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

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m (→‎Solution 3 (Ansatz))
 
(14 intermediate revisions by 2 users not shown)
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=== Solution 1 ===
 
=== Solution 1 ===
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
−
p(0,0) &= a_0 = 0 \\
+
p(0,0) &= a_0 \\
−
p(1,0) &= a_0 + a_1 + a_3 + a_6 = a_1 + a_3 + a_6 = 0 \\
+
&= 0 \\
−
p(-1,0) &= -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>
 
\end{align*}</cmath>
  
Line 46: Line 50:
 
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>.
 
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>5 + 16 + 19 = \boxed{40}</math>.
+
The answer is <math>5 + 16 + 19 = \boxed{040}</math>.
 +
 
 +
<asy>
 +
unitsize(1.2 cm);
 +
 
 +
real upperhyper (real x) {
 +
return(sqrt((3*x^2 - 3*x + 2)/2));
 +
}
 +
 
 +
real lowerhyper (real x) {
 +
return(-sqrt((3*x^2 - 3*x + 2)/2));
 +
}
 +
 
 +
int i;
 +
 
 +
for (i = -3; i <= 3; ++i) {
 +
draw((-3,i)--(3,i),gray(0.7));
 +
draw((i,-3)--(i,3),gray(0.7));
 +
}
 +
 
 +
draw((0,-3)--(0,3),red);
 +
draw((1,-3)--(1,3),red);
 +
draw((-3,-4/3)--(3,8/3),red);
 +
draw((-3,0)--(3,0),blue);
 +
draw(graph(upperhyper,-1.863,2.863),blue);
 +
draw(graph(lowerhyper,-1.836,2.863),blue);
 +
 
 +
dot("$(0,0)$", (0,0), NE, fontsize(8));
 +
dot("$(1,0)$", (1,0), NE, fontsize(8));
 +
dot("$(-1,0)$", (-1,0), NW, fontsize(8));
 +
dot("$(0,1)$", (0,1), SW, fontsize(8));
 +
dot("$(0,-1)$", (0,-1), NW, fontsize(8));
 +
dot("$(1,1)$", (1,1), SE, fontsize(8));
 +
dot("$(1,-1)$", (1,-1), NE, fontsize(8));
 +
dot("$(2,2)$", (2,2), SE, fontsize(8));
 +
dot((5/19,16/19), green);
 +
</asy>
  
 
=== Solution 2 ===
 
=== Solution 2 ===
−
Consider the cross section of <math>z = p(x, y)</math> on the plane <math>z = 0</math>. We realize that we could construct the lines/curves in the cross section such that their equations multiply to match the form of <math>p(x, y)</math> and they go over the eight given points. A simple way to do this would be to use the equations <math>x = 0</math>, <math>x = 1</math>, and <math>y = \frac{2}{3}x + \frac{2}{3}</math>, giving us  
+
Consider the cross section of <math>z = p(x, y)</math> on the plane <math>z = 0</math>. We realize that we could construct the lines/curves in the cross section such that their equations multiply to match the form of <math>p(x, y)</math> (same degree of <math>x</math> and <math>y</math> in terms) and they include the eight given points. One simple way to do this would be to use the equations <math>x = 0</math>, <math>x = 1</math>, and <math>y = \frac{2}{3}x + \frac{2}{3}</math>, giving us  
  
 
<math>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</math>.
 
<math>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</math>.
Line 55: Line 95:
 
Another way to do this would to use the line <math>y = x</math> and the ellipse, <math>x^2 + xy + y^2 = 1</math>. This would give
 
Another way to do this would to use the line <math>y = x</math> and the ellipse, <math>x^2 + xy + y^2 = 1</math>. This would give
  
−
<math>p_2(x, y) = \left(x - y\right)\left(x^2 + xy + y^2 - 1\right) = -x + y + x^3 - y^3</math>.
+
<math>p_2(x, y) = \left(x - y\right)\left(x^2 + xy + y^2 - 1\right) = -x + y + x^3 - y^3</math>. (But
 +
 
 +
(Another way would be to use the hyperbola from Solution 1. Interesting that different curves both work.)
  
−
At this point, we see that <math>p_1</math> and <math>p_2</math> both must have <math>\left(\frac{a}{c}, \frac{b}{c}\right)</math> as a zero. A quick graph of the 4 lines and the ellipse used to create <math>p_1</math> and <math>p_2</math> gives nine intersection points. Eight of them are the given ones, and the ninth is <math>\left(\frac{5}{9}, \frac{16}{9}\right)</math>. The last intersection point can be found by finding the intersection points of <math>y = \frac{2}{3}x + \frac{2}{3}</math> and <math>x^2 + xy + y^2 = 1</math>.
+
At this point, we consider that <math>p_1</math> and <math>p_2</math> both must have <math>\left(\frac{a}{c}, \frac{b}{c}\right)</math> as a zero. A quick graph of the 4 lines and the ellipse used to create <math>p_1</math> and <math>p_2</math> gives nine intersection points. Eight of them are the given ones, and the ninth is <math>\left(\frac{5}{9}, \frac{16}{9}\right)</math>. The last intersection point can be found by finding the intersection points of <math>y = \frac{2}{3}x + \frac{2}{3}</math> and <math>x^2 + xy + y^2 = 1</math>.
 
