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AMC 10 USA Math Olympiad

Number Theory – AMC 10A, 2018 – Problem 10

Try this beautiful problem from AMC 10A, 2018 based on number theory. You may use sequential hints to help you solve the problem.

Try this beautiful problem from AMC 10A, 2018 based on Number theory.

Problem – Number Theory


Let’s try this problem number 10 from AMC 10A, 2018 based on Number Theory.

Suppose that the real number $x$ satisfies $\sqrt {49-x^2}$ – $\sqrt {25-x^2}$ = $3$.

What is the value of $\sqrt {49-x^2}$ + $\sqrt {25-x^2}$?

  • 8
  • $\sqrt 33 + 8$
  • 9
  • $2\sqrt10+4$
  • 12

Key Concepts


Number Theory

Real number

Square root

Check the Answer


Answer: 8

AMC 10 A – 2018 – Problem No.10

Mathematics can be fun by Perelman

Try with Hints


As a first hint we can start from here :

In order to get rid of the square roots, we multiply by the conjugate. Its value is the solution.The \(x^2\) terms cancel out.

\((\sqrt {49 – x^2} +\sqrt {25 – x^2}) (\sqrt {49 – x^2}) -(\sqrt {25 – x^2})\)

= 49 -\(x^2 – 25 + x^2\)

=24

Given that \(\sqrt {49 – x^2}) -(\sqrt {25 – x^2})\) = 3

\(\sqrt {49 – x^2} +\sqrt {25 – x^2}\) = \(\frac {24}{3}\)

= 8

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