Categories

Arithmetic and geometric mean | AIME I, 2000 Question 6

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2000 based on Arithmetic and geometric mean with Algebra.

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2000 based on Arithmetic and geometric mean with Algebra.

Arithmetic and geometric mean with Algebra – AIME 2000

Find the number of ordered pairs (x,y) of integers is it true that $0 \lt y \lt 10^{6}$ and that the arithmetic mean of x and y is exactly 2 more than the geometric mean of x and y.

• is 107
• is 997
• is 840
• cannot be determined from the given information

Key Concepts

Algebra

Equations

Ordered pair

AIME, 2000, Question 3

Elementary Algebra by Hall and Knight

Try with Hints

First hint

given that $\frac{x+y}{2}=2+({xy})^\frac{1}{2}$ then solving we have $y^\frac{1}{2}$-$x^\frac{1}{2}$=+2 and-2

Second Hint

given that $y \gt x$ then $y^\frac{1}{2}$-$x^\frac{1}{2}$=+2 and here maximum integer value of $y^\frac{1}{2}$=$10^{3}-1$=999 whose corresponding $x^\frac{1}{2}$=997 and decreases upto $y^\frac{1}{2}$=3 whose corresponding $x^\frac{1}{2}$=1

Final Step

then number of pairs ($x^\frac{1}{2}$,$y^\frac{1}{2}$)=number of pairs of (x,y)=997.

.

Subscribe to Cheenta at Youtube

This site uses Akismet to reduce spam. Learn how your comment data is processed.