Finally, just add the values of <math>a</math>, <math>b</math>, and <math>c</math> to get <math>5 + 16 + 19 = \boxed{040}</math>
 
Finally, just add the values of <math>a</math>, <math>b</math>, and <math>c</math> to get <math>5 + 16 + 19 = \boxed{040}</math>
 +
 +
<asy>
 +
unitsize(1.2 cm);
 +
 +
real upperellipse(real x) {
 +
return((sqrt(4- 3*x^2 )-x)/2);
 +
}
 +
 +
real lowerellipse(real x) {
 +
return((-sqrt(4- 3*x^2 )-x)/2);
 +
}
 +
 +
int i;
 +
 +
for (i = -3; i <= 3; ++i) {
 +
draw((-3,i)--(3,i),gray(0.7));
 +
draw((i,-3)--(i,3),gray(0.7));
 +
}
 +
 +
draw((0,-3)--(0,3),red);
 +
draw((1,-3)--(1,3),red);
 +
draw((-3,-4/3)--(3,8/3),red);
 +
 +
draw((-3, -3)--(3,3),blue);
 +
draw(graph(upperellipse,-1.1547,1.1547),blue);
 +
draw(graph(lowerellipse, -1.1547,1.1547),blue);
 +
 +
dot("$(0,0)$", (0,0), NE, fontsize(8));
 +
dot("$(1,0)$", (1,0), NE, fontsize(8));
 +
dot("$(-1,0)$", (-1,0), NW, fontsize(8));
 +
dot("$(0,1)$", (0,1), SW, fontsize(8));
 +
dot("$(0,-1)$", (0,-1), NW, fontsize(8));
 +
dot("$(1,1)$", (1,1), SE, fontsize(8));
 +
dot("$(1,-1)$", (1,-1), NE, fontsize(8));
 +
dot("$(2,2)$", (2,2), SE, fontsize(8));
 +
dot((5/19,16/19), green);
 +
</asy>
 +
 +
== Solution 3 (Ansatz) ==
 +
 +
We can plug in the values to obtain
 +
 +
<cmath>p(0,0)=0\Longrightarrow a_0=0</cmath>
 +
 +
<cmath>p(1,0)=0\Longrightarrow a_1+a_3+a_6=0</cmath>
 +
 +
<cmath>p(0,1)=0\Longrightarrow a_2+a_5+a_9=0</cmath>
 +
 +
<cmath>p(-1,0)=0\Longrightarrow a_1-a_3+a_6=0</cmath>
 +
 +
<cmath>p(0,-1)=0\Longrightarrow a_2-a-5+a_9=0</cmath>
 +
 +
<cmath>p(1,1)=0\Longrightarrow a_4+a_7+a_8=0</cmath>
 +
 +
<cmath>p(1,-1)=0\Longrightarrow a_4+a_7-a_8=0</cmath>
 +
 +
<cmath>p(2,2)=0\Longrightarrow2a_4=3a_6+3a_9\Leftrightarrow2a_4+3a_1+3a_2=0.</cmath>
 +
 +
Now, this means that
 +
 +
<cmath>p(x,y)=a_1x+a_2y+a_4xy-a_1x^3+a_7x^2y+a_8xy^2-a_2y^3.</cmath>
 +
 +
After some simplifying, we obtain
 +
 +
<cmath>p(x,y)=a_1(x-x^3)+a_2(y-y^3)+a_4xy(1-x).</cmath>
 +
 +
Since <math>p(x,y)=0</math>, <math>3a_1+3a_2+2a_4=0,</math> and we suspect that:
 +
 +
<cmath>x-x^3=y-y^3</cmath>
 +
 +
and
 +
 +
<cmath>\frac{x-x^3}{xy(1-x)}=\frac{3}{2}\Leftrightarrow\frac{1+x}{y}=\frac{3}{2} \Leftrightarrow\frac{2}{3}+\frac{2x}{3}=y</cmath>.
 +
 +
Plugging this into the first equation, and factoring, and cancelling <math>(x+1)</math>, and simplifying, we get <math>19x^2 -43x+10=0</math>, so we find that <math>(x,y)=\left(\frac{5}{19},\frac{16}{19}\right)\Longrightarrow a+b+c=5+16+19=\boxed{40}.</math>
 +
 +
~~pinkpig
  
 
== See also ==
 
== See also ==

Latest revision as of 23:07, 10 April 2024

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 $5 + 16 + 19 = \boxed{040}$.

