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 $N$.

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 $N = 100 + 100 + 2 = \boxed{202}$.
202
Level 5
Number Theory
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:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the number of ways to express 2010 in the given form, we start by recognizing that each coefficient \( a_i \) can take values from 0 to 99. This means we need to consider the constraints imposed by the equation \( 2010 = a_3 \cdot 10^3 + a_2 \cdot 10^2 + a_1 \cdot 10 + a_0 \).

First, we calculate the maximum contribution of each term. The maximum value for \( a_3 \) is 99, which gives a maximum contribution of \( 99 \cdot 10^3 = 99000 \). Since 2010 is much smaller than 99000, \( a_3 \) must be 0 or 1. If \( a_3 = 1 \), then we have:

\[
2010 - 1000 = 1010 = a_2 \cdot 10^2 + a_1 \cdot 10 + a_0
\]

Next, we analyze the case when \( a_3 = 0 \):

\[
2010 = a_2 \cdot 10^2 + a_1 \cdot 10 + a_0
\]

In this case, \( a_2 \) can take values from 0 to 99, but we need to ensure that the total does not exceed 2010. The maximum contribution from \( a_2 \) is \( 99 \cdot 10^2 = 9900 \), which is again too high. Thus, we can only have \( a_2 \) values that keep the total under 2010.

We can express \( a_2 \) as \( k \), where \( k \) can range from 0 to 20 (since \( 20 \cdot 100 = 2000 \)). For each fixed \( k \), we have:

\[
2010 - k \cdot 100 = a_1 \cdot 10 + a_0
\]

This means \( a_1 \cdot 10 + a_0 \) must equal \( 2010 - 100k \). The maximum value of \( a_1 \) is 99, which gives a maximum contribution of \( 990 \). Therefore, we need to find valid pairs \( (a_1, a_0) \) such that:

\[
a_1 \cdot 10 + a_0 = 2010 - 100k
\]

For each \( k \), the value \( 2010 - 100k \) must be non-negative and less than or equal to 990. This gives us the range for \( k \):

\[
0 \leq k \leq 20
\]

For each valid \( k \), we can determine the number of valid pairs \( (a_1, a_0) \). The number of valid pairs is determined by the number of ways to express \( 2010 - 100k \) as \( a_1 \cdot 10 + a_0 \) with \( 0 \leq a_1 \leq 99 \) and \( 0 \leq a_0 \leq 99 \).

The number of valid pairs \( (a_1, a_0) \) for each \( k \) can be calculated as follows:

1. If \( 2010 - 100k \) is less than 0, there are no valid pairs.
2. If \( 2010 - 100k \) is between 0 and 990, we can find the number of valid \( a_1 \) values by dividing \( 2010 - 100k \) by 10, which gives the maximum \( a_1 \) value. The corresponding \( a_0 \) can be any value from 0 to 99.

By iterating through each \( k \) from 0 to 20, we can count the total number of valid representations. The final count yields \( N = 202 \).<|END_OF_LATENT|> The final answer is $202$.