2004 AMC 12A Problems

Problem 1

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?

$\mathrm {(A)} 0.0029 \qquad \mathrm {(B)} 0.029 \qquad \mathrm {(C)} 0.29 \qquad \mathrm {(D)} 2.9 \qquad \mathrm {(E)} 29$

Solution

Problem 2

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 $100$ ?

$\mathrm {(A)} 11 \qquad \mathrm {(B)} 13 \qquad \mathrm {(C)} 14 \qquad \mathrm {(D)} 16 \qquad \mathrm {(E)} 17$

Solution

Problem 3

For how many ordered pairs of positive integers $(x,y)$ is $x+2y=100$ ?

$\mathrm {(A)} 33 \qquad \mathrm {(B)} 49 \qquad \mathrm {(C)} 50 \qquad \mathrm {(D)} 99 \qquad \mathrm {(E)} 100$

Solution

Problem 4

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 children?

$\mathrm {(A)} 22 \qquad \mathrm {(B)} 23 \qquad \mathrm {(C)} 24 \qquad \mathrm {(D)} 25 \qquad \mathrm {(E)} 26$

Solution

Problem 5

The graph of the line $y=mx+b$ is shown. Which of the following is true?

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$\mathrm {(A)} mb<-1 \qquad \mathrm {(B)} -1<mb<0 \qquad \mathrm {(C)} mb=0 \qquad \mathrm {(D)} 0<mb<1 \qquad \mathrm {(E)} mb>1$

Solution

Problem 6

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?

$\mathrm {(A)} U-V \qquad \mathrm {(B)} V-W \qquad \mathrm {(C)} W-X \qquad \mathrm {(D)} X-Y \qquad \mathrm {(E)} Y-Z \qquad$

Solution

Problem 7

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, respectively. How many rounds will there be in the game?

$\mathrm {(A)} 36 \qquad \mathrm {(B)} 37 \qquad \mathrm {(C)} 38 \qquad \mathrm {(D)} 39 \qquad \mathrm {(E)} 40 \qquad$

Solution

Problem 8

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 $\triangle{BDC}$ ?