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Difference between revisions of "1984 AIME Problems"
(→‎Problem 4)
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== Problem 4 ==
 
== Problem 4 ==
 +
Let <math>S</math> be a list of positive integers - not necessarily distinct - in which the number <math>68</math> appears. The arithmetic mean of the numbers in <math>S</math> is <math>56</math>. However, if <math>68</math> is removed, the arithmetic mean of the numbers is <math>55</math>. What's the largest number that can appear in <math>S</math>?
  
 
[[1984 AIME Problems/Problem 4|Solution]]
 
[[1984 AIME Problems/Problem 4|Solution]]

Revision as of 22:40, 20 January 2007

Problem 1

Find the value of $\displaystyle a_2+a_4+a_6+a_8+\ldots+a_{98}$ if $\displaystyle a_1$, $\displaystyle a_2$, $\displaystyle a_3\ldots$ is an arithmetic progression with common difference 1, and $\displaystyle a_1+a_2+a_3+\ldots+a_{98}=137$.

Solution

Problem 2

The integer $n$ is the smallest positive multiple of $15$ such that every digit of $n$ is either $8$ or $0$. Compute $\frac{n}{15}$.

Solution

Problem 3

A point $P$ is chosen in the interior of $\triangle ABC$ such that when lines are drawn through $P$ parallel to the sides of $\triangle ABC$, the resulting smaller triangles $t_{1}$, $t_{2}$, and $t_{3}$ in the figure, have areas $4$, $9$, and $49$, respectively. Find the area of $\triangle ABC$.

Solution

Problem 4

Let $S$ be a list of positive integers - not necessarily distinct - in which the number $68$ appears. The arithmetic mean of the numbers in $S$ is $56$. However, if $68$ is removed, the arithmetic mean of the numbers is $55$. What's the largest number that can appear in $S$?

Solution

Problem 5

Solution

Problem 6

Solution

Problem 7

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

See also

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