question string | answer string | difficulty int64 | problem_id string |
|---|---|---|---|
Two is $10 \%$ of $x$ and $20 \%$ of $y$. What is $x - y$?
$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 20$ | D | 2 | 2005A-P1 |
The equations $2x + 7 = 3$ and $bx - 10 = - 2$ have the same solution for $x$. What is the value of $b$?
$\textbf{(A) } -8 \qquad \textbf{(B) } -4 \qquad \textbf{(C) } -2 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 8$ | B | 2 | 2005A-P2 |
A rectangle with a diagonal of length $x$ is twice as long as it is wide. What is the area of the rectangle?
$\textbf{(A) } \frac{1}{4}x^2 \qquad \textbf{(B) } \frac{2}{5}x^2 \qquad \textbf{(C) } \frac{1}{2}x^2 \qquad \textbf{(D) } x^2 \qquad \textbf{(E) } \frac{3}{2}x^2$ | B | 2 | 2005A-P3 |
A store normally sells windows at $\$100$ each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?
$\textbf{(A) } 100 \qquad \textbf{(B) } 200 \qq... | A | 2 | 2005A-P4 |
The average (mean) of $20$ numbers is $30$, and the average of $30$ other numbers is $20$. What is the average of all $50$ numbers?
$\textbf{(A) } 23 \qquad \textbf{(B) } 24 \qquad \textbf{(C) } 25 \qquad \textbf{(D) } 26 \qquad \textbf{(E) } 27$ | B | 2 | 2005A-P5 |
Josh and Mike live $13$ miles apart. Yesterday Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met?
... | B | 2 | 2005A-P6 |
Square $EFGH$ is inside the square $ABCD$ so that each side of $EFGH$ can be extended to pass through a vertex of $ABCD$. Square $ABCD$ has side length $\sqrt{50}$, $E$ is between $B$ and $H$, and $BE = 1$. What is the area of the inner square $EFGH$?
```asy
unitsize(4cm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
p... | C | 2 | 2005A-P7 |
Let $A$, $M$, and $C$ be digits with
$$
(100A+10M+C)(A+M+C) = 2005.
$$
What is $A$?
$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 5$ | D | 2 | 2005A-P8 |
There are two values of $a$ for which the equation $4x^2 + ax + 8x + 9 = 0$ has only one solution for $x$. What is the sum of those values of $a$?
$\textbf{(A) } -16 \qquad \textbf{(B) } -8 \qquad \textbf{(C) } 0 \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 20$ | A | 2 | 2005A-P9 |
A wooden cube $n$ units on a side is painted red on all six faces and then cut into $n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $n$?
$\textbf{(A) } 3 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 7$ | B | 2 | 2005A-P10 |
How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?
$\textbf{(A) } 41 \qquad \textbf{(B) } 42 \qquad \textbf{(C) } 43 \qquad \textbf{(D) } 44 \qquad \textbf{(E) } 45$ | E | 3 | 2005A-P11 |
A line passes through $A(1,1)$ and $B(100,1000)$. How many other points with integer coordinates are on the line and strictly between $A$ and $B$?
$\textbf{(A) } 0 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 9$ | D | 3 | 2005A-P12 |
In the five-sided star shown, the letters $A$, $B$, $C$, $D$ and $E$ are replaced by the numbers $3$, $5$, $6$, $7$ and $9$, although not necessarily in that order. The sums of the numbers at the ends of the line segments $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, $\overline{DE}$ and $\overline{EA}$ form an ari... | D | 3 | 2005A-P13 |
On a standard die one of the dots is removed at random with each dot equally likely to be chosen. The die is then rolled. What is the probability that the top face has an odd number of dots?
$\textbf{(A) } \frac{5}{11} \qquad \textbf{(B) } \frac{10}{21} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \frac{11}{2... | D | 3 | 2005A-P14 |
Let $\overline{AB}$ be a diameter of a circle and $C$ be a point on $\overline{AB}$ with $2 \cdot AC = BC$. Let $D$ and $E$ be points on the circle such that $\overline{DC} \perp \overline{AB}$ and $\overline{DE}$ is a second diameter. What is the ratio of the area of $\triangle DCE$ to the area of $\triangle ABD$?
``... | C | 3 | 2005A-P15 |
Three circles of radius $s$ are drawn in the first quadrant of the $xy$-plane. The first circle is tangent to both axes, the second is tangent to the first circle and the $x$-axis, and the third is tangent to the first circle and the $y$-axis. A circle of radius $r > s$ is tangent to both axes and to the second and thi... | D | 3 | 2005A-P16 |
A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure $1$. The cube is then cut in the same manner along the dashed lines shown in Figure $2$. This creates nine pieces. What is the volume of the piece that contains vertex $W$?
