2005 AMC 12A Problems: Difference between revisions
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== Problem 22 == | == Problem 22 == | ||
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 $r$? | |||
\[ | |||
\text{(A) } 8 \qquad \text{(B) } 10 \qquad \text{(C) } 12 \qquad \text{(D) } 14 \qquad \text{(E) } 16 | |||
\] | |||
[[2005 AMC 12A Problems/Problem 22|Solution]] | [[2005 AMC 12A Problems/Problem 22|Solution]] | ||
== Problem 23 == | == Problem 23 == | ||
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? | |||
\[ | |||
\text {(A) } \frac{2}{25} \qquad \text {(B) } \frac{31}{300} \qquad \text {(C) } \frac{13}{100} \qquad \text {(D) } \frac{7}{50} \qquad \text {(E) } \frac{1}{2} | |||
\] | |||
[[2005 AMC 12A Problems/Problem 23|Solution]] | [[2005 AMC 12A Problems/Problem 23|Solution]] | ||
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== Problem 25 == | == Problem 25 == | ||
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 $S$? | |||
\[ | |||
\text {(A) } 72 \qquad \text {(B) } 76 \qquad \text {(C) } 80 \qquad \text {(D) } 84 \qquad \text {(E) } 88 | |||
\] | |||
[[2005 AMC 12A Problems/Problem 25|Solution]] | [[2005 AMC 12A Problems/Problem 25|Solution]] | ||
Revision as of 19:09, 19 September 2007
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
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 $r$? \[ \text{(A) } 8 \qquad \text{(B) } 10 \qquad \text{(C) } 12 \qquad \text{(D) } 14 \qquad \text{(E) } 16 \] Solution
Problem 23
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? \[ \text {(A) } \frac{2}{25} \qquad \text {(B) } \frac{31}{300} \qquad \text {(C) } \frac{13}{100} \qquad \text {(D) } \frac{7}{50} \qquad \text {(E) } \frac{1}{2} \] Solution
Problem 24
Problem 25
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 $S$? \[ \text {(A) } 72 \qquad \text {(B) } 76 \qquad \text {(C) } 80 \qquad \text {(D) } 84 \qquad \text {(E) } 88 \] Solution
