Let $N, x$ and $y$ be positive integers such that $N = x + y,$ $2 < x < 10$ and $14 < y < 23$. If $N > 25$, then how many distinct values are possible for $N?$

We need to find all possible values of N where N = x + y, given the constraints on x and y.

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Let us first figure out what values x and y can actually take:

Since $2 < x < 10$ and x is a positive integer: x can be 3, 4, 5, 6, 7, 8, or 9

Since $14 < y < 23$ and y is a positive integer: y can be 15, 16, 17, 18, 19, 20, 21, or 22

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To find all possible values of N = x + y, we need to determine the minimum and maximum values:

Minimum value of N:

Smallest x = 3, smallest y = 15

Minimum N = 3 + 15 = 18

Maximum value of N:

Largest x = 9, largest y = 22

Maximum N = 9 + 22 = 31

So N can range from 18 to 31.

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The problem states that N > 25, so we only need to consider values from 26 to 31.

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Now we need to verify that each value from 26 to 31 can actually be achieved with valid combinations of x and y:

N = 26: x = 4, y = 22 (both within constraints)

N = 27: x = 5, y = 22 (both within constraints)

N = 28: x = 6, y = 22 (both within constraints)

N = 29: x = 7, y = 22 (both within constraints)

N = 30: x = 8, y = 22 (both within constraints)

N = 31: x = 9, y = 22 (both within constraints)

Why this works: For any N in this range, we can always find at least one valid combination. For example, if we keep y = 22 (the maximum), then x just needs to be N - 22, and this x will always fall within our allowed range of 3 to 9.

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The possible values of N are: 26, 27, 28, 29, 30, 31

That's 6 distinct values.

Answer: 6