Difference between revisions of "1984 AIME Problems/Problem 15"

Revision as of 13:41, 2 April 2020

Problem

Determine $w^2+x^2+y^2+z^2$ if

$\frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1$
$\frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1$
$\frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1$
$\frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1$

Solution 1

Rewrite the system of equations as $\frac{x^{2}}{t-1}+\frac{y^{2}}{t-3^{2}}+\frac{z^{2}}{t-5^{2}}+\frac{w^{2}}{t-7^{2}}=1.$ This equation is satisfied when $t = 4,16,36,64$ , as then the equation is equivalent to the given equations. After clearing fractions, for each of the values $t=4,16,36,64$ , we have the equation $x^2(t-9)(t-25)(t-49)+y^2(t-1)(t-25)(t-49)$ $+z^2(t-1)(t-9)(t-49)+w^2(t-1)(t-9)(t-25) = (t-1)(t-9)(t-25)(t-49)$ . We can move the expression $(t-1)(t-9)(t-25)(t-49)$ to the left hand side to obtain the difference of the polynomials: $x^2(t-9)(t-25)(t-49)+y^2(t-1)(t-25)(t-49)$ $+z^2(t-1)(t-9)(t-49)+w^2(t-1)(t-9)(t-25)$ and $(t-1)(t-9)(t-25)(t-49)$

Since the polynomials are equal at $t=4,16,36,64$ , we can express the difference of the two polynomials with a quartic polynomial that has roots at $t=4,16,36,64$ , so

$x^2(t-9)(t-25)(t-49)+y^2(t-1)(t-25)(t-49)$ $+z^2(t-1)(t-9)(t-49)+w^2(t-1)(t-9)(t-25) - (t-1)(t-9)(t-25)(t-49) = -(t-4)(t-16)(t-36)(t-64)$

Note the leading coefficient of the RHS is $-1$ because it must match the leading coefficient of the LHS, which is $-1$ .

Now we can plug in $t=1$ into the polynomial equation. Most terms drop, and we end up with

$x^2(-8)(-24)(-48)=-(-3)(-15)(-35)(-63)$

so that

$x^2=\frac{3\cdot 15\cdot 35\cdot 63}{8\cdot 24\cdot 48}=\frac{3^2\cdot 5^2\cdot 7^2}{2^{10}}$

Similarly, we can plug in $t=9,25,49$ and get

$\begin{align*} y^2&=\frac{5\cdot 7\cdot 27\cdot 55}{8\cdot 16\cdot 40}=\frac{3^3\cdot 5\cdot 7\cdot 11}{2^{10}}\\ z^2&=\frac{21\cdot 9\cdot 11\cdot 39}{24\cdot 16\cdot 24}=\frac{3^2\cdot 7\cdot 11\cdot 13}{2^{10}}\\ w^2&=\frac{45\cdot 33\cdot 13\cdot 15}{48\cdot 40\cdot 24}=\frac{3^2\cdot 5\cdot 11\cdot 13}{2^{10}}\end{align*}$

Now adding them up,

$\begin{align*}z^2+w^2&=\frac{3^2\cdot 11\cdot 13(7+5)}{2^{10}}=\frac{3^3\cdot 11\cdot 13}{2^8}\\ x^2+y^2&=\frac{3^2\cdot 5\cdot 7(5\cdot 7+3\cdot 11)}{2^{10}}=\frac{3^2\cdot 5\cdot 7\cdot 17}{2^8}\end{align*}$

with a sum of

$\frac{3^2(3\cdot 11\cdot 13+5\cdot 7\cdot 17)}{2^8}=3^2\cdot 4=\boxed{036}.$

/*Lengthy proof that any two cubic polynomials in $t$ which are equal at 4 values of $t$ are themselves equivalent: Let the two polynomials be $A(t)$ and $B(t)$ and let them be equal at $t=a,b,c,d$ . Thus we have $A(a) - B(a) = 0, A(b) - B(b) = 0, A(c) - B(c) = 0, A(d) - B(d) = 0$ . Also the polynomial $A(t) - B(t)$ is cubic, but it equals 0 at 4 values of $t$ . Thus it must be equivalent to the polynomial 0, since if it were nonzero it would necessarily be able to be factored into $(t-a)(t-b)(t-c)(t-d)($ some nonzero polynomial $)$ which would have a degree greater than or equal to 4, contradicting the statement that $A(t) - B(t)$ is cubic. Because $A(t) - B(t) = 0, A(t)$ and $B(t)$ are equivalent and must be equal for all $t$ .

Post script for the puzzled: This solution which is seemingly unnecessarily redundant in that it computes $x^2,y^2,z^2,$ and $w^2$ separately before adding them to obtain the final answer is appealing because it gives the individual values of $x^2,y^2,z^2,$ and $w^2$ which can be plugged into the given equations to check.

Solution 2

As in Solution 1, we have

$(t-1)(t-9)(t-25)(t-49)-x^2(t-9)(t-25)(t-49)-y^2(t-1)(t-25)(t-49)$ $-z^2(t-1)(t-9)(t-49)-w^2(t-1)(t-9)(t-25)$

$=(t-4)(t-16)(t-36)(t-64)$

Now the coefficient of $t^3$ on both sides must be equal. Therefore we have $1+9+25+49+x^2+y^2+z^2+w^2=4+16+36+64\implies x^2+y^2+z^2+w^2=\boxed{036}$ .

Solution 3

Let $a + b + c + d = \frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2} = 1$ , where each of $a, b, c, d$ corresponds to the term in the same position in this original first equation. Dividing the $n$ th term in each equation by the $n$ th term in the equation before it, we find that the ratio between the two is dependent only on the first number in the denominator of each term (namely $2, 4, 6,$ and $8$ ), and can be written as a product of linear factors. Computing these ratios, we can rewrite our system linearly as the following:

$a + b + c + d = 1$
$\frac{1}{5}a-\frac{5}{7}b+\frac{7}{3}c+\frac{15}{11}d=1$
$\frac{3}{35}a-\frac{5}{27}b-\frac{21}{11}c+\frac{45}{13}d=1$
$\frac{1}{21}a-\frac{1}{11}b-\frac{7}{13}c-3d=1$

Solving this painstaking system, you'll find each of $a, b, c, d$ can be written in reduced form with a denominator of $1024$ , as shown in Solution 1. Substituting these values back in and solving for $x^2, y^2, z^2,$ and $w^2$ and summing them, we get a final value of $\boxed{036}$ .

~anellipticcurveoverq

@@ Line 57: / Line 57: @@
 </div>
-Solving this painstaking system, you'll find each of <math>a, b, c, d</math> can be written in reduced form with a denominator of <math>1024</math>, as shown above. Substituting these values back in and solving for <math>x^2, y^2, z^2,</math> and <math>w^2</math> and summing them, we get a final value of <math>\boxed{036}</math>.
+Solving this painstaking system, you'll find each of <math>a, b, c, d</math> can be written in reduced form with a denominator of <math>1024</math>, as shown in Solution 1. Substituting these values back in and solving for <math>x^2, y^2, z^2,</math> and <math>w^2</math> and summing them, we get a final value of <math>\boxed{036}</math>.
 ~anellipticcurveoverq

1984 AIME (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Last Question
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