Difference between revisions of "2010 AIME I Problems/Problem 10"

Revision as of 19:23, 4 July 2013

Problem

Let $N$ be the number of ways to write $2010$ in the form $2010 = a_3 \cdot 10^3 + a_2 \cdot 10^2 + a_1 \cdot 10 + a_0$ , where the $a_i$ 's are integers, and $0 \le a_i \le 99$ . An example of such a representation is $1\cdot 10^3 + 3\cdot 10^2 + 67\cdot 10^1 + 40\cdot 10^0$ . Find $N$ .

Solution

If we choose $a_3$ and $a_1$ such that $(10^3)(a_3) + (10)(a_1) \leq 2010$ there is a unique choice of $a_2$ and $a_0$ that makes the equality hold. So $N$ is just the number of combinations of $a_3$ and $a_1$ we can pick. If $a_3 = 0$ or $a_3 = 1$ we can let $a_1$ be anything from $0$ to $99$ . If $a_3 = 2$ then $a_1 = 0$ or $a_1 = 1$ . Thus $N = 100 + 100 + 2 = \fbox{202}$ .

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	[[Category:Intermediate Number Theory Problems]]		[[Category:Intermediate Number Theory Problems]]
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Difference between revisions of "2010 AIME I Problems/Problem 10"

Revision as of 19:23, 4 July 2013

Problem

Solution

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