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Difference between revisions of "2008 AIME I Problems/Problem 13"

(Made proof of final equations more rigorous)
m
Line 13: Line 13:
 
=== Solution 1 ===
 
=== Solution 1 ===
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
−
p(0,0) &= a_0 = 0\\
+
p(0,0) &= a_0 = 0 \\
−
p(1,0) &= a_0 + a_1 + a_3 + a_6 = a_1 + a_3 + a_6 = 0\\
+
p(1,0) &= a_0 + a_1 + a_3 + a_6 = a_1 + a_3 + a_6 = 0 \\
−
p(-1,0) &= -a_1 + a_3 - a_6 = 0\end{align*}</cmath>
+
p(-1,0) &= -a_1 + a_3 - a_6 = 0
 +
\end{align*}</cmath>
  
 
Adding the above two equations gives <math>a_3 = 0</math>, and so we can deduce that <math>a_6 = -a_1</math>.
 
Adding the above two equations gives <math>a_3 = 0</math>, and so we can deduce that <math>a_6 = -a_1</math>.
  
 
Similarly, plugging in <math>(0,1)</math> and <math>(0,-1)</math> gives <math>a_5 = 0</math> and <math>a_9 = -a_2</math>. Now,
 
Similarly, plugging in <math>(0,1)</math> and <math>(0,-1)</math> gives <math>a_5 = 0</math> and <math>a_9 = -a_2</math>. Now,
−
<cmath>\begin{align*}p(1,1) &= a_0 + a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7 + a_8 + a_9\\
+
<cmath>\begin{align*}
−
&= 0 + a_1 + a_2 + 0 + a_4 + 0 - a_1 + a_7 + a_8 - a_2 = a_4 + a_7 + a_8 = 0\\
+
p(1,1) &= a_0 + a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7 + a_8 + a_9 \\
−
p(1,-1) &= a_0 + a_1 - a_2 + 0 - a_4 + 0 - a_1 - a_7 + a_8 + a_2\\ &= -a_4 - a_7 + a_8 = 0\end{align*}</cmath>
+
&= 0 + a_1 + a_2 + 0 + a_4 + 0 - a_1 + a_7 + a_8 - a_2 \\
 +
&= a_4 + a_7 + a_8 \\
 +
&= 0 \\
 +
p(1,-1) &= a_0 + a_1 - a_2 + 0 - a_4 + 0 - a_1 - a_7 + a_8 + a_2 \\
 +
&= -a_4 - a_7 + a_8 \\
 +
&= 0
 +
\end{align*}</cmath>
 
Therefore <math>a_8 = 0</math> and <math>a_7 = -a_4</math>. Finally,
 
Therefore <math>a_8 = 0</math> and <math>a_7 = -a_4</math>. Finally,
−
<math>p(2,2) = 0 + 2a_1 + 2a_2 + 0 + 4a_4 + 0 - 8a_1 - 8a_4 +0 - 8a_2 = -6 a_1 - 6 a_2 - 4 a_4 = 0</math>
+
<cmath>\begin{align*}
 +
p(2,2) &= 0 + 2a_1 + 2a_2 + 0 + 4a_4 + 0 - 8a_1 - 8a_4 +0 - 8a_2 \\
 +
&= -6 a_1 - 6 a_2 - 4 a_4 \\
 +
&= 0
 +
\end{align*}</cmath>
 
So, <math>3a_1 + 3a_2 + 2a_4 = 0</math>, or equivalently <math>a_4 = -\frac{3(a_1 + a_2)}{2}</math>.
 
So, <math>3a_1 + 3a_2 + 2a_4 = 0</math>, or equivalently <math>a_4 = -\frac{3(a_1 + a_2)}{2}</math>.
  
