Difference between revisions of "1988 AIME Problems/Problem 8"

Revision as of 09:01, 19 June 2021

Problem

The function $f$ , defined on the set of ordered pairs of positive integers, satisfies the following properties: $f(x, x) = x,\; f(x, y) = f(y, x), {\rm \ and\ } (x+y)f(x, y) = yf(x, x+y).$ Calculate $f(14,52)$ .

Solution 1 (Algebra: Generalized)

Let $z = x+y$ . By the substitution $z=x+y,$ we rewrite the third property in terms of $x$ and $z,$ then solve for $f(x,z):$ $\begin{align*} zf(x,z-x) &= (z-x)f(x,z) \\ f(x,z) &= \frac{z}{z-x} \cdot f(x,z-x). \end{align*}$ Using the properties of $f,$ we have $\begin{align*} f(14,52) &= \frac{52}{38} \cdot f(14,38) \\ &= \frac{52}{38} \cdot \frac{38}{24} \cdot f(14,24) \\ &= \frac{52}{38} \cdot \frac{38}{24} \cdot \frac{24}{10} \cdot f(14,10)\\ &= \frac{52}{38} \cdot \frac{38}{24} \cdot \frac{24}{10} \cdot f(10,14)\\ &= \frac{52}{38} \cdot \frac{38}{24} \cdot \frac{24}{10} \cdot \frac{14}{4} \cdot f(10,4)\\ &= \frac{52}{38} \cdot \frac{38}{24} \cdot \frac{24}{10} \cdot \frac{14}{4} \cdot f(4,10)\\ &= \frac{52}{38} \cdot \frac{38}{24} \cdot \frac{24}{10} \cdot \frac{14}{4} \cdot \frac{10}{6} \cdot f(4,6)\\ &= \frac{52}{38} \cdot \frac{38}{24} \cdot \frac{24}{10} \cdot \frac{14}{4} \cdot \frac{10}{6} \cdot \frac{6}{2} \cdot f(4,2)\\ &= \frac{52}{38} \cdot \frac{38}{24} \cdot \frac{24}{10} \cdot \frac{14}{4} \cdot \frac{10}{6} \cdot \frac{6}{2} \cdot f(2,4)\\ &= \frac{52}{38} \cdot \frac{38}{24} \cdot \frac{24}{10} \cdot \frac{14}{4} \cdot \frac{10}{6} \cdot \frac{6}{2} \cdot \frac{4}{2} \cdot f(2,2)\\ &= \frac{52}{38} \cdot \frac{38}{24} \cdot \frac{24}{10} \cdot \frac{14}{4} \cdot \frac{10}{6} \cdot \frac{6}{2} \cdot \frac{4}{2} \cdot 2\\ &=\boxed{364}. \end{align*}$ ~MRENTHUSIASM (credit given to AoPS)

Solution 2 (Algebra)

Since all of the function's properties contain a recursive definition except for the first one, we know that $f(x,x) = x$ in order to obtain an integer answer. So, we have to transform $f(14,52)$ to this form by exploiting the other properties. The second one doesn't help us immediately, so we will use the third one.

Note that $f(14,52) = f(14,14 + 38) = \frac{52}{38}\cdot f(14,38).$

Repeating the process several times, $\begin{eqnarray*} f(14,52) & = & f(14,14 + 38) \\ & = & \frac{52}{38}\cdot f(14,38) \\ & = & \frac{52}{38}\cdot \frac{38}{24}\cdot f(14,14 + 24) \\ & = & \frac{52}{24}\cdot f(14,24) \\ & = & \frac{52}{10}\cdot f(10,14) \\ & = & \frac{52}{10}\cdot \frac{14}{4}\cdot f(10,4) \\ & = & \frac{91}{5}\cdot f(4,10) \\ & = & \frac{91}{3}\cdot f(4,6) \\ & = & 91\cdot f(2,4) \\ & = & 91\cdot 2 \cdot f(2,2) \\ & = & \boxed{364}. \end{eqnarray*}$

Solution 3 (Number Theory)

Notice that $f(x,y) = \mathrm{lcm}(x,y)$ satisfies all three properties:

For the first two properties, it is clear that $\mathrm{lcm}(x,x) = x$ and $\mathrm{lcm}(x,y) = \mathrm{lcm}(y,x)$ .

For the third property, using the identities $\gcd(x,y) \cdot \mathrm{lcm}(x,y) = x\cdot y$ and $\gcd(x,x+y) = \gcd(x,y)$ gives $\begin{align*} y \cdot \mathrm{lcm}(x,x+y) &= \dfrac{y \cdot x(x+y)}{\gcd(x,x+y)} \\ &= \dfrac{(x+y) \cdot xy}{\gcd(x,y)} \\ &= (x+y) \cdot \mathrm{lcm}(x,y). \end{align*}$ Hence, $f(x,y) = \mathrm{lcm}(x,y)$ is a solution to the functional equation.

Since this is an AIME problem, there is exactly one correct answer, and thus, exactly one possible value of $f(14,52)$ .

Therefore, we have $\begin{align*} f(14,52) &= \mathrm{lcm}(14,52) \\ &= \mathrm{lcm}(2 \cdot 7,2^2 \cdot 13) \\ &= 2^2 \cdot 7 \cdot 13 \\ &= \boxed{364}. \end{align*}$

@@ Line 10: / Line 10: @@
 f(x,z) &= \frac{z}{z-x} \cdot f(x,z-x).
 \end{align*}</cmath>
-Using the properties, we have
+Using the properties of <math>f,</math> we have
 <cmath>\begin{align*}
 f(14,52) &= \frac{52}{38} \cdot f(14,38) \\

1988 AIME (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
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