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Difference between revisions of "2021 AIME I Problems/Problem 15"

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==Problem==
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Let <math>S</math> be the set of positive integers <math>k</math> such that the two parabolas<cmath>y=x^2-k~~\text{and}~~x=2(y-20)^2-k</cmath>intersect in four distinct points, and these four points lie on a circle with radius at most <math>21</math>. Find the sum of the least element of <math>S</math> and the greatest element of <math>S</math>.
  
 +
==Diagram==
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Graph in Desmos: https://www.desmos.com/calculator/37hsgxbygj
 +
 +
~MRENTHUSIASM
 +
 +
== Solution 1 (Inequalities and Circles) ==
 +
Note that <math>y=x^2-k</math> is an upward-opening parabola with the vertex at <math>(0,-k),</math> and <math>x=2(y-20)^2-k</math> is a rightward-opening parabola with the vertex at <math>(-k,20).</math> We consider each condition separately:
 +
<ol style="margin-left: 1.5em;">
 +
  <li>The two parabolas intersect at four distinct points.<p></li>
 +
By a quick sketch, we have two subconditions:
 +
<ol style="margin-left: 1.5em;" type="A">
 +
  <li>The point <math>(-k,20)</math> is on or below the parabola <math>y=x^2-k.</math><p>
 +
We need <math>20\leq(-k)^2-k,</math> from which <math>k\geq5.</math> <p>
 +
Moreover, the point <math>(-k,20)</math> is on the parabola <math>y=x^2-k</math> when <math>k=5.</math> We will prove that the two parabolas intersect at four distinct points at this value of <math>k:</math><p>
 +
Substituting <math>y=x^2-5</math> into <math>x=2(y-20)^2-5,</math> we get <math>x=2\left(\left(x^2-5\right)-20\right)^2-5.</math> Expanding and rearranging give <cmath>2x^4-100x^2-x+1245=0. \hspace{20mm}(\bigstar)</cmath>
 +
By either the graphs of the parabolas or the Rational Root Theorem, we conclude that <math>x=-5</math> is a root of <math>(\bigstar).</math> So, we factor its left side: <cmath>(x+5)\left(2x^3-10x^2-50x+249\right)=0.</cmath>
 +
By either the graphs of the parabolas or Descartes' Rule of Signs, we conclude that <math>2x^3-10x^2-50x+249=0</math> has two positive roots and one negative root such that <math>x\neq-5.</math> So, <math>(\bigstar)</math> has four distinct real roots, or the two parabolas intersect at four distinct points.<p>
 +
For Subcondition A, we deduce that <math>k\geq5.</math> <p>
 +
<u>Remark for Subcondition A</u><p>
 +
Recall that if <math>1\leq k\leq 4,</math> then the point <math>(-k,20)</math> is above the parabola <math>y=x^2-k.</math> It follows that for <math>-k\leq x\leq0:</math>
 +
<ul style="list-style-type:square;">
 +
<li>The maximum value of <math>y</math> for the parabola <math>y=x^2-k</math> occurs at <math>x=-k,</math> from which <math>y=k^2-k\leq12.</math></li><p>
 +
<li>The minimum value of <math>y</math> for the parabola <math>x=2(y-20)^2-k</math> occurs at <math>x=0,</math> from which <math>y=20-\sqrt{\frac k2}>18.</math></li><p>
 +
</ul>
 +
Clearly, the parabola <math>x=2(y-20)^2-k</math> and the left half of the parabola <math>y=x^2-k</math> do not intersect. Therefore, the two parabolas do not intersect at four distinct points.
 +
</li>
 +
  <li>The point <math>(0,-k)</math> is on or below the parabola <math>x=2(y-20)^2-k.</math><p>
 +
The lower half of the parabola <math>x=2(y-20)^2-k</math> is <math>y=20-\sqrt{\frac{x+k}{2}}.</math> We need <math>-k\leq20-\sqrt{\frac k2},</math> which holds for all values of <math>k.</math><p>
 +
For Subcondition B, we deduce that <math>k</math> can be any positive integer.