−
Substituting these equations into the original polynomial <math>p</math>, we find that at <math>(\frac{a}{c}, \frac{b}{c})</math>,
+
Substituting these equations into the original polynomial <math>p</math>, we find that at <math>\left(\frac{a}{c}, \frac{b}{c}\right)</math>,
−
<cmath>\begin{align*}
+
<cmath>a_1x + a_2y + a_4xy - a_1x^3 - a_4x^2y - a_2y^3 = 0 \iff</cmath>
−
a_1x + a_2y + a_4xy - a_1x^3 - a_4x^2y - a_2y^3 &= 0 \iff
+
<cmath>a_1x + a_2y - \frac{3(a_1 + a_2)}{2}xy - a_1x^3 + \frac{3(a_1 + a_2)}{2}x^2y - a_2y^3 = 0 \iff</cmath>
−
a_1x + a_2y + -\frac{3(a_1 + a_2)}{2}xy - a_1x^3 + \frac{3(a_1 + a_2)}{2}x^2y - a_2y^3 &= 0 \iff
+
<cmath>a_1x(x - 1)\left(x + 1 - \frac{3}{2}y\right) + a_2y\left(y^2 - 1 - \frac{3}{2}x(x - 1)\right) = 0</cmath>.
−
a_1x(x - 1)(x + 1 - \frac{3}{2}y) + a_2y(y^2 - 1 - \frac{3}{2}x(x - 1)) &= 0
+
The remaining coefficients <math>a_1</math> and <math>a_2</math> are now completely arbitrary because the original equations impose no more restrictions on them. Hence, for the final equation to hold for all possible <math>p</math>, we must have <math>x(x - 1)\left(x + 1 - \frac{3}{2}y\right) = y\left(y^2 - 1 - \frac{3}{2}x(x - 1)\right) = 0</math>.  
−
\end{align*}</cmath>.
 
−
The remaining coefficients <math>a_1</math> and <math>a_2</math> are now completely arbitrary because the original equations impose no more restrictions on them. Hence, for the final equation to hold for all possible <math>p</math>, we must have <math>x(x - 1)(x + 1 - \frac{3}{2}y) = y(y^2 - 1 - \frac{3}{2}x(x - 1)) = 0</math>.  
 
  
−
As the answer format implies that the <math>x</math>-coordinate of the root is non-integral, <math>x(x - 1)(x + 1 - \frac{3}{2}y) = 0 \iff x + 1 - \frac{3}{2}y = 0 \iff y = \frac{2}{3}(x + 1)</math>. The format also implies that <math>y</math> is positive, so <math>y(y^2 - 1 - \frac{3}{2}x(x - 1)) = 0 \iff y^2 - 1 - \frac{3}{2}x(x - 1) = 0</math>. Substituting <math>y</math> into this equation and factoring the quadratic yields <math>(19x - 5)(x - 2) = 0</math>, in which the only non-integral root is <math>x = \frac{5}{19}</math>, so <math>y = \frac{16}{19}</math>.
+
As the answer format implies that the <math>x</math>-coordinate of the root is non-integral, <math>x(x - 1)\left(x + 1 - \frac{3}{2}y\right) = 0 \iff x + 1 - \frac{3}{2}y = 0 \iff y = \frac{2}{3}(x + 1)\ (1)</math>. The format also implies that <math>y</math> is positive, so <math>y\left(y^2 - 1 - \frac{3}{2}x(x - 1)\right) = 0 \iff y^2 - 1 - \frac{3}{2}x(x - 1) = 0\ (2)</math>. Substituting <math>(1)</math> into <math>(2)</math> and reducing to a quadratic yields <math>(19x - 5)(x - 2) = 0</math>, in which the only non-integral root is <math>x = \frac{5}{19}</math>, so <math>y = \frac{16}{19}</math>.
  
 
The answer is <math>a + b + c = \boxed{40}</math>.
 
The answer is <math>a + b + c = \boxed{40}</math>.

Revision as of 12:24, 3 January 2021

Problem

Let

\[p(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + a_6x^3 + a_7x^2y + a_8xy^2 + a_9y^3.\]

Suppose that

\[p(0,0) = p(1,0) = p( - 1,0) = p(0,1) = p(0, - 1)\\ = p(1,1) = p(1, - 1) = p(2,2) = 0.\]

There is a point $\left(\frac {a}{c},\frac {b}{c}\right)$ for which $p\left(\frac {a}{c},\frac {b}{c}\right) = 0$ for all such polynomials, where $a$, $b$, and $c$ are positive integers, $a$ and $c$ are relatively prime, and $c > 1$. Find $a + b + c$.

Solution

Solution 1

\begin{align*} p(0,0) &= a_0 = 0 \\ p(1,0) &= a_0 + a_1 + a_3 + a_6 = a_1 + a_3 + a_6 = 0 \\ p(-1,0) &= -a_1 + a_3 - a_6 = 0 \end{align*}

Adding the above two equations gives $a_3 = 0$, and so we can deduce that $a_6 = -a_1$.

