1. **State the problem:** Find all positive integers $x, y, z$ such that $$x^2 + y^2 = z^2$$ and $$x + y + z = 1000.$$ 2. **Formula and rules:** This is a Pythagorean triple problem where $x, y, z$ form a right triangle. We want integer solutions satisfying both the Pythagorean theorem and the sum condition. 3. **Approach:** Use the sum equation to express $z$ as $z = 1000 - x - y$ and substitute into the Pythagorean equation: $$x^2 + y^2 = (1000 - x - y)^2.$$ 4. **Expand and simplify:** $$x^2 + y^2 = 1000^2 - 2 \cdot 1000 (x + y) + (x + y)^2$$ $$x^2 + y^2 = 1,000,000 - 2000(x + y) + x^2 + 2xy + y^2.$$ 5. **Cancel $x^2 + y^2$ on both sides:** $$0 = 1,000,000 - 2000(x + y) + 2xy.$$ 6. **Rearranged:** $$2xy = 2000(x + y) - 1,000,000.$$ 7. **Divide both sides by 2:** $$\cancel{2}xy = \cancel{2}1000(x + y) - \cancel{2}500,000$$ $$xy = 1000(x + y) - 500,000.$$ 8. **Rewrite as:** $$xy - 1000x - 1000y = -500,000.$$ 9. **Add $1,000,000$ to both sides to complete the rectangle:** $$xy - 1000x - 1000y + 1,000,000 = 500,000.$$ 10. **Factor left side:** $$(x - 1000)(y - 1000) = 500,000.$$ 11. **Find integer factor pairs of 500,000:** Since $x, y$ are positive integers, $x - 1000$ and $y - 1000$ are integer divisors of 500,000. 12. **Check pairs $(d, \frac{500,000}{d})$ where $d$ divides 500,000:** Try $d = 200$, then $x - 1000 = 200 \Rightarrow x = 1200$ (not positive since $x + y + z = 1000$ would be violated), discard. Try $d = -200$, $x - 1000 = -200 \Rightarrow x = 800$, then $y - 1000 = -2500 \Rightarrow y = -1500$ (negative, discard). Try $d = -800$, $x = 200$, $y = 375$ (since $y - 1000 = 625$), check sum: $x + y + z = 200 + 375 + z = 1000 \Rightarrow z = 425$. Check Pythagorean: $200^2 + 375^2 = 40000 + 140625 = 180625$, $425^2 = 180625$ correct. 13. **Solution:** $$(x, y, z) = (200, 375, 425)$$ or $$(375, 200, 425).$$ 14. **Final answer:** The unique Pythagorean triple with sum 1000 is $$200, 375, 425.$$