 +
</li>
 +
</ol>
 +
<b>For Condition 1, we obtain <math>\boldsymbol{k\geq5}</math> by taking the intersection of Subconditions A and B.</b>
 +
  <li>The four points of intersection lie on a circle with radius at most <math>21.</math><p>
 +
For equations of circles, the coefficients of <math>x^2</math> and <math>y^2</math> must be the same. So, we add the equation <math>y=x^2-k</math> to half the equation <math>x=2(y-20)^2-k:</math> <cmath>y+\frac12x=x^2+(y-20)^2-\frac32k.</cmath>
 +
We expand, rearrange, and complete the squares:
 +
<cmath>\begin{align*}
 +
y+\frac12x&=x^2+y^2-40y+400-\frac32k \\
 +
\frac32k-400&=\left(x^2-\frac12x\right)+\left(y^2-41y\right) \\
 +
\frac32k-400+\frac{1}{16}+\frac{1681}{4}&=\left(x-\frac14\right)^2+\left(y-\frac{41}{2}\right)^2.
 +
\end{align*}</cmath>
 +
We need <math>\frac32k-400+\frac{1}{16}+\frac{1681}{4}\leq21^2,</math> from which <math>k\leq\left\lfloor\frac{6731}{24}\right\rfloor=280.</math><p>
 +
<b>For Condition 2, we obtain <math>\boldsymbol{k\leq280.}</math></b>
 +
</li>
 +
</ol>
 +
Taking the intersection of Conditions 1 and 2 produces <math>5\leq k\leq280.</math> Therefore, the answer is <math>5+280=\boxed{285}.</math>
 +
 +
~MRENTHUSIASM
 +
 +
== Solution 2 (Translations, Inequalities, Circles) ==
 +
 +
Make the translation <math>y \rightarrow y+20</math> to obtain <math>20+y=x^2-k</math> and <math>x=2y^2-k</math>. Multiply the first equation by <math>2</math> and sum, we see that <math>2(x^2+y^2)=3k+40+2y+x</math>. Completing the square gives us <math>\left(y- \frac{1}{2}\right)^2+\left(x - \frac{1}{4}\right)^2 = \frac{325+24k}{16}</math>; this explains why the two parabolas intersect at four points that lie on a circle*. For the upper bound, observe that <math>LHS \leq 21^2=441 \rightarrow 24k \leq 6731</math>, so <math>k \leq 280</math>.
 +
 +
For the lower bound, we need to ensure there are <math>4</math> intersections to begin with. (Here I'm using the un-translated coordinates.) Draw up a graph, and realize that two intersections are guaranteed, on the so called "right branch" of <math>y=x^2-k</math>. As we increase the value of <math>k</math>, two more intersections appear on the "left branch":
 +
 +
<math>k=4</math> does not work because the "leftmost" point of <math>x=2(y-20)^2-4</math> is <math>(-4,20)</math> which lies to the right of <math>\left(-\sqrt{24}, 20\right)</math>, which is on the graph <math>y=x^2-4</math>. While technically speaking this doesn't prove that there are no intersections (why?), drawing the graph should convince you that this is the case. Clearly, <math>k<4</math> does not work.
 +
 +
<math>k=5</math> does work because the two graphs intersect at <math>(-5,20)</math>, and by drawing the graph, you realize this is not a tangent point and there is in fact another intersection nearby, due to slope. Therefore, the answer is <math>5+280=\boxed{285}</math>.
 +
 +
*In general (assuming four intersections exist), when two conics intersect, if one conic can be written as <math>ax^2+by^2=f(x,y)</math> and the other as <math>cx^2+dy^2=g(x,y)</math> for polynomials <math>f</math> and <math>g</math> of degree at most <math>1</math>, whenever <math>(a,b),(c,d)</math> are linearly independent (L.I.), we can combine the two equations and then complete the square to achieve <math>(x-p)^2+(y-q)^2=r^2</math>. We can also combine these two equations to form a parabola, or a hyperbola, or an ellipse. When <math>(a,b),(c,d)</math> are not L.I., the intersection points instead lie on a line, which is a circle of radius infinity. When the two conics only have <math>3,2</math> or <math>1</math> intersection point(s), the statement that all these points lie on a circle is trivially true.
