Difference between revisions of "1995 AHSME Problems"
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== Problem 25 == | == Problem 25 == | ||
| + | A list of five positive integers has mean 12 and range 18. The mode and median are both 8. How many different values are possible for the second largest element of the list? | ||
| + | |||
| + | |||
| + | <math> \mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ 12 } </math> | ||
[[1995 AMC 12 Problems/Problem 25|Solution]] | [[1995 AMC 12 Problems/Problem 25|Solution]] | ||
| Line 104: | Line 108: | ||
== Problem 27 == | == Problem 27 == | ||
| + | Consider the triangular array of numbers with 0,1,2,3,... along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown. | ||
| + | \[ | ||
| + | \begin{tabular}{ccccccccccc} & & & & & 0 & & & & & \\ | ||
| + | & & & & 1 & & 1 & & & & \\ | ||
| + | & & & 2 & & 2 & & 2 & & & \\ | ||
| + | & & 3 & & 4 & & 4 & & 3 & & \\ | ||
| + | & 4 & & 7 & & 8 & & 7 & & 4 & \\ | ||
| + | 5 & & 11 & & 15 & & 15 & & 11 & & 5 & \end{tabular} | ||
| + | \] | ||
| + | Let <math>f(n)</math> denote the sum of the numbers in row <math>n</math>. What is the remainder when <math>f(100)</math> is divided by 100? | ||
| + | |||
| + | |||
| + | <math> \mathrm{(A) \ 12 } \qquad \mathrm{(B) \ 30 } \qquad \mathrm{(C) \ 50 } \qquad \mathrm{(D) \ 62 } \qquad \mathrm{(E) \ 74 } </math> | ||
[[1995 AMC 12 Problems/Problem 27|Solution]] | [[1995 AMC 12 Problems/Problem 27|Solution]] | ||
== Problem 28 == | == Problem 28 == | ||
| + | Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of length <math>\sqrt {a}</math> where <math>a</math> is | ||
| + | |||
| + | |||
| + | <math> \mathrm{(A) \ 144 } \qquad \mathrm{(B) \ 156 } \qquad \mathrm{(C) \ 168 } \qquad \mathrm{(D) \ 176 } \qquad \mathrm{(E) \ 184 } </math> | ||
[[1995 AMC 12 Problems/Problem 28|Solution]] | [[1995 AMC 12 Problems/Problem 28|Solution]] | ||
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== Problem 30 == | == Problem 30 == | ||
| + | A large cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is | ||
| + | |||
| + | |||
| + | <math> \mathrm{(A) \ 16 } \qquad \mathrm{(B) \ 17 } \qquad \mathrm{(C) \ 18 } \qquad \mathrm{(D) \ 19 } \qquad \mathrm{(E) \ 20 } </math> | ||
[[1995 AMC 12 Problems/Problem 30|Solution]] | [[1995 AMC 12 Problems/Problem 30|Solution]] | ||
Revision as of 17:48, 6 January 2008
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 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
- 30 Problem 30
- 31 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
Problem 23
Problem 24
Problem 25
A list of five positive integers has mean 12 and range 18. The mode and median are both 8. How many different values are possible for the second largest element of the list?
Problem 26
Problem 27
Consider the triangular array of numbers with 0,1,2,3,... along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown.
\[
\begin{tabular}{ccccccccccc} & & & & & 0 & & & & & \\
& & & & 1 & & 1 & & & & \\
& & & 2 & & 2 & & 2 & & & \\
& & 3 & & 4 & & 4 & & 3 & & \\
& 4 & & 7 & & 8 & & 7 & & 4 & \\
5 & & 11 & & 15 & & 15 & & 11 & & 5 & \end{tabular}
\]
Let 
denote the sum of the numbers in row 
. What is the remainder when 
is divided by 100?
Problem 28
Two parallel chords in a circle have lengths 10 and 14, and the distance between them is 6. The chord parallel to these chords and midway between them is of length 
where 
is
Problem 29
For how many three-element sets of positive integers 
is it true that 
?
Problem 30
A large cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is