Similarly, plugging in $(0,1)$ and $(0,-1)$ gives $a_5 = 0$ and $a_9 = -a_2$. Now, \begin{align*} p(1,1) &= a_0 + a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7 + a_8 + a_9 \\ &= 0 + a_1 + a_2 + 0 + a_4 + 0 - a_1 + a_7 + a_8 - a_2 \\ &= a_4 + a_7 + a_8 \\ &= 0 \\ p(1,-1) &= a_0 + a_1 - a_2 + 0 - a_4 + 0 - a_1 - a_7 + a_8 + a_2 \\ &= -a_4 - a_7 + a_8 \\ &= 0 \end{align*} Therefore $a_8 = 0$ and $a_7 = -a_4$. Finally, \begin{align*} p(2,2) &= 0 + 2a_1 + 2a_2 + 0 + 4a_4 + 0 - 8a_1 - 8a_4 +0 - 8a_2 \\ &= -6 a_1 - 6 a_2 - 4 a_4 \\ &= 0 \end{align*} So, $3a_1 + 3a_2 + 2a_4 = 0$, or equivalently $a_4 = -\frac{3(a_1 + a_2)}{2}$.

Substituting these equations into the original polynomial $p$, we find that at $\left(\frac{a}{c}, \frac{b}{c}\right)$, \[a_1x + a_2y + a_4xy - a_1x^3 - a_4x^2y - a_2y^3 = 0 \iff\] \[a_1x + a_2y - \frac{3(a_1 + a_2)}{2}xy - a_1x^3 + \frac{3(a_1 + a_2)}{2}x^2y - a_2y^3 = 0 \iff\] \[a_1x(x - 1)\left(x + 1 - \frac{3}{2}y\right) + a_2y\left(y^2 - 1 - \frac{3}{2}x(x - 1)\right) = 0\]. The remaining coefficients $a_1$ and $a_2$ are now completely arbitrary because the original equations impose no more restrictions on them. Hence, for the final equation to hold for all possible $p$, we must have $x(x - 1)\left(x + 1 - \frac{3}{2}y\right) = y\left(y^2 - 1 - \frac{3}{2}x(x - 1)\right) = 0$.

As the answer format implies that the $x$-coordinate of the root is non-integral, $x(x - 1)\left(x + 1 - \frac{3}{2}y\right) = 0 \iff x + 1 - \frac{3}{2}y = 0 \iff y = \frac{2}{3}(x + 1)\ (1)$. The format also implies that $y$ is positive, so $y\left(y^2 - 1 - \frac{3}{2}x(x - 1)\right) = 0 \iff y^2 - 1 - \frac{3}{2}x(x - 1) = 0\ (2)$. Substituting $(1)$ into $(2)$ and reducing to a quadratic yields $(19x - 5)(x - 2) = 0$, in which the only non-integral root is $x = \frac{5}{19}$, so $y = \frac{16}{19}$.

The answer is $a + b + c = \boxed{40}$.

Solution 2

Consider the cross section of $z = p(x, y)$ on the plane $z = 0$. We realize that we could construct the lines/curves in the cross section such that their equations multiply to match the form of $p(x, y)$ and they go over the eight given points. A simple way to do this would be to use the equations $x = 0$, $x = 1$, and $y = \frac{2}{3}x + \frac{2}{3}$, giving us

$p_1(x, y) = x\left(x - 1\right)\left( \frac{2}{3}x - y + \frac{2}{3}\right) = \frac{2}{3}x + xy + \frac{2}{3}x^3-x^2y$.

Another way to do this would to use the line $y = x$ and the ellipse, $x^2 + xy + y^2 = 1$. This would give

$p_2(x, y) = \left(x - y\right)\left(x^2 + xy + y^2 - 1\right) = -x + y + x^3 - y^3$.

At this point, we see that $p_1$ and $p_2$ both must have $\left(\frac{a}{c}, \frac{b}{c}\right)$ as a zero. A quick graph of the 4 lines and the ellipse used to create $p_1$ and $p_2$ gives nine intersection points. Eight of them are the given ones, and the ninth is $\left(\frac{5}{9}, \frac{16}{9}\right)$. The last intersection point can be found by finding the intersection points of $y = \frac{2}{3}x + \frac{2}{3}$ and $x^2 + xy + y^2 = 1$. Finally, just add the values of $a$, $b$, and $c$ to get $5 + 16 + 19 = \boxed{040}$

See also

2008 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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