 +
 +
~Ross Gao
 +
 +
==Solution 3 (Parabola's Properties)==
 +
 +
<i><b>Claim</b></i>
 +
[[File:2021 AIME I 15a.png|500px|right]]
 +
Let the axes of two parabolas be perpendicular, their focal parameters be  <math>p_1</math> and <math>p_2</math> and the distances from the foci to the point of intersection of the axes be  <math>x_2</math> and <math>y_1</math>. Suppose that these parabolas intersect at four points.
 +
 +
Then these points lie on the circle centered at point <math>(p_2, p_1)</math> with radius <math>r = \sqrt{2(p_1^2 + p_2^2 + p_1 y_1 + p_2 x_2)}.</math>
 +
 +
<i><b>Proof</b></i>
 +
 +
Let's introduce a coordinate system with the center at the point of intersection of the axes. Let the first (red) parabola have axis <math>x = 0,</math> focal parameter  <math>p_1</math> and focus at point <math>A(0, –y_1), y_1 > 0.</math> Let second (blue) parabola have axis <math>y = 0,</math> focal parameter <math>p_2</math> and focus at point <math>B(–x_2,0), x_2  > 0.</math>
 +
Let us denote the angle between the vector connecting the focus of the first parabola and its point and the positive direction of the ordinate axis <math>2\theta,</math> its length <math>\rho_1(\theta),</math> the angle between the vector connecting the focus of the second parabola and its point and the positive direction of the abscissa axis <math>2\phi,</math> its length <math>\rho_2(\phi).</math> Then
 +
<cmath>\rho_1(\theta) = \frac{p_1}{1 - \cos(2\theta)}, \rho_2(\phi) = \frac{p_2}{1 - \cos(2\phi)}.</cmath>
 +
 +
Abscissa of the point of intersection is
 +
<cmath>\begin{align*} x =\rho_1 \sin(2\theta) = p_1\cot\theta = \rho_2 \cos (2\phi) - x_2 = \frac{p_2}{2} (\cot^2\phi - 1)- x_2,\end{align*}</cmath>
 +
<cmath>\begin{align*} x^2 = p_1^2 \cot ^2 \theta , 2 p_1\cot\theta = p_2 \cos^2 \phi - p_2 - 2x_2 .\end{align*}</cmath>
 +
Ordinate of the point of intersection is
 +
<cmath>\begin{align*} y =\rho_2 \sin 2\phi = p_2\cot\phi = \rho_1 \cos 2\theta - y_1 = \frac{p_1}{2} (\cot^2\theta - 1)- y_1,\end{align*}</cmath>
 +
<cmath>\begin{align*} y^2 = p_2^2 \cot ^2 \phi , 2 p_2\cot\phi = p_1 \cos^2 \theta - p_1 - 2y_1 .\end{align*}</cmath>
 +
The square of the distance from point of intersection to the point <math>(p_2, p_1)</math> is
 +
<cmath>\begin{align*} r^2 = (x-p_2)^2 + (y-p_1)^2 = x^2 + y^2 - 2 p_1 y - 2 p_2 x + p_1^2 + p_2^2 .\end{align*}</cmath>
 +
After simple transformations, we get <math>r^2 = 2(p_1^2 + p_2^2 + p_1 y_1 + p_2 x_2).</math>
 +
 +
Hence, any intersection point has the same distance <math>r</math> from the point  <math>(p_2, p_1).</math>
 +
 +
 +
<i><b>Solution</b></i>
 +
 +
Parameters of the parabola <math>y = x^2 – k</math> are <math>p_1 = \frac{1}{2}, y_1 = 20 + k – \frac{1}{2}.</math>
 +
 +
Parameters of the parabola <math>\frac{x}{2} = (y – 20)^2 – \frac{k}{2}</math> are <math>p_2 = \frac{1}{4}, x_2 = k – \frac{1}{4}  \implies
 +
r^2 = 20 + \frac{3k}{2}.</math>
 +
 +
If <math>r \le 21, k \le \frac{842}{3},</math> then integer  <math>k \le 280.</math>
 +
 +
The vertex of the second parabola is point  <math>(– k,20)</math>  can be on the parabola <math>y = x^2 – k</math> or below the point of the parabola with the same abscissa. So  <cmath>20 \ge (– k)^2 – k  \implies  5 \le k \le 280.</cmath> Therefore, the answer is <math>5+280=\boxed{285}</math>.
 +
 +
'''vladimir.shelomovskii@gmail.com, vvsss'''
 +
 +
==See Also==
 +
{{AIME box|year=2021|n=I|num-b=14|after=Last problem}}
 +
 +
[[Category:Intermediate Algebra Problems]]
 +
{{MAA Notice}}

