Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

m (corrections?)
m (→‎Solution: more underscores)
Line 25: Line 25:
  
 
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
−
<cmath>p(2,2) = 0 + 2a1 + 2a2 + 0 + 4a4 + 0 - 8a1 - 8a4 +0 - 8a2 = -6 a1 - 6 a2 - 4 a4 = 0</cmath>
+
<cmath>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</cmath>
−
So <math>3a1 + 3a2 + 2a4 = 0</math>.
+
So <math>3a_1 + 3a_2 + 2a_4 = 0</math>.
  
−
Now <math>p(x,y) = 0 + x a1 + y a2 + 0 + xy a4 + 0 - x^3 a1 - x^2 y a4 + 0 - y^3 a2</math> <math>= x(1-x)(1+x) a1 + y(1-y)(1+y) a2 + xy (1-x) a4</math>.
+
Now <math>p(x,y) = 0 + x a_1 + y a_2 + 0 + xy a_4 + 0 - x^3 a_1 - x^2 y a_4 + 0 - y^3 a_2</math> <math>= x(1-x)(1+x) a_1 + y(1-y)(1+y) a_2 + xy (1-x) a_4</math>.
  
 
In order for the above to be zero, we must have
 
In order for the above to be zero, we must have
 
<cmath>x(1-x)(1+x) = y(1-y)(1+y)</cmath>
 
<cmath>x(1-x)(1+x) = y(1-y)(1+y)</cmath>
 
and
 
and
−
<cmath>x(1-x)(1+x) = 3/2 xy (1-x)</cmath>.  Canceling terms on the second equation gives us
+
<cmath>x(1-x)(1+x) = 3/2 xy (1-x).</cmath> Canceling terms on the second equation gives us
−
<math>1+x = 3/2 y \Longrightarrow x = 3/2 y - 1</math>. Plugging that into the first equation and solving yieldst <math>x = 5/19, y = 16/19</math>, and <math>5+16+19 = \boxed{040}</math>.
+
<math>1+x = 3/2 y \Longrightarrow x = 3/2 y - 1</math>. Plugging that into the first equation and solving yields <math>x = 5/19, y = 16/19</math>, and <math>5+16+19 = \boxed{040}</math>.
  
 
== See also ==
 
== See also ==

Revision as of 13:49, 23 March 2008

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

\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 \[p(2,2) = 0 + 2a_1 + 2a_2 + 0 + 4a_4 + 0 - 8a_1 - 8a_4 +0 - 8a_2 = -6 a_1 - 6 a_2 - 4 a_4 = 0\] So $3a_1 + 3a_2 + 2a_4 = 0$.

Now $p(x,y) = 0 + x a_1 + y a_2 + 0 + xy a_4 + 0 - x^3 a_1 - x^2 y a_4 + 0 - y^3 a_2$ $= x(1-x)(1+x) a_1 + y(1-y)(1+y) a_2 + xy (1-x) a_4$.

In order for the above to be zero, we must have \[x(1-x)(1+x) = y(1-y)(1+y)\] and \[x(1-x)(1+x) = 3/2 xy (1-x).\] Canceling terms on the second equation gives us $1+x = 3/2 y \Longrightarrow x = 3/2 y - 1$. Plugging that into the first equation and solving yields $x = 5/19, y = 16/19$, and $5+16+19 = \boxed{040}$.

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
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2008_AIME_I_Problems/Problem_13&oldid=24270"