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Difference between revisions of "2008 AMC 10B Problems"

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A basketball player made 5 baskets during a game. Each basket was worth either 2 or 3 points. How many different numbers could represent the total points scored by the player?
 
A basketball player made 5 baskets during a game. Each basket was worth either 2 or 3 points. How many different numbers could represent the total points scored by the player?
  
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<math>\textbf{(A)} 2 \qquad \textbf{(B)} 3 \qquad \textbf{(C)} 4 \qquad \textbf{(D)} 5 \qquad \textbf{(E)} 6</math>
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<math>\mathrm{(A)}\ 2\qquad\mathrm{(B)}\ 3\qquad\mathrm{(C)}\ 4\qquad\mathrm{(D)}\ 5\qquad\mathrm{(E)}\ 6</math>
 +
 
 +
[[2008 AMC 10B Problems/Problem 1|Solution]]
  
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([[2008 AMC 10B Problems/Problem 1|Solution]])
 
 
==Problem 2==
 
==Problem 2==
 
A <math>4\times 4</math> block of calendar dates has the numbers <math>1</math> through <math>4</math> in the first row, <math>8</math> though <math>11</math> in the second, <math>15</math> though <math>18</math> in the third, and <math>22</math> through <math>25</math> in the fourth. The order of the numbers in the second and the fourth rows are reversed. The numbers on each diagonal are added. What will be the positive difference between the diagonal sums?
 
A <math>4\times 4</math> block of calendar dates has the numbers <math>1</math> through <math>4</math> in the first row, <math>8</math> though <math>11</math> in the second, <math>15</math> though <math>18</math> in the third, and <math>22</math> through <math>25</math> in the fourth. The order of the numbers in the second and the fourth rows are reversed. The numbers on each diagonal are added. What will be the positive difference between the diagonal sums?
  
−
<math>\textbf{(A)} 2 \qquad \textbf{(B)} 4 \qquad \textbf{(C)} 6 \qquad \textbf{(D)} 8 \qquad \textbf{(E)} 10</math>
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<math>\mathrm{(A)}\ 2\qquad\mathrm{(B)}\ 4\qquad\mathrm{(C)}\ 6\qquad\mathrm{(D)}\ 8\qquad\mathrm{(E)}\ 10</math>
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 +
[[2008 AMC 10B Problems/Problem 2|Solution]]
  
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([[2008 AMC 10B Problems/Problem 2|Solution]])
 
 
==Problem 3==
 
==Problem 3==
 
Assume that <math>x</math> is a [[positive]] [[real number]]. Which is equivalent to <math>\sqrt[3]{x\sqrt{x}}</math>?
 
Assume that <math>x</math> is a [[positive]] [[real number]]. Which is equivalent to <math>\sqrt[3]{x\sqrt{x}}</math>?
  
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<math>\textbf{(A)} x^{1/6} \qquad \textbf{(B)} x^{1/4} \qquad \textbf{(C)} </math>x^{3/8} \qquad \textbf{(D)} x^{1/2} \qquad \textbf{(E)} x<math>
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<math>\mathrm{(A)}\ x^{1/6}\qquad\mathrm{(B)}\ x^{1/4}\qquad\mathrm{(C)}\ x^{3/8}\qquad\mathrm{(D)}\ x^{1/2}\qquad\mathrm{(E)}\ x</math>
  
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([[2008 AMC 10B Problems/Problem 3|Solution]])
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[[2008 AMC 10B Problems/Problem 3|Solution]]
  
 
==Problem 4==
 
==Problem 4==
 
A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least <dollar/>15,000 and that the total of all players' salaries for each team cannot exceed <dollar/>700,000. What is the maximum possible salary, in dollars, for a single player?
 
A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least <dollar/>15,000 and that the total of all players' salaries for each team cannot exceed <dollar/>700,000. What is the maximum possible salary, in dollars, for a single player?
  
