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AIME I Algebra Arithmetic Math Olympiad USA Math Olympiad

Odd and Even integers | AIME I, 1997 | Question 1

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1997 based on Odd and Even integers.

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1997 based on Odd and Even integers.

Odd and Even Integers – AIME I, 1997


Find the number of integers between 1 and 1000 that can be expressed as the difference of squares of two non-negative integers.

  • is 107
  • is 750
  • is 840
  • cannot be determined from the given information

Key Concepts


Integers

Divisibility

Difference of squares

Check the Answer


But try the problem first…

Answer: is 750.

Source
Suggested Reading

AIME I, 1997, Question 1

Elementary Number Theory by David Burton

Try with Hints


First hint

Let x be a non-negetive integer \((x+1)^{2}-x^{2}=2x+1\)

Second Hint

Let y be a non-negetive integer \((y+1)^{2}-(y-1)^{2}=4y\)

Final Step

Numbers 2(mod 4) cannot be obtained as difference of squares then number of such numbers =500+250=750.

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