Latest revision as of 01:31, 13 November 2023

Problem

Let $S$ be the set of positive integers $k$ such that the two parabolas\[y=x^2-k~~\text{and}~~x=2(y-20)^2-k\]intersect in four distinct points, and these four points lie on a circle with radius at most $21$. Find the sum of the least element of $S$ and the greatest element of $S$.

Diagram

Graph in Desmos: https://www.desmos.com/calculator/37hsgxbygj

~MRENTHUSIASM

Solution 1 (Inequalities and Circles)

Note that $y=x^2-k$ is an upward-opening parabola with the vertex at $(0,-k),$ and $x=2(y-20)^2-k$ is a rightward-opening parabola with the vertex at $(-k,20).$ We consider each condition separately:

  1. The two parabolas intersect at four distinct points.

    2. By a quick sketch, we have two subconditions:

    1. The point $(-k,20)$ is on or below the parabola $y=x^2-k.$

      We need $20\leq(-k)^2-k,$ from which $k\geq5.$

      Moreover, the point $(-k,20)$ is on the parabola $y=x^2-k$ when $k=5.$ We will prove that the two parabolas intersect at four distinct points at this value of $k:$

      Substituting $y=x^2-5$ into $x=2(y-20)^2-5,$ we get $x=2\left(\left(x^2-5\right)-20\right)^2-5.$ Expanding and rearranging give \[2x^4-100x^2-x+1245=0. \hspace{20mm}(\bigstar)\] By either the graphs of the parabolas or the Rational Root Theorem, we conclude that $x=-5$ is a root of $(\bigstar).$ So, we factor its left side: \[(x+5)\left(2x^3-10x^2-50x+249\right)=0.\] By either the graphs of the parabolas or Descartes' Rule of Signs, we conclude that $2x^3-10x^2-50x+249=0$ has two positive roots and one negative root such that $x\neq-5.$ So, $(\bigstar)$ has four distinct real roots, or the two parabolas intersect at four distinct points.

      For Subcondition A, we deduce that $k\geq5.$

      Remark for Subcondition A

      Recall that if $1\leq k\leq 4,$ then the point $(-k,20)$ is above the parabola $y=x^2-k.$ It follows that for $-k\leq x\leq0:$

      • The maximum value of $y$ for the parabola $y=x^2-k$ occurs at $x=-k,$ from which $y=k^2-k\leq12.$

      • The minimum value of $y$ for the parabola $x=2(y-20)^2-k$ occurs at $x=0,$ from which $y=20-\sqrt{\frac k2}>18.$

      Clearly, the parabola $x=2(y-20)^2-k$ and the left half of the parabola $y=x^2-k$ do not intersect. Therefore, the two parabolas do not intersect at four distinct points.

