Mock AIME 2 Pre 2005 Problems

Problem 1

Compute the largest integer $k$ such that $2004^k$ divides $2004!$ .

Problem 2

$x$ is a real number with the property that $x+\tfrac1x = 3$ . Let $S_m = x^m + \tfrac{1}{x^m}$ . Determine the value of $S_7$ .

Solution

Problem 3

In a box, there are $4$ green balls, $4$ blue balls, $2$ red balls, a brown ball, a white ball, and a black ball. These balls are randomly drawn out of the box one at a time (without replacement) until two of the same color have been removed. This process requires that at most $7$ balls be removed. The probability that $7$ balls are drawn can be expressed as $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m+n$ .

Solution

Problem 4

Let $S = \{5^k | k \in \textbf{Z}, 0 \le k \le 2004 \}$ . Given that $5^{2004} = 5443 \cdots 0625$ has $1401$ digits, how many elements of $S$ begin with the digit $1$ ?

Solution

Problem 5

Let $S$ be the set of integers $n > 1$ for which $\tfrac1n = 0.d_1d_2d_3d_4\ldots$ , an infinite decimal that has the property that $d_i = d_{i+12}$ for all positive integers $i$ . Given that $9901$ is prime, how many positive integers are in $S$ ? (The $d_i$ are digits.)

Solution

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Mock AIME 2 Pre 2005 Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5