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Difference between revisions of "2001 AMC 10 Problems"
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1. The median of the list
+
==Problem 1==
−
<math>n; n + 3; n + 4; n + 5; n + 6; n + 8; n + 10; n + 12; n + 15</math>
+
 
 +
The median of the list <math>n, n + 3, n + 4, n + 5, n + 6, n + 8, n + 10, n + 12, n + 15</math>
 
is 10. What is the mean?
 
is 10. What is the mean?
  
 
<math>\mathrm{(A)}\ 4 \qquad\mathrm{(B)}\ 6 \qquad\mathrm{(C)}\ 7 \qquad\mathrm{(D)}\ 10 \qquad\mathrm{(E)}\ 11</math>
 
<math>\mathrm{(A)}\ 4 \qquad\mathrm{(B)}\ 6 \qquad\mathrm{(C)}\ 7 \qquad\mathrm{(D)}\ 10 \qquad\mathrm{(E)}\ 11</math>
  
−
2. A number <math>x</math> is <math>2</math> more than the product of its reciprocal and its additive inverse. In which interval does the number lie?
+
==Problem 2==
 +
A number <math>x</math> is <math>2</math> more than the product of its reciprocal and its additive inverse. In which interval does the number lie?
  
 
<math>\mathrm{(A)}\ -4\leq x\leq -2 \qquad\mathrm{(B)}\ -2<x\leq 0 \qquad\mathrm{(C)}\ 0<x\leq 2</math>
 
<math>\mathrm{(A)}\ -4\leq x\leq -2 \qquad\mathrm{(B)}\ -2<x\leq 0 \qquad\mathrm{(C)}\ 0<x\leq 2</math>
Line 11: Line 13:
 
<math>\mathrm{(D)}\ 2<x\leq 4 \qquad\mathrm{(E)}\ 4<x\leq 6</math>
 
<math>\mathrm{(D)}\ 2<x\leq 4 \qquad\mathrm{(E)}\ 4<x\leq 6</math>
  
−
3. The sum of two numbers is <math>S</math>. Suppose <math>3</math> is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?
+
==Problem 3==
 +
 
 +
The sum of two numbers is <math>S</math>. Suppose <math>3</math> is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?
  
 
<math>\mathrm{(A)}\ 2S+3 \qquad\mathrm{(B)}\ 3S+2 \qquad\mathrm{(C)}\ 3S+6 \qquad\mathrm{(D)}\ 2S+6 \qquad\mathrm{(E)}\ 2S+12</math>
 
<math>\mathrm{(A)}\ 2S+3 \qquad\mathrm{(B)}\ 3S+2 \qquad\mathrm{(C)}\ 3S+6 \qquad\mathrm{(D)}\ 2S+6 \qquad\mathrm{(E)}\ 2S+12</math>
  
−
4. What is the maximum number for the possible points of intersection of a circle and a triangle?
+
==Problem 4==
 +
 
 +
What is the maximum number for the possible points of intersection of a circle and a triangle?
  
 
<math>\mathrm{(A)}\ 2 \qquad\mathrm{(B)}\ 3 \qquad\mathrm{(C)}\ 4 \qquad\mathrm{(D)}\ 5 \qquad\mathrm{(E)}\ 6</math>
 