Image:2005 AMC 12A Problem 17.png
$\text... | A | 3 | 2005A-P17 |
Call a number "prime-looking" if it is composite but not divisible by $2$, $3$, or $5$. The three smallest prime-looking numbers are $49$, $77$, and $91$. There are $168$ prime numbers less than $1000$. How many prime-looking numbers are there less than $1000$?
$\textbf{(A) } 100 \qquad \textbf{(B) } 102 \qquad \textb... | A | 3 | 2005A-P18 |
A faulty car odometer proceeds from digit $3$ to digit $5$, always skipping the digit $4$, regardless of position. For example, after traveling one mile the odometer changed from $000039$ to $000050$. If the odometer now reads $002005$, how many miles has the car actually traveled?
$\textbf{(A) } 1404 \qquad \textbf{(... | B | 3 | 2005A-P19 |
For each $x$ in $[0,1]$, define
$$
f(x) = \begin{cases}
2x, \qquad\qquad \mathrm{if} \quad 0 \leq x \leq \frac{1}{2}\\
2-2x, \qquad \mathrm{if} \quad \frac{1}{2} < x \leq 1.
\end{cases}
$$
Let $f^{[2]}(x) = f(f(x))$, and $f^{[n + 1]}(x) = f^{[n]}(f(x))$ for each integer $n \geq 2$. For how many values of $x$ in $[0... | E | 3 | 2005A-P20 |
How many ordered triples of integers $(a,b,c)$, with $a \geq 2$, $b \geq 1$, and $c \geq 0$, satisfy both $\log_{a}b = c^{2005}$ and $a + b + c = 2005$?
$\textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 4$ | C | 4 | 2005A-P21 |
A rectangular box $P$ is inscribed in a sphere of radius $r$. The surface area of $P$ is $384$, and the sum of the lengths of its $12$ edges is $112$. What is $r$?
$\textbf{(A) } 8 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 14 \qquad \textbf{(E) } 16$ | B | 4 | 2005A-P22 |
Two distinct numbers $a$ and $b$ are chosen randomly from the set $\{2, 2^2, 2^3, \ldots, 2^{25}\}$. What is the probability that $\log_{a}b$ is an integer?
$\textbf{(A) } \frac{2}{25} \qquad \textbf{(B) } \frac{31}{300} \qquad \textbf{(C) } \frac{13}{100} \qquad \textbf{(D) } \frac{7}{50} \qquad \textbf{(E) } \frac{1... | B | 4 | 2005A-P23 |
Let $P(x) = (x - 1)(x - 2)(x - 3)$. For how many polynomials $Q(x)$ does there exist a polynomial $R(x)$ of degree $3$ such that $P(Q(x)) = P(x) \cdot R(x)$?
$\textbf{(A) } 19 \qquad \textbf{(B) } 22 \qquad \textbf{(C) } 24 \qquad \textbf{(D) } 27 \qquad \textbf{(E) } 32$ | B | 4 | 2005A-P24 |
Let $S$ be the set of all points with coordinates $(x,y,z)$, where $x$, $y$, and $z$ are each chosen from the set $\{0, 1, 2\}$. How many equilateral triangles have all their vertices in $S$?
$\textbf{(A) } 72 \qquad \textbf{(B) } 76 \qquad \textbf{(C) } 80 \qquad \textbf{(D) } 84 \qquad \textbf{(E) } 88$ | C | 4 | 2005A-P25 |
A scout troop buys $1000$ candy bars at a price of five for $2$ dollars. They sell all the candy bars at the price of two for $1$ dollar. What was their profit, in dollars?
$\mathrm{(A)}\ 100 \qquad
\mathrm{(B)}\ 200 \qquad
\mathrm{(C)}\ 300 \qquad
\mathrm{(D)}\ 400 \qquad
\mathrm{(E)}\ 500$ | A | 2 | 2005B-P1 |
A positive number $x$ has the property that $x\%$ of $x$ is $4$. What is $x$?
$\mathrm{(A)}\ 2 \qquad
\mathrm{(B)}\ 4 \qquad
\mathrm{(C)}\ 10 \qquad
\mathrm{(D)}\ 20 \qquad
\mathrm{(E)}\ 40$ | D | 2 | 2005B-P2 |
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?
