Difference between revisions of "2005 AMC 12A Problems"
m |
Archimedes1 (talk | contribs) (22,23,25) |
||
| Line 84: | Line 84: | ||
== 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]] | ||
| Line 96: | Line 102: | ||
== 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 20:09, 19 September 2007
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also
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
