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Difference between revisions of "2003 AMC 12A Problems"
(added problems 1-8, 12-13, 15, and 18)
Line 1: Line 1:
 
== Problem 1 ==
 
== Problem 1 ==
 +
What is the difference between the sum of the first <math>2003</math> even counting numbers and the sum of the first <math>2003</math> odd counting numbers?
 +
 +
<math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 2\qquad \mathrm{(D) \ } 2003\qquad \mathrm{(E) \ } 4006 </math>
  
 
[[2003 AMC 12A/Problem 1|Solution]]
 
[[2003 AMC 12A/Problem 1|Solution]]
  
 
== Problem 2 ==
 
== Problem 2 ==
 +
Members of the Rockham Soccer Leauge buy socks and T-shirts. Socks cost &#36;4 per pair and each T-shirt costs &#36;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 &#36;2366, how many members are in the Leauge?
 +
 +
<math> \mathrm{(A) \ } 77\qquad \mathrm{(B) \ } 91\qquad \mathrm{(C) \ } 143\qquad \mathrm{(D) \ } 182\qquad \mathrm{(E) \ } 286 </math>
  
 
[[2003 AMC 12A/Problem 2|Solution]]
 
[[2003 AMC 12A/Problem 2|Solution]]
  
 
== Problem 3 ==
 
== Problem 3 ==
 +
A solid box is <math>15</math> cm by <math>10</math> cm by <math>8</math> cm. A new solid is formed by removing a cube <math>3</math> cm on a side from each corner of this box. What percent of the original volume is removed?
 +
 +
<math> \mathrm{(A) \ } 4.5\qquad \mathrm{(B) \ } 9\qquad \mathrm{(C) \ } 12\qquad \mathrm{(D) \ } 18\qquad \mathrm{(E) \ } 24 </math>
  
 
[[2003 AMC 12A/Problem 3|Solution]]
 
[[2003 AMC 12A/Problem 3|Solution]]
  
 
== Problem 4 ==
 
== Problem 4 ==
 +
It takes Mary <math>30</math> minutes to walk uphill <math>1</math> km from her home to school, but it takes her only <math>10</math> minutes to walk from school to her home along the same route. What is her average speed, in km/hr, for the round trip?
 +
 +
<math> \mathrm{(A) \ } 3\qquad \mathrm{(B) \ } 3.125\qquad \mathrm{(C) \ } 3.5\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 4.5 </math>
  
 
[[2003 AMC 12A/Problem 4|Solution]]
 
[[2003 AMC 12A/Problem 4|Solution]]
  
 
== Problem 5 ==
 
== Problem 5 ==
 +
The sum of the two 5-digit numbers <math>AMC10</math> and <math>AMC12</math> is <math>123422</math>. What is <math>A+M+C</math>?
 +
 +
<math> \mathrm{(A) \ } 10\qquad \mathrm{(B) \ } 11\qquad \mathrm{(C) \ } 12\qquad \mathrm{(D) \ } 13\qquad \mathrm{(E) \ } 14 </math>
  
 
[[2003 AMC 12A/Problem 5|Solution]]
 
[[2003 AMC 12A/Problem 5|Solution]]
  
 
== Problem 6 ==
 
== Problem 6 ==
 +
Define <math>x \heartsuit y</math> to be <math>|x-y|</math> for all real numbers <math>x</math> and <math>y</math>. Which of the following statements is not true?
 +
 +
<math> \mathrm{(A) \ } x \heartsuit y = y \heartsuit x </math> for all <math>x</math> and <math>y</math>
 +
 +
<math>\mathrm{(B) \ } 2(x \heartsuit y) = (2x) \heartsuit (2y) </math> for all <math>x</math> and <math>y</math>
 +
 +
<math>\mathrm{(C) \ } x \heartsuit 0 = x </math> for all <math>x</math>
 +
 +
<math>\mathrm{(D) \ } x \heartsuit x = 0 </math> for all <math>x</math>
 +
 +
<math> \mathrm{(E) \ } x \heartsuit y > 0 </math> if <math>x \neq y</math>
  
 
[[2003 AMC 12A/Problem 6|Solution]]
 
[[2003 AMC 12A/Problem 6|Solution]]
  
 
== Problem 7 ==
 
== Problem 7 ==
 +
How many non-congruent triangles with perimeter <math>7</math> have integer side lengths?
 +
 +
<math> \mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 3\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 5 </math>
  
 
[[2003 AMC 12A/Problem 7|Solution]]
 
[[2003 AMC 12A/Problem 7|Solution]]
  
 
== Problem 8 ==
 
== Problem 8 ==
 +
What is the probability that a randomly drawn positive factor of <math>60</math> is less than <math>7</math>
 +
 +
<math> \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} </math>
  
 
[[2003 AMC 12A/Problem 8|Solution]]
 
[[2003 AMC 12A/Problem 8|Solution]]
Line 44: Line 76:
  
 
== Problem 12 ==
 
== Problem 12 ==
 +
Sally has five red cards numbered <math>1</math> through <math>5</math> and four blue cards numbered <math>3</math> through <math>6</math>. 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?
 +
 +
<math> \mathrm{(A) \ } 8\qquad \mathrm{(B) \ } 9\qquad \mathrm{(C) \ } 10\qquad \mathrm{(D) \ } 11\qquad \mathrm{(E) \ } 12 </math>
  
 
[[2003 AMC 12A/Problem 12|Solution]]
 
[[2003 AMC 12A/Problem 12|Solution]]
  
 
== Problem 13 ==
 
== Problem 13 ==
 +
The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attatched 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]]
 +
 +
<math> \mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 3\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 5\qquad \mathrm{(E) \ } 6 </math>
  