[asy] unitsize(1.2 cm); real upperhyper (real x) { return(sqrt((3*x^2 - 3*x + 2)/2)); } real lowerhyper (real x) { return(-sqrt((3*x^2 - 3*x + 2)/2)); } int i; for (i = -3; i <= 3; ++i) { draw((-3,i)--(3,i),gray(0.7)); draw((i,-3)--(i,3),gray(0.7)); } draw((0,-3)--(0,3),red); draw((1,-3)--(1,3),red); draw((-3,-4/3)--(3,8/3),red); draw((-3,0)--(3,0),blue); draw(graph(upperhyper,-1.863,2.863),blue); draw(graph(lowerhyper,-1.836,2.863),blue); dot("$(0,0)$", (0,0), NE, fontsize(8)); dot("$(1,0)$", (1,0), NE, fontsize(8)); dot("$(-1,0)$", (-1,0), NW, fontsize(8)); dot("$(0,1)$", (0,1), SW, fontsize(8)); dot("$(0,-1)$", (0,-1), NW, fontsize(8)); dot("$(1,1)$", (1,1), SE, fontsize(8)); dot("$(1,-1)$", (1,-1), NE, fontsize(8)); dot("$(2,2)$", (2,2), SE, fontsize(8)); dot((5/19,16/19), green); [/asy]

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 $p(x, y)$ (same degree of $x$ and $y$ in terms) and they include the eight given points. One 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

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

(Another way would be to use the hyperbola from Solution 1. Interesting that different curves both work.)

At this point, we consider 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 $5 + 16 + 19 = \boxed{040}$

[asy] unitsize(1.2 cm); real upperellipse(real x) { return((sqrt(4- 3*x^2 )-x)/2); } real lowerellipse(real x) { return((-sqrt(4- 3*x^2 )-x)/2); } int i; for (i = -3; i <= 3; ++i) { draw((-3,i)--(3,i),gray(0.7)); draw((i,-3)--(i,3),gray(0.7)); } draw((0,-3)--(0,3),red); draw((1,-3)--(1,3),red); draw((-3,-4/3)--(3,8/3),red); draw((-3, -3)--(3,3),blue); draw(graph(upperellipse,-1.1547,1.1547),blue); draw(graph(lowerellipse, -1.1547,1.1547),blue); dot("$(0,0)$", (0,0), NE, fontsize(8)); dot("$(1,0)$", (1,0), NE, fontsize(8)); dot("$(-1,0)$", (-1,0), NW, fontsize(8)); dot("$(0,1)$", (0,1), SW, fontsize(8)); dot("$(0,-1)$", (0,-1), NW, fontsize(8)); dot("$(1,1)$", (1,1), SE, fontsize(8)); dot("$(1,-1)$", (1,-1), NE, fontsize(8)); dot("$(2,2)$", (2,2), SE, fontsize(8)); dot((5/19,16/19), green); [/asy]

Solution 3 (Ansatz)

We can plug in the values to obtain

\[p(0,0)=0\Longrightarrow a_0=0\]

\[p(1,0)=0\Longrightarrow a_1+a_3+a_6=0\]

\[p(0,1)=0\Longrightarrow a_2+a_5+a_9=0\]

\[p(-1,0)=0\Longrightarrow a_1-a_3+a_6=0\]

\[p(0,-1)=0\Longrightarrow a_2-a-5+a_9=0\]

\[p(1,1)=0\Longrightarrow a_4+a_7+a_8=0\]

\[p(1,-1)=0\Longrightarrow a_4+a_7-a_8=0\]

\[p(2,2)=0\Longrightarrow2a_4=3a_6+3a_9\Leftrightarrow2a_4+3a_1+3a_2=0.\]

Now, this means that

\[p(x,y)=a_1x+a_2y+a_4xy-a_1x^3+a_7x^2y+a_8xy^2-a_2y^3.\]

After some simplifying, we obtain

\[p(x,y)=a_1(x-x^3)+a_2(y-y^3)+a_4xy(1-x).\]

Since $p(x,y)=0$, $3a_1+3a_2+2a_4=0,$ and we suspect that:

\[x-x^3=y-y^3\]

and

\[\frac{x-x^3}{xy(1-x)}=\frac{3}{2}\Leftrightarrow\frac{1+x}{y}=\frac{3}{2} \Leftrightarrow\frac{2}{3}+\frac{2x}{3}=y\].

Plugging this into the first equation, and factoring, and cancelling $(x+1)$, and simplifying, we get $19x^2 -43x+10=0$, so we find that $(x,y)=\left(\frac{5}{19},\frac{16}{19}\right)\Longrightarrow a+b+c=5+16+19=\boxed{40}.$

~~pinkpig

See also

2008 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
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