<math>\mathrm{(A)}\ 2 \qquad\mathrm{(B)}\ 3 \qquad\mathrm{(C)}\ 4 \qquad\mathrm{(D)}\ 5 \qquad\mathrm{(E)}\ 6</math>
 +
 +
==Problem 5==
 +
 +
How many of the twelve pentominoes pictured below have at least one line of
 +
symmetry?
 +
 +
<math>\mathrm{(A)}\ 3 \qquad\mathrm{(B)}\ 2 \qquad\mathrm{(C)}\ 5 \qquad\mathrm{(D)}\ 6 \qquad\mathrm{(E)}\ 7</math>
 +
 +
==Problem 6==
 +
 +
Let <math>P(n)</math> and <math>S(n)</math> denote the product and the sum, respectively, of the digits
 +
of the integer <math>n</math>. For example, <math>P(23) = 6</math> and <math>S(23) = 5</math>. Suppose <math>N</math> is a
 +
two-digit number such that <math>N = P(N)+S(N)</math>. What is the units digit of <math>N</math>?
 +
 +
<math>\mathrm{(A)}\ 2 \qquad\mathrm{(B)}\ 3 \qquad\mathrm{(C)}\ 6 \qquad\mathrm{(D)}\ 8 \qquad\mathrm{(E)}\ 9</math>
  
 
=== Solutions ===
 
=== Solutions ===

Revision as of 12:18, 11 February 2009

Problem 1

The median of the list $n, n + 3, n + 4, n + 5, n + 6, n + 8, n + 10, n + 12, n + 15$ is 10. What is the mean?

$\mathrm{(A)}\ 4 \qquad\mathrm{(B)}\ 6 \qquad\mathrm{(C)}\ 7 \qquad\mathrm{(D)}\ 10 \qquad\mathrm{(E)}\ 11$

Problem 2

A number $x$ is $2$ more than the product of its reciprocal and its additive inverse. In which interval does the number lie?

$\mathrm{(A)}\ -4\leq x\leq -2 \qquad\mathrm{(B)}\ -2<x\leq 0 \qquad\mathrm{(C)}\ 0<x\leq 2$

$\mathrm{(D)}\ 2<x\leq 4 \qquad\mathrm{(E)}\ 4<x\leq 6$

Problem 3

The sum of two numbers is $S$. Suppose $3$ is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?

$\mathrm{(A)}\ 2S+3 \qquad\mathrm{(B)}\ 3S+2 \qquad\mathrm{(C)}\ 3S+6 \qquad\mathrm{(D)}\ 2S+6 \qquad\mathrm{(E)}\ 2S+12$

Problem 4

What is the maximum number for the possible points of intersection of a circle and a triangle?

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

Problem 5

How many of the twelve pentominoes pictured below have at least one line of symmetry?

$\mathrm{(A)}\ 3 \qquad\mathrm{(B)}\ 2 \qquad\mathrm{(C)}\ 5 \qquad\mathrm{(D)}\ 6 \qquad\mathrm{(E)}\ 7$

Problem 6

Let $P(n)$ and $S(n)$ denote the product and the sum, respectively, of the digits of the integer $n$. For example, $P(23) = 6$ and $S(23) = 5$. Suppose $N$ is a two-digit number such that $N = P(N)+S(N)$. What is the units digit of $N$?

$\mathrm{(A)}\ 2 \qquad\mathrm{(B)}\ 3 \qquad\mathrm{(C)}\ 6 \qquad\mathrm{(D)}\ 8 \qquad\mathrm{(E)}\ 9$

Solutions

1. The median is $n+6$, therefore $n=4$. Computation shows that the sum of all numbers is $99$ and thus the mean is $99/9=11$.

2. The reciprocal of $x$ is $\frac 1x$ and the additive inverse is $-x$. (Note that $x$ must be non-zero to have a reciprocal.) The product of these two is $\frac 1x \cdot (-x) = -1$. Thus $x$ is $2$ more than $-1$. Therefore $x=1$.

3. The original two numbers are $x$ and $y$, with $x+y=S$. The new two numbers are $2(x+3)$ and $2(y+3)$. Their sum is $2(x+3)+2(y+3)=2x+2y+12=2(x+y)+12 = 2S+12$.

4. Each side of the triangle can only intersect the circle twice, so the maximum is at most 6. This can be achieved: [asy] unitsize(0.3cm); draw( circle((0,-0.2),2.2) ); draw( (-2,-2)--(2,-2)--(0,3)--cycle ); [/asy]

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