the students in an algebra class took a $100$-point test. Five students scored $100$, each student scored at least $60$, and the mean score was $76$. What is the smallest possible number of students in the class?
$\mathrm{(A)}\ 10
\qquad
\mathrm{(B)}\ 11
\qquad
\mathrm{(C)}\ 12
\qquad
\mathrm{(D)}\ 13
\qquad
\math... | D | 3 | 2004B-P11 |
In the sequence $2001$, $2002$, $2003$, $\ldots$ , each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is $2001 + 2002 - 2003 = 2000$. What is the
$2004^\textrm{th}$ term in this sequence?
$\mathrm{(A) \ } -2004 \qquad \... | C | 3 | 2004B-P12 |
If $f(x) = ax+b$ and $f^{-1}(x) = bx+a$ with $a$ and $b$ real, what is the value of $a+b$?
$\mathrm{(A)}\ -2
\qquad\mathrm{(B)}\ -1
\qquad\mathrm{(C)}\ 0
\qquad\mathrm{(D)}\ 1
\qquad\mathrm{(E)}\ 2$ | A | 3 | 2004B-P13 |
In $\triangle ABC$, $AB=13$, $AC=5$, and $BC=12$. Points $M$ and $N$ lie on $AC$ and $BC$, respectively, with $CM=CN=4$. Points $J$ and $K$ are on $AB$ so that $MJ$ and $NK$ are perpendicular to $AB$. What is the area of pentagon $CMJKN$?
```asy
unitsize(0.5cm);
defaultpen(0.8);
pair C=(0,0), A=(0,5), B=(12,0), M=(0,4... | D | 3 | 2004B-P14 |
The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages?
$\mathrm{(A) \ } 9 \qquad \mathrm{(B) \ } 18 \qquad \mathrm{(C) \ } 27 \qquad \mathrm{(D) \ } 36\qquad \mathrm{(E) \ }... | B | 3 | 2004B-P15 |
A function $f$ is defined by $f(z) = i\overline{z}$, where $i=\sqrt{-1}$ and $\overline{z}$ is the complex conjugate of $z$. How many values of $z$ satisfy both $|z| = 5$ and $f(z) = z$?
$\mathrm{(A)}\ 0
\qquad\mathrm{(B)}\ 1
\qquad\mathrm{(C)}\ 2
\qquad\mathrm{(D)}\ 4
\qquad\mathrm{(E)}\ 8$ | C | 3 | 2004B-P16 |
For some real numbers $a$ and $b$, the equation
$$
8x^3 + 4ax^2 + 2bx + a = 0
$$
has three distinct positive roots. If the sum of the base-$2$ logarithms of the roots is $5$, what is the value of $a$?
$\mathrm{(A)}\ -256
\qquad\mathrm{(B)}\ -64
\qquad\mathrm{(C)}\ -8
\qquad\mathrm{(D)}\ 64
\qquad\mathrm{(E)}\ 256$ | A | 3 | 2004B-P17 |
Points $A$ and $B$ are on the parabola $y=4x^2+7x-1$, and the origin is the midpoint of $AB$. What is the length of $AB$?
$\mathrm{(A)}\ 2\sqrt5
\qquad
\mathrm{(B)}\ 5+\frac{\sqrt2}{2}
\qquad
\mathrm{(C)}\ 5+\sqrt2
\qquad
\mathrm{(D)}\ 7
\qquad
\mathrm{(E)}\ 5\sqrt2$ | E | 3 | 2004B-P18 |
A truncated cone has horizontal bases with radii $18$ and $2$. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?
$\mathrm{(A)}\ 6
\qquad\mathrm{(B)}\ 4\sqrt{5}
\qquad\mathrm{(C)}\ 9
\qquad\mathrm{(D)}\ 10
\qquad\mathrm{(E)}\ 6\sqrt{3}$ | A | 3 | 2004B-P19 |
Each face of a cube is painted either red or blue, each with probability $1/2$. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?
\textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \fra... | B | 3 | 2004B-P20 |
The graph of $2x^2 + xy + 3y^2 - 11x - 20y + 40 = 0$ is an ellipse in the first quadrant of the $xy$-plane. Let $a$ and $b$ be the maximum and minimum values of $\frac yx$ over all points $(x,y)$ on the ellipse. What is the value of $a+b$?
$\mathrm{(A)}\ 3
\qquad\mathrm{(B)}\ \sqrt{10}
\qquad\mathrm{(C)}\ \frac 72
\qq... | C | 4 | 2004B-P21 |
The square
<center>$\begin{tabular}{|c|c|c|} \hline 50 & \textit{b} & \textit{c} \\
\hline \textit{d} & \textit{e} & \textit{f} \\
\hline \textit{g} & \textit{h} & 2 \\
\hline \end{tabular}$</center>
is a multiplicative magic square. That is, the product of the numbers in each row, column, and diagonal is the same. If ... | C | 4 | 2004B-P22 |
The polynomial $x^3 - 2004 x^2 + mx + n$ has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of $n$ are possible?
$\mathrm{(A)}\ 250,000
\qquad\mathrm{(B)}\ 250,250
\qquad\mathrm{(C)}\ 250,500
\qquad\mathrm{(D)}\ 250,750
\q... | C | 4 | 2004B-P23 |
In $\triangle ABC$, $AB = BC$, and $\overline{BD}$ is an altitude. Point $E$ is on the extension of $\overline{AC}$ such that $BE = 10$. The values of $\tan \angle CBE$, $\tan \angle DBE$, and $\tan \angle ABE$ form a geometric progression, and the values of $\cot \angle DBE,$ $\cot \angle CBE,$ $\cot \angle DBC$ form ... | B | 4 | 2004B-P24 |
Given that $2^{2004}$ is a $604$-digit number whose first digit is $1$, how many elements of the set $S = \{2^0,2^1,2^2,\ldots ,2^{2003}\}$ have a first digit of $4$?
$\mathrm{(A)}\ 194
\qquad\mathrm{(B)}\ 195
\qquad\mathrm{(C)}\ 196
\qquad\mathrm{(D)}\ 197
\qquad\mathrm{(E)}\ 198$ | B | 4 | 2004B-P25 |
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