Triangle PQR has sides PQ = 983 units and PR = 893 units. If all side lengths of the triangle are positive integers, how many distinct integer values are possible for side QR so that PQR can form a valid triangle?

Introduction / Context:
This question is about the triangle inequality and counting the number of valid integer side lengths for a triangle when two sides are fixed. For any triangle with sides a, b and c, each side must be less than the sum of the other two and greater than their difference. Using these inequalities, we can derive a range of possible values for the third side and then count how many integers lie within that range. The numbers in this problem are large, but the underlying logic is straightforward.

Given Data / Assumptions:
- Triangle PQR has side PQ = 983 units. - Side PR = 893 units. - Side QR is an unknown positive integer. - PQR must form a valid non degenerate triangle. - All three sides are required to satisfy the triangle inequality.

Concept / Approach:
Let the unknown side QR be x. The triangle inequality states that for sides a, b and c, each of the following must hold: a + b > c, a + c > b and b + c > a. When a and b are fixed and c is the variable side x, these conditions reduce to a simple double inequality. For sides 983, 893 and x, we obtain |983 − 893| < x < 983 + 893. This gives a strict range of values for x. Once we know this interval, the task becomes counting the number of integer values of x that satisfy it.

Step-by-Step Solution:
Step 1: Let QR = x, where x is a positive integer. Step 2: Apply the triangle inequality: the difference of two side lengths must be less than the third side, and the sum must be greater than the third side. Step 3: Compute the absolute difference of the fixed sides: |983 − 893| = |90| = 90. Step 4: Compute the sum of the fixed sides: 983 + 893 = 1876. Step 5: The allowed range for x is 90 < x < 1876. Step 6: Since x must be an integer, the smallest possible integer value is 91 and the largest is 1875. Step 7: Count the number of integers from 91 to 1875 inclusive. Step 8: The count is 1875 − 91 + 1 = 1785. Step 9: Therefore there are 1785 distinct integer values possible for side QR.

Verification / Alternative check:
We can verify the counting by expressing it generally. For integers strictly between two numbers L and U, the count is U − L − 1. Here L = 90 and U = 1876, so the count should be 1876 − 90 − 1 = 1785, which matches the previous calculation. Additionally, a quick check of the endpoints confirms they cannot be used. If x = 90, then 983 = 893 + 90, which degenerates the triangle into a straight line. If x = 1876, then x equals the sum of the other two sides and again the points become collinear. Hence those endpoints must be excluded, reinforcing that only integers strictly between 90 and 1876 are valid.

Why Other Options Are Wrong:
Option 1876 counts all integers up to the sum of the given sides and ignores the lower bound from the triangle inequality. Option 90 incorrectly suggests that only differences of the sides matter and misinterprets the range. Option 1786 would be the result if one accidentally included one of the endpoints 91 or 1875 twice or miscalculated the interval length.

Common Pitfalls:
Some learners forget that the triangle inequality involves strict inequalities and mistakenly allow the third side to equal the sum or difference of the other two, which produces a degenerate triangle. Others miscount the integer values in the interval by either omitting one endpoint or adding one extra. It is useful to remember the formula for counting integers in an inclusive range and to check extreme cases to ensure they produce valid triangles.

Final Answer:
The number of possible integer values for side QR is 1785.