Okay, let's see. The problem says that positive integers a, b, and 2009 form a geometric sequence with an integer ratio, and a < b < 2009. We need to find a. Alright, let's start by recalling what a geometric sequence is. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio. So, if a, b, 2009 are in geometric sequence, then b = a * r and 2009 = b * r, where r is the common ratio. Since all terms are positive integers and the ratio is an integer, r must be a positive integer greater than 1 (since a < b < 2009, the ratio can't be 1 or less). 

So, starting with the given information, we can write:

b = a * r  
2009 = b * r = a * r^2

Therefore, 2009 must be equal to a multiplied by r squared. Since a and r are positive integers, and r is at least 2, we can think of this as a factorization problem. We need to find integers a and r such that a = 2009 / r^2, and a must also be a positive integer. Additionally, since a < b = a * r < 2009, we have constraints on the possible values of r.

First, let's factorize 2009 to understand its prime components. Breaking down 2009:

2009 ÷ 7 = 287  
287 ÷ 7 = 41  
41 is a prime number.

So, 2009 = 7^2 * 41^1. Therefore, the prime factorization is 7² × 41. 

Now, since 2009 = a * r², and both a and r are positive integers, r² must be a divisor of 2009. Let's find all possible values of r² that divide 2009.

Given that 2009 = 7² × 41, the exponents in r² must be less than or equal to the exponents in 2009 for each prime factor. Since r² is a square, the exponents in its prime factorization must be even. Let's consider the possible exponents for each prime in r².

For prime 7: exponent in 2009 is 2. So in r², exponent can be 0 or 2 (since 2 is even and the next even number is 4, which is higher than 2, so not possible).  
For prime 41: exponent in 2009 is 1. Since 1 is odd, in r², the exponent for 41 must be 0 (since we can't have an odd exponent in a square). Therefore, the only possible square divisor of 2009 is 7² × 1 = 49. 

Wait, that can't be right. Let me check again. If r² divides 2009, and 2009 is 7² × 41, then r² must be a divisor of 7² × 41. Since r² is a square, the exponents in its prime factors must be even. For each prime factor, the exponent in r² can be at most the exponent in 2009. For 7, exponent in 2009 is 2, so r² can have exponent 0 or 2. For 41, exponent in 2009 is 1, so r² can have exponent 0 (since 1 is odd, and exponents in squares must be even). Therefore, the possible square divisors are:

- 7^0 × 41^0 = 1
- 7^2 × 41^0 = 49

So, r² can be either 1 or 49. Therefore, the possible values of r are 1 and 7. However, since the ratio r must be greater than 1 (because a < b < 2009, so r can't be 1), the only possible ratio is 7. Therefore, r = 7.

Then, a = 2009 / r² = 2009 / 49. Let's compute that. 49 × 41 = 2009? Wait, 49 × 40 = 1960, plus 49 is 2009. Yes, so 49 × 41 = 2009. Therefore, a = 41.

But wait, let's check if this satisfies the conditions. If a = 41, then b = 41 × 7 = 287, and the next term would be 287 × 7 = 2009. So the sequence is 41, 287, 2009. Checking the inequalities: 41 < 287 < 2009, which is true. So that works.

But hold on, are there any other possible ratios? Earlier, I thought r² could only be 1 or 49, but maybe I missed something. Let's confirm again. The prime factors of 2009 are 7² and 41. So any square divisor must have even exponents in the primes. For 7, exponent can be 0 or 2. For 41, exponent must be 0. Therefore, the only square divisors are 1 and 49. So r can only be 1 or 7. Since r must be greater than 1, only 7 is possible. Therefore, the only possible a is 41.

Wait, but let me think again. What if r is not an integer? But the problem states the ratio is an integer. So r must be an integer. Therefore, the only possible ratio is 7, leading to a = 41.

But just to be thorough, let's check if there are any other factors. For example, if 41 is a prime factor, could there be a square factor that includes 41? But since 41 is to the first power in 2009, any square divisor must have even exponents for 41, which would require 41^2, but 2009 doesn't have 41 squared. Therefore, the square divisor can't include 41. So indeed, only 1 and 49 are possible.

Therefore, the answer is a = 41.

But wait, let me check once more. Suppose someone suggests that maybe r is 41, but then r² would be 41², which is 1681. Then a = 2009 / 1681 ≈ 1.2, which is not an integer. So that's invalid. Similarly, if r is 7, we get a = 41, which works. If r is 1, then a = 2009, but then b = 2009, which violates b < 2009. So that's invalid.

Therefore, the only valid ratio is 7, leading to a = 41. So the answer is 41.

But wait, let me check another angle. Suppose someone argues that 2009 could be expressed as a product of three terms in a geometric progression with a different ratio. For example, if the ratio is a composite number. Wait, but the ratio has to be an integer, and the terms must be integers. So as long as the ratio is an integer, and the first term is integer, then all terms are integers. So the key is that 2009 must be divisible by r squared, and r squared must be a divisor of 2009.

Since 2009 factors into 7² × 41, and r squared must divide it, the possible r squared values are 1, 7², and 41. Wait, but 41 is not a square. Hmm, but 41 is a prime, so 41 squared is 1681, which is larger than 2009. So 41 cannot be a factor here. Therefore, the only possible square divisors are 1 and 49. Therefore, r can only be 1 or 7.

Hence, the only possible a is 41. Therefore, the answer is 41.

But let me check once more. Suppose someone made a mistake in the initial steps. For example, if they thought that the ratio could be 41, but then the next term would be 2009, which is not less than 2009. So that's invalid. If the ratio is 2, then 2009 divided by 2 squared is 2009 / 4 = 502.25, which is not an integer. So that's invalid. Similarly, ratio 3: 2009 / 9 ≈ 223.222, not integer. Ratio 4: 2009 / 16 ≈ 125.5625, not integer. Ratio 5: 2009 / 25 ≈ 80.36, not integer. Ratio 6: 2009 / 36 ≈ 55.805, not integer. Ratio 7: 2009 / 49 = 41, which works. So that's correct.

Therefore, the answer is 41.
Therefore, the value of $a$ is $\boxed{41}$.
</think>