If $x$ is a positive integer such that $(x-1)(x-3)(x-5)\cdots(x-93)<0$, how many values can $x$ take?

This one took me a while. The key is to realize that the product has 47 factors, each of the form (x - odd number). The sign changes at each odd integer from 1 to 93. Since x is positive and the product is negative, we need an odd number of negative factors. But we also have to consider the intervals between these odd integers, not just the integers themselves. I initially forgot to check the intervals and got the wrong answer. Once I did that, it made sense.

I got 23 at first because I only counted the intervals between 1 and 93, but forgot that x can be greater than 93 as well. When x > 93, all factors are positive, so product is positive, so not negative. So that's not it. Also, x=1 gives zero, so not negative. So we only consider x from 2 to 93. But careful: the number of negative factors depends on how many odd numbers are greater than x. For x between 2 and 3, say x=2, factors: (2-1)=positive, (2-3)=negative, and so on. So the number of negative factors is the number of odd integers greater than x. Since there are 47 odd integers from 1 to 93, for x between 2 and 3, the odd integers greater than x are 3,5,...,93, which is 46 numbers. So product has 46 negative factors, even, so positive. So not negative. So we need an odd number of negative factors. So we need the number of odd integers greater than x to be odd. That happens when x is between certain odds. Let's see: if x is between 1 and 3 (but not 1 or 3), then odd integers greater than x: 3,5,...,93 -> 46, even. If x between 3 and 5: odd greater than x: 5,7,...,93 -> 45, odd. So product negative. So x can be 4? Actually x must be integer, so x=4 is in (3,5). So x=4 works. Similarly, x=6,8,...,92? Wait, check: x=92 is between 91 and 93? 91 and 93 are both odd, so between them, odd greater than x: 93 only? Actually, if x=92, odd integers greater than 92: only 93, so 1 negative factor, product negative. So x=92 works. So the working x are even numbers from 4 to 92? That's 45 numbers? But choices don't have 45. So I must be off by one. Let's list intervals: (1,3): even # negatives, so positive. (3,5): odd # negatives, so negative. So x=4 works. (5,7): even # negatives? Let's check x=6: odd greater than 6: 7,9,...,93. Count: (93-7)/2 +1 = 44? Actually 7 to 93 step 2: (93-7)/2=43, plus 1 =44, even. So positive. So x=6 does NOT work. So pattern: works only in intervals (3,5), (7,9), (11,13), ... up to (91,93)? Let's see: the intervals where # negatives is odd are those with an odd number of odds greater than x. The total odds from 1 to 93 is 47. For x in (1,3), odds greater than x: 46 (even). For x in (3,5): 45 (odd). For x in (5,7): 44 (even). So it alternates. So working intervals are (3,5), (7,9), (11,13), ..., (91,93). That's from 3 to 91 step 4? Actually starting at 3, then 7, then 11, ... up to 91? 91 is 3 + 4*22 = 91, so 23 intervals? But each interval contains exactly one even integer? (3,5) contains 4; (7,9) contains 8; (11,13) contains 12; ... (91,93) contains 92. So that's 23 even integers. But wait, what about x=2? x=2 is in (1,3) but that interval gives even negatives, so positive. So not working. So total working x: 4,8,12,...,92. That's 23 numbers. But the answer choices don't have 23? Actually (C) is 23. So I think answer is 23. But I've seen some people say 22. Let's double-check: x=92 works? x=92: factors: (92-1)=91 positive, (92-3)=89 positive, ..., (92-91)=1 positive, (92-93)=-1 negative. So only one negative factor, product negative. Yes. So 23 seems correct. But wait, the product has 47 factors? Let's count: from 1 to 93 odd numbers: 1,3,5,...,93. That's (93-1)/2 +1 = 47. So yes. So 23 working x. But the choices include 22 and 23. I think 23 is correct. But why would 22 be there? Maybe if you exclude x=92? But x=92 works. So I'm leaning 23.

I think the answer is 22. Here's why: The product is negative when the number of negative factors is odd. The factors are (x-1), (x-3), ..., (x-93). For a given x, a factor (x - k) is negative if x < k. So the number of negative factors is the number of odd integers greater than x. Let that be N. We need N odd. The odd integers are 1,3,...,93. Total 47. For x in (1,3), N=46 (even). For x in (3,5), N=45 (odd). So x in (3,5) works. But x must be integer, so x=4 works. Similarly, x in (7,9) works: x=8. In general, x in (4k-1, 4k+1) for k=1,2,...? Let's see: (3,5) is k=1: 4k-1=3, 4k+1=5. (7,9) is k=2: 4k-1=7, 4k+1=9. So up to k=23? 4*23-1=91, 4*23+1=93. So k=1 to 23 gives 23 intervals. Each interval contains exactly one even integer? (3,5) contains 4; (7,9) contains 8; (11,13) contains 12; ... (91,93) contains 92. So that's 23 even integers. But wait, what about x=2? x=2 is in (1,3) which gives even N, so not working. So 23. But then why is 22 an option? Maybe because x=92 gives N=1, which is odd, so works. So 23. I think 23 is correct. But I've seen some solutions say 22 because they exclude x=92? That doesn't make sense. Unless x=92 is not allowed because x-93 is negative, but that's fine. So I think 23. But the answer key might say 22. Let's double-check: For x=92, factors: (92-1)=91, (92-3)=89, ..., (92-91)=1, (92-93)=-1. So only one negative, product negative. So works. So 23. I'm confused why 22 is there.

I think the trick is to realize that the product is negative when x is between an odd number and the next odd number, but only for certain intervals. Actually, the sign of the product changes at each odd integer. Since there are 47 factors, the sign alternates. For x < 1, all factors negative? But x positive, so x>=1. At x=1, product is zero. For x in (1,3), the number of negative factors is 46 (even), so positive. For x in (3,5), 45 (odd), so negative. So the intervals where product is negative are (3,5), (7,9), (11,13), ..., (91,93). That's 23 intervals. But x is integer, so in each interval there is exactly one even integer? Actually, (3,5) contains integer 4; (7,9) contains 8; ... (91,93) contains 92. So 23 integers. But wait, what about x=2? x=2 is in (1,3) which is positive, so no. So 23. But the answer choices have 23 and 22. I think 23 is correct. However, I recall a similar problem where the answer was 22 because they excluded the last interval? But why? Maybe because x=92 gives x-93 negative, but that's fine. So I think 23. But let's see: if x=92, product is negative, so it counts. So 23. I'll go with 23.