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.

## 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

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.

.

Categories

## Ordered pair Problem | AIME I, 1987 | Question 1

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

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

Number of ordered pair (1+1)(4+1)(9+1)(2+1)

=(2)(5)(10)(3)

=300.

Categories

## GCD and Ordered pair | AIME I, 1995 | Question 8

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1995 based on GCD and Ordered pair.

## GCD and Ordered pair – AIME I, 1995

Find number of ordered pairs of positive integers (x,y) with $y \lt x \leq 100$ are both $\frac{x}{y}$ and $\frac{x+1}{y+1}$ integers.

• is 107
• is 85
• is 840
• cannot be determined from the given information

### Key Concepts

Integers

GCD

Ordered pair

AIME I, 1995, Question 8

Elementary Number Theory by David Burton

## Try with Hints

here y|x and (y+1)|(x+1) $\Rightarrow gcd(y,x)=y, gcd(y+1,x+1)=y+1$

$\Rightarrow gcd(y,x-y)=y, gcd(y+1,x-y)=y+1$

$\Rightarrow y,y+1|(x-y) and gcd (y,y+1)=1$

$\Rightarrow y(y+1)|(x-y)$

here number of multiples of y(y+1) from 0 to 100-y $(x \leq 100)$ are

[$\frac{100-y}{y(y+1)}$]

$\Rightarrow \displaystyle\sum_{y=1}^{99}[\frac{100-y}{y(y+1)}$]=49+16+8+4+3+2+1+1+1=85.