2003 AIME II problem 10 | AIME | Math for fun and glory | Khan Academy

TL;DR

Find the maximum sum of two positive integers that differ by 60, where the sum of their square roots is the square root of an integer that is not a perfect square.

Transcript

Two positive integers differ by 60. The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers? So the second sentence is a little confusing, but let's take a step-by-step. So two positive integers differ by 60. So let's say they're a and b. So we have a and b. T... Read More

Key Insights

• 😃 The positive integers a and b differ by 60.
• ❎ The sum of the square roots of a and b equals the square root of an integer that is not a perfect square.
• ❎ The square root of the product of a and b must be an integer.
• ❎ The product of a and b must be a perfect square.
• #️⃣ Both a and b must be even numbers.
• 🧑‍🏭 The sum of two even factors of a product is also even.

Questions & Answers

Q: What are the given constraints for finding the maximum sum of two integers?

The positive integers a and b must differ by 60, and the sum of their square roots must equal the square root of an integer that is not a perfect square.

Q: Why does the square root of the product of a and b have to be an integer?

The square root of the product of a and b must be an integer because a, b, and c (the integer that is not a perfect square) are all integers. If integers plus an integer equal another integer, the sum of the integers in the middle must be an integer.

Q: How does the square root of ab being an integer contradict the given constraints?

If the square root of ab is an integer, then ab would be a perfect square, which is not allowed according to the given constraints. The square root of ab must not simplify any further.

Q: How can we find the maximum possible sum of a and b?

To find the maximum sum, we need to maximize b. By trying different factors of 900 that meet the constraints, we can gradually increase b and find the corresponding a value that maximizes the sum.

Summary & Key Takeaways

• Two positive integers, a and b, differ by 60.

• The sum of the square roots of a and b is the square root of an integer that is not a perfect square.

• To maximize the sum of a and b, we need to maximize b and find an a that meets the given constraints.