$\mathrm{(A)}\ \frac15 \qquad
\mathrm{(B)}\ \frac13 \qquad
\mathrm{(C)}\ \frac25 \qq... | C | 2 | 2005B-P3 |
At the beginning of the school year, Lisa's goal was to earn an A on at least $80\%$ of her $50$ quizzes for the year. She earned an A on $22$ of the first $30$ quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A?
$\mathrm{(A)}\ 1 \qquad
\mathrm{(B)... | B | 2 | 2005B-P4 |
An $8$-foot by $10$-foot floor is tiled with square tiles of size $1$ foot by $1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $1/2$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?
```asy
unitsize(2... | A | 2 | 2005B-P5 |
In $\triangle ABC$, we have $AC=BC=7$ and $AB=2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD=8$. What is $BD$?
$\mathrm{(A)}\ 3 \qquad
\mathrm{(B)}\ 2\sqrt{3} \qquad
\mathrm{(C)}\ 4 \qquad
\mathrm{(D)}\ 5 \qquad
\mathrm{(E)}\ 4\sqrt{2}$ | A | 2 | 2005B-P6 |
What is the area enclosed by the graph of $|3x|+|4y|=12$?
$\mathrm{(A)}\ 6 \qquad
\mathrm{(B)}\ 12 \qquad
\mathrm{(C)}\ 16 \qquad
\mathrm{(D)}\ 24 \qquad
\mathrm{(E)}\ 25$ | D | 2 | 2005B-P7 |
For how many values of $a$ is it true that the line $y = x + a$ passes through the
vertex of the parabola $y = x^2 + a^2$ ?
$\mathrm{(A)}\ 0 \qquad
\mathrm{(B)}\ 1 \qquad
\mathrm{(C)}\ 2 \qquad
\mathrm{(D)}\ 10 \qquad
\mathrm{(E)}\ \text{infinitely many}$ | C | 2 | 2005B-P8 |
On a certain math exam, $10\%$ of the students got $70$ points, $25\%$ got $80$ points, $20\%$ got $85$ points, $15\%$ got $90$ points, and the rest got $95$ points. What is the difference between the mean and the median score on this exam?
$\mathrm{(A)}\ } \qquad \mathrm{(B)}\ } \qquad \mathrm{(C)}\ } \qquad \mathrm{... | B | 2 | 2005B-P9 |
The first term of a sequence is $2005$. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the $2005^{\text{th}}$ term of the sequence?
$\mathrm{(A)}\ } \qquad \mathrm{(B)}\ } \qquad \mathrm{(C)}\ } \qquad \mathrm{(D)}\ } \qquad \mathrm{(E)}\ }$ | E | 2 | 2005B-P10 |
An envelope contains eight bills: $2$ ones, $2$ fives, $2$ tens, and $2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $$20$ or more?
$\mathrm{(A)}\ }} \qquad \mathrm{(B)}\ }} \qquad \mathrm{(C)}\ }} \qquad \mathrm{(D)}\ }} \qquad \mathrm{(E)}\ }}$ | D | 3 | 2005B-P11 |
The quadratic equation $x^2+mx+n$ has roots twice those of $x^2+px+m$, and none of $m,n,$ and $p$ is zero. What is the value of $n/p$?
$\mathrm{(A)}\ } \qquad \mathrm{(B)}\ } \qquad \mathrm{(C)}\ } \qquad \mathrm{(D)}\ } \qquad \mathrm{(E)}\ }$ | D | 3 | 2005B-P12 |
Suppose that $4^{x_1}=5$, $5^{x_2}=6$, $6^{x_3}=7$, ... , $127^{x_{124}}=128$. What is $x_1x_2...x_{124}$?
$\mathrm{(A)}\ } \qquad \mathrm{(B)}\ }} \qquad \mathrm{(C)}\ } \qquad \mathrm{(D)}\ }} \qquad \mathrm{(E)}\ }$ | D | 3 | 2005B-P13 |
A circle having center $(0,k)$, with $k>6$,is tangent to the lines $y=x$, $y=-x$ and $y=6$. What is the radius of this circle?
$\mathrm{(A)}\ 6\sqrt{2}-6 \qquad
\mathrm{(B)}\ 6 \qquad
\mathrm{(C)}\ 6\sqrt{2} \qquad
\mathrm{(D)}\ 12 \qquad
\mathrm{(E)}\ 6+6\sqrt{2}$ | E | 3 | 2005B-P14 |
The sum of four two-digit numbers is $221$. None of the eight digits is $0$ and no two of them are the same. Which of the following is '''not''' included among the eight digits?
