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This problem trains for: SAT-I, AMC-8, GMAT, AMC-10.

The sum of 35 positive integers is 687. Which of the following statements are always true?

The arithmetic mean is 19 because it is the sum of the numbers divided by their count. This is easily verified as 19. Therefore, I is true.

Since the sum is odd, there must be at least one odd number in the set of numbers. If by removing a number we happen to remove an odd one, the sum will become even, which would contradict the statement. Therefore, II is false.

Assume that no two numbers are equal in this sequence. Assume that they are consecutive numbers starting at 1. In this case, their sum is given by the formula: Since the above sum is smaller than 687 then the numbers cannot be consecutive starting at 1. But this would be the most favorable case of making them all different, since there are no integer numbers between two consecutive integers (remember they are all positive). By the Pigeonhole principle, there must be at least two numbers in the set that are equal . Therefore, III is true.

By removing two numbers smaller than the average, the average will increase. Therefore, IV is false.

If the numbers are all equal and they must sum up to 687, then each of them must be equal to 19 and this is also the mode of the sequence, since it is the most frequently occurring number. Therefore, V is true.