−
</math>\textbf{(A)} 270,000 \qquad \textbf{(B)} 385,000 \qquad \textbf{(C)} 400,000 \qquad \textbf{(D)} 430,000 \qquad \textbf{(E)} 700,000<math>
+
<math>\mathrm{(A)}\ 270,000\qquad\mathrm{(B)}\ 385,000\qquad\mathrm{(C)}\ 400,000\qquad\mathrm{(D)}\ 430,000\qquad\mathrm{(E)}\ 700,000</math>
−
([[2008 AMC 10B Problems/Problem 4|Solution]])
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 +
[[2008 AMC 10B Problems/Problem 4|Solution]]
 +
 
 
==Problem 5==
 
==Problem 5==
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For [[real number]]s </math>a<math> and </math>b<math>, define </math>a\<math> b=(a-b)^2</math>. What is <math>(x-y)^2\</math> (y-x)^2<math>?
+
For [[real number]]s <math>a</math> and <math>b</math>, define <math>a\</math> b=(a-b)^2<math>. What is </math>(x-y)^2\<math> (y-x)^2</math>?
  
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</math>\textbf{(A)} 0 \qquad \textbf{(B)} x^2+y^2 \qquad \textbf{(C)} 2x^2 \qquad \textbf{(D)} 2y^2 \qquad \textbf{(E)} 4xy$
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<math>\mathrm{(A)}\ 0\qquad\mathrm{(B)}\ x^2+y^2\qquad\mathrm{(C)}\ 2x^2\qquad\mathrm{(D)}\ 2y^2\qquad\mathrm{(E)}\ 4xy</math>
 +
 
 +
[[2008 AMC 10B Problems/Problem 5|Solution]]
  
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([[2008 AMC 10B Problems/Problem 5|Solution]])
 
 
==Problem 6==
 
==Problem 6==
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{{problem}}
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[[2008 AMC 10B Problems/Problem 6|Solution]]
  
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([[2008 AMC 10B Problems/Problem 6|Solution]])
 
 
==Problem 7==
 
==Problem 7==
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{{problem}}
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[[2008 AMC 10B Problems/Problem 7|Solution]]
  
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([[2008 AMC 10B Problems/Problem 7|Solution]])
 
 
==Problem 8==
 
==Problem 8==
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{{problem}}
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[[2008 AMC 10B Problems/Problem 8|Solution]]
  
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([[2008 AMC 10B Problems/Problem 8|Solution]])
 
 
==Problem 9==
 
==Problem 9==
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{{problem}}
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[[2008 AMC 10B Problems/Problem 9|Solution]]
  
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([[2008 AMC 10B Problems/Problem 9|Solution]])
 
 
==Problem 10==
 
==Problem 10==
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{{problem}}
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[[2008 AMC 10B Problems/Problem 10|Solution]]
  
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([[2008 AMC 10B Problems/Problem 10|Solution]])
 
 
==Problem 11==
 
==Problem 11==
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{{problem}}
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[[2008 AMC 10B Problems/Problem 11|Solution]]
  
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([[2008 AMC 10B Problems/Problem 11|Solution]])
 
 
==Problem 12==
 
==Problem 12==
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{{problem}}
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[[2008 AMC 10B Problems/Problem 12|Solution]]
  
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([[2008 AMC 10B Problems/Problem 12|Solution]])
 
 
==Problem 13==
 
==Problem 13==
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{{problem}}
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[[2008 AMC 10B Problems/Problem 13|Solution]]
  
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([[2008 AMC 10B Problems/Problem 13|Solution]])
 
 
==Problem 14==
 
==Problem 14==
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{{problem}}
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[[2008 AMC 10B Problems/Problem 14|Solution]]
  
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([[2008 AMC 10B Problems/Problem 14|Solution]])
 
 
==Problem 15==
 
==Problem 15==
  
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([[2008 AMC 10B Problems/Problem 15|Solution]])
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[[2008 AMC 10B Problems/Problem 15|Solution]]
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==Problem 16==
 
==Problem 16==
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{{problem}}
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[[2008 AMC 10B Problems/Problem 16|Solution]]
  
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([[2008 AMC 10B Problems/Problem 16|Solution]])
 
 
==Problem 17==
 
==Problem 17==
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{{problem}}
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[[2008 AMC 10B Problems/Problem 17|Solution]]
  
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([[2008 AMC 10B Problems/Problem 17|Solution]])
 
 
==Problem 18==
 
==Problem 18==
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{{problem}}
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[[2008 AMC 10B Problems/Problem 18|Solution]]
  
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([[2008 AMC 10B Problems/Problem 18|Solution]])
 