$\mathrm{(A)}\ 1 \qquad
\mathrm{(B)}\ 2 \qquad
\mathrm{(C)}\ 3 \qquad
\mathrm{(D)}\ 4 \qquad
\mathrm{(E)}\ 5$ | D | 3 | 2005B-P15 |
Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres?
$\mathrm {(A)}\ \sqrt{2} \qquad
\mathrm {(B)}\ \sqrt{3} \qquad
\mathrm {(C)}\ 1+\sqrt{2}\qquad
\mathrm {(D)}\ 1+\sqrt{3}\qquad
\mat... | D | 3 | 2005B-P16 |
How many distinct four-tuples $(a, b, c, d)$ of rational numbers are there with
$a \cdot \log_{10} 2+b \cdot \log_{10} 3 +c \cdot \log_{10} 5 + d \cdot \log_{10} 7 = 2005$?
$\mathrm{(A)}\ 0 \qquad
\mathrm{(B)}\ 1 \qquad
\mathrm{(C)}\ 17 \qquad
\mathrm{(D)}\ 2004 \qquad
\mathrm{(E)}\ \text{infinitely many}$ | B | 3 | 2005B-P17 |
Let $A(2,2)$ and $B(7,7)$ be points in the plane. Define $R$ as the region in the first quadrant consisting of those points $C$ such that $\triangle ABC$ is an acute triangle. What is the closest integer to the area of the region $R$?
$\mathrm{(A)}\ 25 \qquad
\mathrm{(B)}\ 39 \qquad
\mathrm{(C)}\ 51 \qquad
\mathrm{(D)... | C | 3 | 2005B-P18 |
Let $x$ and $y$ be two-digit integers such that $y$ is obtained by reversing the digits of $x$. The integers $x$ and $y$ satisfy $x^{2}-y^{2}=m^{2}$ for some positive integer $m$. What is $x+y+m$?
$\mathrm{(A)}\ 88 \qquad
\mathrm{(B)}\ 112 \qquad
\mathrm{(C)}\ 116 \qquad
\mathrm{(D)}\ 144 \qquad
\mathrm{(E)}\ 154 \qqu... | E | 3 | 2005B-P19 |
Let $a,b,c,d,e,f,g$ and $h$ be distinct elements in the set
$$
\{-7,-5,-3,-2,2,4,6,13\}.
$$
What is the minimum possible value of
$$
(a+b+c+d)^{2}+(e+f+g+h)^{2}?
$$
$\mathrm{(A)}\ 30 \qquad
\mathrm{(B)}\ 32 \qquad
\mathrm{(C)}\ 34 \qquad
\mathrm{(D)}\ 40 \qquad
\mathrm{(E)}\ 50$ | C | 3 | 2005B-P20 |
A positive integer $n$ has $60$ divisors and $7n$ has $80$ divisors. What is the greatest integer $k$ such that $7^k$ divides $n$?
$\mathrm{(A)}\ } \qquad \mathrm{(B)}\ } \qquad \mathrm{(C)}\ } \qquad \mathrm{(D)}\ } \qquad \mathrm{(E)}\ }$ | C | 4 | 2005B-P21 |
A sequence of complex numbers $z_{0}, z_{1}, z_{2}, ...$ is defined by the rule
$$
z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},
$$
where $\overline {z_{n}}$ is the complex conjugate of $z_{n}$ and $i^{2}=-1$. Suppose that $|z_{0}|=1$ and $z_{2005}=1$. How many possible values are there for $z_{0}$?
\textbf{(A)}\ 1 \... | E | 4 | 2005B-P22 |
Let $S$ be the set of ordered triples $(x,y,z)$ of real numbers for which
$$
\log_{10}(x+y) = z \text{ and } \log_{10}(x^{2}+y^{2}) = z+1.
$$
There are real numbers $a$ and $b$ such that for all ordered triples $(x,y,z)$ in $S$ we have $x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.$ What is the value of $a+b?$
\text... | B | 4 | 2005B-P23 |
All three vertices of an equilateral triangle are on the parabola $y=x^2$, and one of its sides has a slope of $2$. The $x$-coordinates of the three vertices have a sum of $m/n$, where $m$ and $n$ are relatively prime positive integers. What is the value of $m+n$?
$\mathrm{(A)}\ } \qquad \mathrm{(B)}\ } \qquad \mathrm... | A | 4 | 2005B-P24 |
Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability. What is the probability that no two ants arrive at the same vertex?
