Three of the five positive integers p, q, r, s, t are even and two of them are odd (not necessarily in order). Consider the following: 1. p + q + r - s - t is definitely even. 2. 2p + q + 2r - 2s + t is definitely odd. Which of the above statements is/are correct?

To solve this, we look at the properties of even and odd numbers.

Statement 1: p plus q plus r minus s minus t. In any combination of addition or subtraction, the result depends on the number of odd integers. We are given that there are exactly two odd integers and three even integers. Since the sum or difference of an even number of odd integers is always even, and even integers do not change the parity, the result will always be even regardless of which variables are odd. Therefore, Statement 1 is definitely even and correct.

Statement 2: 2p plus q plus 2r minus 2s plus t. In this expression, the terms 2p, 2r, and 2s will always be even because any integer multiplied by 2 is even. The parity of the entire expression depends only on q plus t. Since we do not know which variables are odd or even, q and t could both be even, both be odd, or one of each. If q and t are both odd, their sum is even. If q and t are both even, their sum is even. If one is odd and one is even, their sum is odd. Since the result could be either even or odd, Statement 2 is not definitely odd and is incorrect.

Therefore, only Statement 1 is correct. The answer is A.