    2. The point $(0,-k)$ is on or below the parabola $x=2(y-20)^2-k.$

      The lower half of the parabola $x=2(y-20)^2-k$ is $y=20-\sqrt{\frac{x+k}{2}}.$ We need $-k\leq20-\sqrt{\frac k2},$ which holds for all values of $k.$

      For Subcondition B, we deduce that $k$ can be any positive integer.

    For Condition 1, we obtain $\boldsymbol{k\geq5}$ by taking the intersection of Subconditions A and B.

  2. The four points of intersection lie on a circle with radius at most $21.$

    For equations of circles, the coefficients of $x^2$ and $y^2$ must be the same. So, we add the equation $y=x^2-k$ to half the equation $x=2(y-20)^2-k:$ \[y+\frac12x=x^2+(y-20)^2-\frac32k.\] We expand, rearrange, and complete the squares: \begin{align*} y+\frac12x&=x^2+y^2-40y+400-\frac32k \\ \frac32k-400&=\left(x^2-\frac12x\right)+\left(y^2-41y\right) \\ \frac32k-400+\frac{1}{16}+\frac{1681}{4}&=\left(x-\frac14\right)^2+\left(y-\frac{41}{2}\right)^2. \end{align*} We need $\frac32k-400+\frac{1}{16}+\frac{1681}{4}\leq21^2,$ from which $k\leq\left\lfloor\frac{6731}{24}\right\rfloor=280.$

    For Condition 2, we obtain $\boldsymbol{k\leq280.}$

Taking the intersection of Conditions 1 and 2 produces $5\leq k\leq280.$ Therefore, the answer is $5+280=\boxed{285}.$

~MRENTHUSIASM

Solution 2 (Translations, Inequalities, Circles)

Make the translation $y \rightarrow y+20$ to obtain $20+y=x^2-k$ and $x=2y^2-k$. Multiply the first equation by $2$ and sum, we see that $2(x^2+y^2)=3k+40+2y+x$. Completing the square gives us $\left(y- \frac{1}{2}\right)^2+\left(x - \frac{1}{4}\right)^2 = \frac{325+24k}{16}$; this explains why the two parabolas intersect at four points that lie on a circle*. For the upper bound, observe that $LHS \leq 21^2=441 \rightarrow 24k \leq 6731$, so $k \leq 280$.

For the lower bound, we need to ensure there are $4$ intersections to begin with. (Here I'm using the un-translated coordinates.) Draw up a graph, and realize that two intersections are guaranteed, on the so called "right branch" of $y=x^2-k$. As we increase the value of $k$, two more intersections appear on the "left branch":

$k=4$ does not work because the "leftmost" point of $x=2(y-20)^2-4$ is $(-4,20)$ which lies to the right of $\left(-\sqrt{24}, 20\right)$, which is on the graph $y=x^2-4$. While technically speaking this doesn't prove that there are no intersections (why?), drawing the graph should convince you that this is the case. Clearly, $k<4$ does not work.

$k=5$ does work because the two graphs intersect at $(-5,20)$, and by drawing the graph, you realize this is not a tangent point and there is in fact another intersection nearby, due to slope. Therefore, the answer is $5+280=\boxed{285}$.

  • In general (assuming four intersections exist), when two conics intersect, if one conic can be written as $ax^2+by^2=f(x,y)$ and the other as $cx^2+dy^2=g(x,y)$ for polynomials $f$ and $g$ of degree at most $1$, whenever $(a,b),(c,d)$ are linearly independent (L.I.), we can combine the two equations and then complete the square to achieve $(x-p)^2+(y-q)^2=r^2$. We can also combine these two equations to form a parabola, or a hyperbola, or an ellipse. When $(a,b),(c,d)$ are not L.I., the intersection points instead lie on a line, which is a circle of radius infinity. When the two conics only have $3,2$ or $1$ intersection point(s), the statement that all these points lie on a circle is trivially true.