$\mathr... | A | 4 | 2005B-P25 |
Sandwiches at Joe's Fast Food cost $3$ dollars each and sodas cost $2$ dollars each. How many dollars will it cost to purchase $5$ sandwiches and $8$ sodas?
$\mathrm{(A) \ } 31\qquad \mathrm{(B) \ } 32\qquad \mathrm{(C) \ } 33\qquad \mathrm{(D) \ } 34\qquad \mathrm{(E) \ } 35$ | A | 2 | 2006A-P1 |
Define $x\otimes y=x^3-y$. What is $h\otimes (h\otimes h)$?
$\mathrm{(A) \ } -h\qquad \mathrm{(B) \ } 0\qquad \mathrm{(C) \ } h\qquad \mathrm{(D) \ } 2h\qquad \mathrm{(E) \ } h^3$ | C | 2 | 2006A-P2 |
The ratio of Mary's age to Alice's age is $3:5$. Alice is $30$ years old. How old is Mary?
$\mathrm{(A) \ } 15\qquad \mathrm{(B) \ } 18\qquad \mathrm{(C) \ } 20\qquad \mathrm{(D) \ } 24\qquad \mathrm{(E) \ } 50$ | B | 2 | 2006A-P3 |
A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?
$\mathrm{(A) \ } 17\qquad \mathrm{(B) \ } 19\qquad \mathrm{(C) \ } 21\qquad \mathrm{(D) \ } 22\qquad \mathrm{(E) \ } 23$ | E | 2 | 2006A-P4 |
Doug and Dave shared a pizza with $8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $8$ dollars, and there was an additional cost of $2$ dollars for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. ... | D | 2 | 2006A-P5 |
The $8\times 18$ rectangle $ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $y$?
```asy
unitsize(3mm);
defaultpen(fontsize(10pt)+linewidth(.8pt));
dotfactor=4;
draw((0,4)--(18,4)--(18,-4)--(0,-4)--cycle);
draw((6,4)--... | A | 2 | 2006A-P6 |
Mary is $20\%$ older than Sally, and Sally is $40\%$ younger than Danielle. The sum of their ages is $23.2$ years. How old will Mary be on her next birthday?
$\mathrm{(A) \ } 7\qquad \mathrm{(B) \ } 8\qquad \mathrm{(C) \ } 9\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 11$ | B | 2 | 2006A-P7 |
How many sets of two or more consecutive positive integers have a sum of $15$?
$\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 3\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 5$ | C | 2 | 2006A-P8 |
Oscar buys $13$ pencils and $3$ erasers for $\textdollar 1.00$. A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total cost, in cents, of one pencil and one eraser?
$\mathrm{(A) \ } 10\qquad \mathrm{(B) \ } 12\qquad \mathrm{(C) \ } 15\qquad \mathrm{(D) \ } 18\qquad \mathrm{(... | A | 2 | 2006A-P9 |
For how many real values of $x$ is $\sqrt{120-\sqrt{x}}$ an integer?
$\mathrm{(A) \ } 3\qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 9\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 11$ | E | 2 | 2006A-P10 |
Which of the following describes the graph of the equation $(x+y)^2=x^2+y^2$?
$\mathrm{(A)}\ \text{the empty set}\qquad\mathrm{(B)}\ \text{one point}\qquad\mathrm{(C)}\ \text{two lines}\qquad\mathrm{(D)}\ \text{a circle}\qquad\mathrm{(E)}\ \text{the entire plane}$ | C | 3 | 2006A-P11 |
A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outside diameter of 20 cm. The outside diameter of each of the outer rings is 1 cm less than that of the ring above it. The bottom ring has an outside diameter of 3 cm. What is the distance, in cm, from the top of the top ring to the b... | B | 3 | 2006A-P12 |
<!-- <center>Image:2006_AMC_12A_Problem_13.gif</center> -->
The vertices of a $3-4-5$ right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles?
```asy
unitsize(5mm);
defaultpen(fontsize(10pt)+linewidth(.8pt));
pair B=(0,0), C=(5,0);
pair A... | E | 3 | 2006A-P13 |
Two farmers agree that pigs are worth $300$ dollars and that goats are worth $210$ dollars. When one farmer owes the other money, he pays the debt in pigs or goats, with "change" received in the form of goats or pigs as necessary. (For example, a $390$ dollar debt could be paid with two pigs, with one goat received in ... | C | 3 | 2006A-P14 |
Suppose $\cos x=0$ and $\cos (x+z)=1/2$. What is the smallest possible positive value of $z$?