 
[[2003 AMC 12A/Problem 13|Solution]]
 
[[2003 AMC 12A/Problem 13|Solution]]
Line 56: Line 96:
  
 
== Problem 15 ==
 
== Problem 15 ==
 +
A semicircle of diameter <math>1</math> sits at the top of a semicircle of diameter <math>2</math>, 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]]
 +
 +
<math> \mathrm{(A) \ } \frac{1}{6}\pi-\frac{\sqrt{3}}{4}\qquad \mathrm{(B) \ } \frac{\sqrt{3}}{4}-\frac{1}{12}\pi\qquad \mathrm{(C) \ } \frac{\sqrt{3}}{4}-\frac{1}{24}\pi\qquad \mathrm{(D) \ } \frac{\sqrt{3}}{4}+\frac{1}{24}\pi\qquad \mathrm{(E) \ } \frac{\sqrt{3}}{4}+\frac{1}{12}\pi </math>
  
 
[[2003 AMC 12A/Problem 15|Solution]]
 
[[2003 AMC 12A/Problem 15|Solution]]
Line 68: Line 113:
  
 
== Problem 18 ==
 
== Problem 18 ==
 +
Let <math>n</math> be a <math>5</math>-digit number, and let <math>q</math> and <math>r</math> be the quotient and the remainder, respectively, when <math>n</math> is divided by <math>100</math>. For how many values of <math>n</math> is <math>q+r</math> divisible by <math>11</math>?
 +
 +
<math> \mathrm{(A) \ } 8180\qquad \mathrm{(B) \ } 8181\qquad \mathrm{(C) \ } 8182\qquad \mathrm{(D) \ } 9000\qquad \mathrm{(E) \ } 9090 </math>
  
 
[[2003 AMC 12A/Problem 18|Solution]]
 
[[2003 AMC 12A/Problem 18|Solution]]

Revision as of 23:00, 9 November 2006

Problem 1

What is the difference between the sum of the first $2003$ even counting numbers and the sum of the first $2003$ odd counting numbers?

$\mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 2\qquad \mathrm{(D) \ } 2003\qquad \mathrm{(E) \ } 4006$

Solution

Problem 2

Members of the Rockham Soccer Leauge 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 the Leauge?

$\mathrm{(A) \ } 77\qquad \mathrm{(B) \ } 91\qquad \mathrm{(C) \ } 143\qquad \mathrm{(D) \ } 182\qquad \mathrm{(E) \ } 286$

Solution

Problem 3

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?

$\mathrm{(A) \ } 4.5\qquad \mathrm{(B) \ } 9\qquad \mathrm{(C) \ } 12\qquad \mathrm{(D) \ } 18\qquad \mathrm{(E) \ } 24$

Solution

Problem 4

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?

$\mathrm{(A) \ } 3\qquad \mathrm{(B) \ } 3.125\qquad \mathrm{(C) \ } 3.5\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 4.5$

Solution

Problem 5

The sum of the two 5-digit numbers $AMC10$ and $AMC12$ is $123422$. What is $A+M+C$?

$\mathrm{(A) \ } 10\qquad \mathrm{(B) \ } 11\qquad \mathrm{(C) \ } 12\qquad \mathrm{(D) \ } 13\qquad \mathrm{(E) \ } 14$

Solution

Problem 6

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 $x$

$\mathrm{(D) \ } x \heartsuit x = 0$ for all $x$

$\mathrm{(E) \ } x \heartsuit y > 0$ if $x \neq y$

Solution

Problem 7

How many non-congruent triangles with perimeter $7$ have integer side lengths?

$\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 3\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 5$

Solution

Problem 8

What is the probability that a randomly drawn positive factor of $60$ is less than $7$

$\mathrm{(A) \ } \frac{1}{10}\qquad \mathrm{(B) \ } \frac{1}{6}\qquad \mathrm{(C) \ } \frac{1}{4}\qquad \mathrm{(D) \ } \frac{1}{3}\qquad \mathrm{(E) \ } \frac{1}{2}$

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

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?

$\mathrm{(A) \ } 8\qquad \mathrm{(B) \ } 9\qquad \mathrm{(C) \ } 10\qquad \mathrm{(D) \ } 11\qquad \mathrm{(E) \ } 12$

Solution

Problem 13

The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attatched 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?

2003amc10a10.gif

$\mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 3\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 5\qquad \mathrm{(E) \ } 6$

Solution

Problem 14

Solution

Problem 15

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.

2003amc10a19.gif

$\mathrm{(A) \ } \frac{1}{6}\pi-\frac{\sqrt{3}}{4}\qquad \mathrm{(B) \ } \frac{\sqrt{3}}{4}-\frac{1}{12}\pi\qquad \mathrm{(C) \ } \frac{\sqrt{3}}{4}-\frac{1}{24}\pi\qquad \mathrm{(D) \ } \frac{\sqrt{3}}{4}+\frac{1}{24}\pi\qquad \mathrm{(E) \ } \frac{\sqrt{3}}{4}+\frac{1}{12}\pi$

Solution

Problem 16

Solution

Problem 17

Solution

Problem 18

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 $11$?

$\mathrm{(A) \ } 8180\qquad \mathrm{(B) \ } 8181\qquad \mathrm{(C) \ } 8182\qquad \mathrm{(D) \ } 9000\qquad \mathrm{(E) \ } 9090$

Solution

Problem 19

Solution

Problem 20

Solution

Problem 21

Solution

Problem 22

Solution

Problem 23

Solution

Problem 24

Solution

Problem 25

Solution

See also

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