Difference between revisions of "2000 AIME I Problems/Problem 6"

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Problem

For how many ordered pairs $(x,y)$ of integers is it true that $0 < x < y < 10^{6}$ and that the arithmetic mean of $x$ and $y$ is exactly $2$ more than the geometric mean of $x$ and $y$ ?

Solution

Solution 1

$\begin{eqnarray*} \frac{x+y}{2} &=& \sqrt{xy} + 2\\ x+y-4 &=& 2\sqrt{xy}\\ y - 2\sqrt{xy} + x &=& 4\\ \sqrt{y} - \sqrt{x} &=& \pm 2\end{eqnarray*}$

Because $y > x$ , we only consider $+2$ .

For simplicity, we can count how many valid pairs of $(\sqrt{x},\sqrt{y})$ that satisfy our equation.

The maximum that $\sqrt{y}$ can be is $10^3 - 1 = 999$ because $\sqrt{y}$ must be an integer (this is because $\sqrt{y} - \sqrt{x} = 2$ , an integer). Then $\sqrt{x} = 997$ , and we continue this downward until $\sqrt{y} = 3$ , in which case $\sqrt{x} = 1$ . The number of pairs of $(\sqrt{x},\sqrt{y})$ , and so $(x,y)$ is then $\boxed{997}$ .

Solution 2

Let $a^2$ = $x$ and $b^2$ = $y$

Then $\frac{a^2 + b^2}{2} = \sqrt{{a^2}{b^2}} +2$ $a^2 + b^2 = 2ab + 4$ $(a-b)^2 = 4$ $(a-b) = \pm 2$

This makes counting a lot easier since now we just have to find all pairs $(a,b)$ that differ by 2.

Because $\sqrt{10^6} = 10^3$ , then we can use all positive integers less than 1000 for $a$ and $b$ .

Without loss of generality, let's say $a < b$ .

We can count even and odd pairs separately to make things easier*:

Odd: $(1,3) , (3,5) , (5,7) . . . (997,999)$

Even: $(2,4) , (4,6) , (6,8) . . . (996,998)$

This makes 499 odd pairs and 498 even pairs, for a total of $\boxed{997}$ pairs.

$*$ Note: We are counting the pairs for the values of $a$ and $b$ , which, when squared, translate to the pairs of $(x,y)$ we are trying to find.

Solution 3(2-Liner)

Our equation is $x+y-4=2\sqrt{xy} \implies \sqrt{y}-\sqrt{x}=2$ since $y>x$ . As a result $y$ must be a perfect square and cannot be $10^6, 4, 1$ so the answer is $\boxed{997}$ .

@@ Line 53: / Line 53: @@
 <math>*</math>Note: We are counting the pairs for the values of <math>a</math> and <math>b</math>, which, when squared, translate to the pairs of <math>(x,y)</math> we are trying to find.
 ===Solution 3(2-Liner)===
-Our equation is <math>x+y-4=2\sqrt{xy} \implies </math>\sqrt{y}-\sqrt{x}=2<math> since </math>y>x<math>. As a result </math>y<math> must be a perfect square and cannot be </math>10^6, 4, 1<math> so the answer is </math>\boxed{997}$.
+Our equation is <math>x+y-4=2\sqrt{xy} \implies \sqrt{y}-\sqrt{x}=2</math> since <math>y>x</math>. As a result <math>y</math> must be a perfect square and cannot be <math>10^6, 4, 1</math> so the answer is <math>\boxed{997}</math>.
 == See also ==

2000 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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