$\mathrm{(A) \ } \frac{\pi}{6}\qquad \mathrm{(B) \ } \frac{\pi}{3}\qquad \mathrm{(C) \ } \frac{\pi}{2}\qquad \mathrm{(D) \ } \frac{5\pi}{6}\qquad \mathrm{(E) \ } \frac{7\pi}{6}$ | A | 3 | 2006A-P15 |
Circles with centers $A$ and $B$ have radii $3$ and $8$, respectively. A common internal tangent intersects the circles at $C$ and $D$, respectively. Lines $AB$ and $CD$ intersect at $E$, and $AE=5$. What is $CD$?
<!-- center -->
```asy
unitsize(2.5mm);
defaultpen(fontsize(10pt)+linewidth(.8pt));
dotfactor=3;
pair A=(... | B | 3 | 2006A-P16 |
Square $ABCD$ has side length $s$, a circle centered at $E$ has radius $r$, and $r$ and $s$ are both rational. The circle passes through $D$, and $D$ lies on $\overline{BE}$. Point $F$ lies on the circle, on the same side of $\overline{BE}$ as $A$. Segment $AF$ is tangent to the circle, and $AF=\sqrt{9+5\sqrt{2}}$. Wha... | B | 3 | 2006A-P17 |
The function $f$ has the property that for each real number $x$ in its domain, $1/x$ is also in its domain and
$f(x)+f\left(\frac{1}{x}\right)=x$
What is the largest set of real numbers that can be in the domain of $f$?
$\mathrm{(A) \ } \{x|x\ne 0\}\qquad \mathrm{(B) \ } \{x|x<0\}\qquad \mathrm{(C) \ } \{x|x>0\}\qqu... | E | 3 | 2006A-P18 |
Circles with centers $(2,4)$ and $(14,9)$ have radii $4$ and $9$, respectively. The equation of a common external tangent to the circles can be written in the form $y=mx+b$ with $m>0$. What is $b$?
<!-- center -->
```asy
size(150);
defaultpen(linewidth(0.7)+fontsize(8));
draw(circle((2,4),4));draw(circle((14,9),9));
... | E | 3 | 2006A-P19 |
A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that... | C | 3 | 2006A-P20 |
Let
$S_1=\{(x,y)|\log_{10}(1+x^2+y^2)\le 1+\log_{10}(x+y)\}$
and
$S_2=\{(x,y)|\log_{10}(2+x^2+y^2)\le 2+\log_{10}(x+y)\}$.
What is the ratio of the area of $S_2$ to the area of $S_1$?
$\mathrm{(A) \ } 98\qquad \mathrm{(B) \ } 99\qquad \mathrm{(C) \ } 100\qquad \mathrm{(D) \ } 101\qquad \mathrm{(E) \ } 102$ | E | 4 | 2006A-P21 |
A circle of radius $r$ is concentric with and outside a regular hexagon of side length $2$. The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is $1/2$. What is $r$?
$\mathrm{(A) \ } 2\sqrt{2}+2\sqrt{3}\qquad \mathrm{(B) \ } 3\sqrt{3}+\sqrt{2}\qquad \mathrm{(C) \ ... | D | 4 | 2006A-P22 |
Given a finite sequence $S=(a_1,a_2,\ldots ,a_n)$ of $n$ real numbers, let $A(S)$ be the sequence
$\left(\frac{a_1+a_2}{2},\frac{a_2+a_3}{2},\ldots ,\frac{a_{n-1}+a_n}{2}\right)$
of $n-1$ real numbers. Define $A^1(S)=A(S)$ and, for each integer $m$, $2\le m\le n-1$, define $A^m(S)=A(A^{m-1}(S))$. Suppose $x>0$, and l... | B | 4 | 2006A-P23 |
The expression
$(x+y+z)^{2006}+(x-y-z)^{2006}$
is simplified by expanding it and combining like terms. How many terms are in the simplified expression?
$\mathrm{(A) \ } 6018\qquad \mathrm{(B) \ } 671,676\qquad \mathrm{(C) \ } 1,007,514\qquad \mathrm{(D) \ } 1,008,016\qquad \mathrm{(E) \ } 2,015,028$ | D | 4 | 2006A-P24 |
How many non-empty subsets $S$ of $\lbrace 1,2,3,\ldots ,15\rbrace$ have the following two properties?
$(1)$ No two consecutive integers belong to $S$.