~Ross Gao

Solution 3 (Parabola's Properties)

Claim

2021 AIME I 15a.png

Let the axes of two parabolas be perpendicular, their focal parameters be $p_1$ and $p_2$ and the distances from the foci to the point of intersection of the axes be $x_2$ and $y_1$. Suppose that these parabolas intersect at four points.

Then these points lie on the circle centered at point $(p_2, p_1)$ with radius $r = \sqrt{2(p_1^2 + p_2^2 + p_1 y_1 + p_2 x_2)}.$

Proof

Let's introduce a coordinate system with the center at the point of intersection of the axes. Let the first (red) parabola have axis $x = 0,$ focal parameter $p_1$ and focus at point $A(0, –y_1), y_1 > 0.$ Let second (blue) parabola have axis $y = 0,$ focal parameter $p_2$ and focus at point $B(–x_2,0), x_2 > 0.$ Let us denote the angle between the vector connecting the focus of the first parabola and its point and the positive direction of the ordinate axis $2\theta,$ its length $\rho_1(\theta),$ the angle between the vector connecting the focus of the second parabola and its point and the positive direction of the abscissa axis $2\phi,$ its length $\rho_2(\phi).$ Then \[\rho_1(\theta) = \frac{p_1}{1 - \cos(2\theta)}, \rho_2(\phi) = \frac{p_2}{1 - \cos(2\phi)}.\]

Abscissa of the point of intersection is \begin{align*} x =\rho_1 \sin(2\theta) = p_1\cot\theta = \rho_2 \cos (2\phi) - x_2 = \frac{p_2}{2} (\cot^2\phi - 1)- x_2,\end{align*} \begin{align*} x^2 = p_1^2 \cot ^2 \theta , 2 p_1\cot\theta = p_2 \cos^2 \phi - p_2 - 2x_2 .\end{align*} Ordinate of the point of intersection is \begin{align*} y =\rho_2 \sin 2\phi = p_2\cot\phi = \rho_1 \cos 2\theta - y_1 = \frac{p_1}{2} (\cot^2\theta - 1)- y_1,\end{align*} \begin{align*} y^2 = p_2^2 \cot ^2 \phi , 2 p_2\cot\phi = p_1 \cos^2 \theta - p_1 - 2y_1 .\end{align*} The square of the distance from point of intersection to the point $(p_2, p_1)$ is \begin{align*} r^2 = (x-p_2)^2 + (y-p_1)^2 = x^2 + y^2 - 2 p_1 y - 2 p_2 x + p_1^2 + p_2^2 .\end{align*} After simple transformations, we get $r^2 = 2(p_1^2 + p_2^2 + p_1 y_1 + p_2 x_2).$

Hence, any intersection point has the same distance $r$ from the point $(p_2, p_1).$


Solution

Parameters of the parabola $y = x^2 – k$ are $p_1 = \frac{1}{2}, y_1 = 20 + k – \frac{1}{2}.$

Parameters of the parabola $\frac{x}{2} = (y – 20)^2 – \frac{k}{2}$ are $p_2 = \frac{1}{4}, x_2 = k – \frac{1}{4} \implies r^2 = 20 + \frac{3k}{2}.$

If $r \le 21, k \le \frac{842}{3},$ then integer $k \le 280.$

The vertex of the second parabola is point $(– k,20)$ can be on the parabola $y = x^2 – k$ or below the point of the parabola with the same abscissa. So \[20 \ge (– k)^2 – k \implies 5 \le k \le 280.\] Therefore, the answer is $5+280=\boxed{285}$.

vladimir.shelomovskii@gmail.com, vvsss

See Also

2021 AIME I (Problems • Answer Key • Resources)
Preceded by
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Last problem
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All AIME Problems and Solutions

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