 
==Problem 19==
 
==Problem 19==
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{{problem}}
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[[2008 AMC 10B Problems/Problem 19|Solution]]
  
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([[2008 AMC 10B Problems/Problem 19|Solution]])
 
 
==Problem 20==
 
==Problem 20==
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{{problem}}
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[[2008 AMC 10B Problems/Problem 20|Solution]]
  
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([[2008 AMC 10B Problems/Problem 20|Solution]])
 
 
==Problem 21==
 
==Problem 21==
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{{problem}}
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[[2008 AMC 10B Problems/Problem 21|Solution]]
  
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([[2008 AMC 10B Problems/Problem 21|Solution]])
 
 
==Problem 22==
 
==Problem 22==
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{{problem}}
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[[2008 AMC 10B Problems/Problem 22|Solution]]
  
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([[2008 AMC 10B Problems/Problem 22|Solution]])
 
 
==Problem 23==
 
==Problem 23==
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{{problem}}
  
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([[2008 AMC 10B Problems/Problem 23|Solution]])
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[[2008 AMC 10B Problems/Problem 23|Solution]]
 
==Problem 24==
 
==Problem 24==
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{{problem}}
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[[2008 AMC 10B Problems/Problem 24|Solution]]
  
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([[2008 AMC 10B Problems/Problem 24|Solution]])
 
 
==Problem 25==
 
==Problem 25==
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{{problem}}
  
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([[2008 AMC 10B Problems/Problem 25|Solution]])
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[[2008 AMC 10B Problems/Problem 25|Solution]]
  
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{{empty}}
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==See also==
 +
* [[AMC 10]]
 +
* [[AMC 10 Problems and Solutions]]
 +
* [[2008 AMC 10B]]
 +
* [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=219 2008 AMC B Math Jam Transcript]
 +
* [[Mathematics competition resources]]

Revision as of 01:28, 25 April 2008

Problem 1

A basketball player made 5 baskets during a game. Each basket was worth either 2 or 3 points. How many different numbers could represent the total points scored by the player?

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

Solution

Problem 2

A $4\times 4$ block of calendar dates has the numbers $1$ through $4$ in the first row, $8$ though $11$ in the second, $15$ though $18$ in the third, and $22$ through $25$ in the fourth. The order of the numbers in the second and the fourth rows are reversed. The numbers on each diagonal are added. What will be the positive difference between the diagonal sums?

$\mathrm{(A)}\ 2\qquad\mathrm{(B)}\ 4\qquad\mathrm{(C)}\ 6\qquad\mathrm{(D)}\ 8\qquad\mathrm{(E)}\ 10$

Solution

Problem 3

Assume that $x$ is a positive real number. Which is equivalent to $\sqrt[3]{x\sqrt{x}}$?

$\mathrm{(A)}\ x^{1/6}\qquad\mathrm{(B)}\ x^{1/4}\qquad\mathrm{(C)}\ x^{3/8}\qquad\mathrm{(D)}\ x^{1/2}\qquad\mathrm{(E)}\ x$

Solution

Problem 4

A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least <dollar/>15,000 and that the total of all players' salaries for each team cannot exceed <dollar/>700,000. What is the maximum possible salary, in dollars, for a single player?

$\mathrm{(A)}\ 270,000\qquad\mathrm{(B)}\ 385,000\qquad\mathrm{(C)}\ 400,000\qquad\mathrm{(D)}\ 430,000\qquad\mathrm{(E)}\ 700,000$

Solution

Problem 5

For real numbers $a$ and $b$, define $a$ b=(a-b)^2$. What is$(x-y)^2\$(y-x)^2$?

$\mathrm{(A)}\ 0\qquad\mathrm{(B)}\ x^2+y^2\qquad\mathrm{(C)}\ 2x^2\qquad\mathrm{(D)}\ 2y^2\qquad\mathrm{(E)}\ 4xy$

Solution

Problem 6

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Problem 7

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Problem 8

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Problem 9

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Problem 10

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Problem 11

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Problem 12

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Problem 13

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Problem 14

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Problem 15

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Problem 16

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Problem 17

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Problem 18

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Problem 19

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Problem 21

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Problem 22

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Problem 23

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Problem 24

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Problem 25

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See also

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