$(2)$ If $S$ contains $k$ elements, then $S$ contains no number less than $k$.
$\mathrm{(A) \ } 277\qquad \mathrm{(B) \ } 311\qquad \mathrm{(C) \ } 376\qquad \mathrm... | E | 4 | 2006A-P25 |
What is $( - 1)^1 + ( - 1)^2 + \cdots + ( - 1)^{2006}$?
$\text {(A) } - 2006 \qquad \text {(B) } - 1 \qquad \text {(C) } 0 \qquad \text {(D) } 1 \qquad \text {(E) } 2006$ | C | 2 | 2006B-P1 |
For real numbers $x$ and $y$, define $x\spadesuit y = (x + y)(x - y)$. What is $3\spadesuit(4\spadesuit 5)$?
$\text {(A) } - 72 \qquad \text {(B) } - 27 \qquad \text {(C) } - 24 \qquad \text {(D) } 24 \qquad \text {(E) } 72$ | A | 2 | 2006B-P2 |
A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score?
$\text {(A) } 10 \qquad \text {(B) } 14 \qquad \text {(C) } 17 \qquad \text {(D) } 20 \qquad \text {(E) } 24$ | A | 2 | 2006B-P3 |
Mary is about to pay for five items at the grocery store. The prices of the items are $\textdollar7.99$, $\textdollar4.99$, $\textdollar2.99$, $\textdollar1.99$, and $\textdollar0.99$. Mary will pay with a twenty-dollar bill. Which of the following is closest to the percentage of the $\textdollar20.00$ that she will re... | A | 2 | 2006B-P4 |
John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up to John?
$\text {(A) } 30 \qquad \text {(B) } 50 \qquad \text {(C) } 60 \qquad \text {(D) } 90 \qquad \text {(E) ... | A | 2 | 2006B-P5 |
Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade?
$\text {(A) } 129 \qquad \text {(B) } 137 \qqua... | B | 2 | 2006B-P6 |
Mr. and Mrs. Lopez have two children. When they get into their family car, two people sit in the front, and the other two sit in the back. Either Mr. Lopez or Mrs. Lopez must sit in the driver's seat. How many seating arrangements are possible?
$\text {(A) } 4 \qquad \text {(B) } 12 \qquad \text {(C) } 16 \qquad \text... | B | 2 | 2006B-P7 |
The lines $x = \frac 14y + a$ and $y = \frac 14x + b$ intersect at the point $(1,2)$. What is $a + b$?
$\text {(A) } 0 \qquad \text {(B) } \frac 34 \qquad \text {(C) } 1 \qquad \text {(D) } 2 \qquad \text {(E) } \frac 94$ | E | 2 | 2006B-P8 |
How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order?
$\text {(A) } 21 \qquad \text {(B) } 34 \qquad \text {(C) } 51 \qquad \text {(D) } 72 \qquad \text {(E) } 150$ | B | 2 | 2006B-P9 |
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?
$\text {(A) } 43 \qquad \text {(B) } 44 \qquad \text {(C) } 45 \qquad \text {(D) } 46 \qquad \text {(E) } 47$ | A | 2 | 2006B-P10 |
Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee?
$\text {(A) ... | E | 3 | 2006B-P11 |
The parabola $y=ax^2+bx+c$ has vertex $(p,p)$ and $y$-intercept $(0,-p)$, where $p\ne 0$. What is $b$?
$\text {(A) } -p \qquad \text {(B) } 0 \qquad \text {(C) } 2 \qquad \text {(D) } 4 \qquad \text {(E) } p$ | D | 3 | 2006B-P12 |
Rhombus $ABCD$ is similar to rhombus $BFDE$. The area of rhombus $ABCD$ is 24, and $\angle BAD = 60^\circ$. What is the area of rhombus $BFDE$?
```asy
defaultpen(linewidth(0.7)+fontsize(11));
unitsize(2cm);
pair A=origin, B=(2,0), C=(3, sqrt(3)), D=(1, sqrt(3)), E=(1, 1/sqrt(3)), F=(2, 2/sqrt(3));
pair point=(3/2, sqr... | C | 3 | 2006B-P13 |
Elmo makes $N$ sandwiches for a fundraiser. For each sandwich he uses $B$ globs of peanut butter at $4$ cents per glob and $J$ blobs of jam at $5$ cents per blob. The cost of the peanut butter and jam to make all the sandwiches is $\textdollar 2.53$. Assume that $B$, $J$ and $N$ are all positive integers with $N>1$. Wh... | D | 3 | 2006B-P14 |
Circles with centers $O$ and $P$ have radii 2 and 4, respectively, and are externally tangent. Points $A$ and $B$ are on the circle centered at $O$, and points $C$ and $D$ are on the circle centered at $P$, such that $\overline{AD}$ and $\overline{BC}$ are common external tangents to the circles. What is the area of he... | B | 3 | 2006B-P15 |
Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$, respectively. What is its area?
