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.
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
Check the Answer
Answer: is 997.
AIME, 2000, Question 3
Elementary Algebra by Hall and Knight
Try with Hints
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
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
then number of pairs (\(x^\frac{1}{2}\),\(y^\frac{1}{2}\))=number of pairs of (x,y)=997.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1987 based on Ordered pair.
Ordered pair Problem - AIME I, 1987
An ordered pair (m,n) of non-negative integers is called simple if the additive m+n in base 10 requires no carrying. Find the number of simple ordered pairs of non-negative integers that sum to 1492.
is 107
is 300
is 840
cannot be determined from the given information
Key Concepts
Integers
Ordered pair
Algebra
Check the Answer
Answer: is 300.
AIME I, 1987, Question 1
Elementary Algebra by Hall and Knight
Try with Hints
for no carrying required
the range of possible values of any digit m is from 0 to 1492 where the value of n is fixed
