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

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:

The two parabolas intersect at four distinct points.

By a quick sketch, we have two subconditions:

The point is on or below the parabola
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.
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.

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

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

@@ Line 3: / Line 3: @@
 ==Diagram==
-Graph in Desmos: https://www.desmos.com/calculator/gz8igmkykn
+Graph in Desmos: https://www.desmos.com/calculator/37hsgxbygj
 ~MRENTHUSIASM
@@ Line 21: / Line 21: @@
 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 in the interval <math>[-k,0]:</math>
+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>
-Together, we conclude that the parabola <math>x=2(y-20)^2-k</math> does not intersect the left half of the parabola <math>y=x^2-k.</math> Therefore, the two parabolas do not intersect at four points.
+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>
@@ Line 62: / Line 62: @@
 *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
+~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==

2021 AIME I (Problems • Answer Key • Resources)
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