$\mathrm{(A)}\ 20\sqrt {3}
\qquad
\mathrm{(B)}\ 22\sqrt {3}
\qquad
\mathrm{(C)}\ 25\sqrt {3}
\qquad
\mathrm{(D)}\ 27\sqrt {3}
\qquad
\mathrm{(E)}\ 50$ | C | 3 | 2006B-P16 |
For a particular peculiar pair of dice, the probabilities of rolling $1$, $2$, $3$, $4$, $5$ and $6$ on each die are in the ratio $1:2:3:4:5:6$. What is the probability of rolling a total of $7$ on the two dice?
$\mathrm{(A)}\ \frac 4{63}
\qquad
\mathrm{(B)}\ \frac 18
\qquad
\mathrm{(C)}\ \frac 8{63}
\qquad
\mathrm{(D... | C | 3 | 2006B-P17 |
An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?
$\mathrm{(A)}\ 120
\qquad
\math... | B | 3 | 2006B-P18 |
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and th... | B | 3 | 2006B-P19 |
Let $x$ be chosen at random from the interval $(0,1)$. What is the probability that
$\lfloor\log_{10}4x\rfloor - \lfloor\log_{10}x\rfloor = 0$?
Here $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.
$\mathrm{(A)}\ \frac 18
\qquad
\mathrm{(B)}\ \frac 3{20}
\qquad
\mathrm{(C)}\ \frac 16
... | C | 3 | 2006B-P20 |
Rectangle $ABCD$ has area $2006$. An ellipse with area $2006\pi$ passes through $A$ and $C$ and has foci at $B$ and $D$. What is the perimeter of the rectangle? (The area of an ellipse is $ab\pi$ where $2a$ and $2b$ are the lengths of the axes.)
$\mathrm{(A)}\ \frac {16\sqrt {2006}}{\pi}
\qquad
\mathrm{(B)}\ \frac {10... | C | 4 | 2006B-P21 |
Suppose $a$, $b$ and $c$ are positive integers with $a+b+c=2006$, and $a!b!c!=m\cdot 10^n$, where $m$ and $n$ are integers and $m$ is not divisible by $10$. What is the smallest possible value of $n$?
$\mathrm{(A)}\ 489
\qquad
\mathrm{(B)}\ 492
\qquad
\mathrm{(C)}\ 495
\qquad
\mathrm{(D)}\ 498
\qquad
\mathrm{(E)}\ 501... | B | 4 | 2006B-P22 |
Isosceles $\triangle ABC$ has a right angle at $C$. Point $P$ is inside $\triangle ABC$, such that $PA=11$, $PB=7$, and $PC=6$. Legs $\overline{AC}$ and $\overline{BC}$ have length $s=\sqrt{a+b\sqrt{2}}$, where $a$ and $b$ are positive integers. What is $a+b$?
```asy
pathpen = linewidth(0.7); pointpen = black;
pen f =... | E | 4 | 2006B-P23 |
Let $S$ be the set of all points $(x,y)$ in the coordinate plane such that $0\leq x\leq \frac\pi 2$ and $0\leq y\leq \frac\pi 2$. What is the area of the subset of $S$ for which $\sin^2 x - \sin x\sin y + \sin^2 y\le \frac 34$?
$\mathrm{(A)}\ \frac {\pi^2}9
\qquad
\mathrm{(B)}\ \frac {\pi^2}8
\qquad
\mathrm{(C)}\ \fra... | C | 4 | 2006B-P24 |
A sequence $a_1,a_2,\dots$ of non-negative integers is defined by the rule $a_{n+2}=|a_{n+1}-a_n|$ for $n\geq 1$. If $a_1=999$, $a_2<999$ and $a_{2006}=1$, how many different values of $a_2$ are possible?
$\mathrm{(A)}\ 165
\qquad
\mathrm{(B)}\ 324
\qquad
\mathrm{(C)}\ 495
\qquad
\mathrm{(D)}\ 499
\qquad
\mathrm{(E)}\... | B | 4 | 2006B-P25 |
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