Difference between revisions of "2021 Fall AMC 12A Problems/Problem 23"
MRENTHUSIASM (talk | contribs) (→Solution 6 (Factored Form): Reformatted.) |
MRENTHUSIASM (talk | contribs) (→Solution 10 (solution 8 but simpler)) |
||
(48 intermediate revisions by 13 users not shown) | |||
Line 6: | Line 6: | ||
<math>\textbf{(A) } \dfrac{5}{16} \qquad\textbf{(B) } \dfrac{1}{2} \qquad\textbf{(C) } \dfrac{5}{8} \qquad\textbf{(D) } 1 \qquad\textbf{(E) } \dfrac{9}{8}</math> | <math>\textbf{(A) } \dfrac{5}{16} \qquad\textbf{(B) } \dfrac{1}{2} \qquad\textbf{(C) } \dfrac{5}{8} \qquad\textbf{(D) } 1 \qquad\textbf{(E) } \dfrac{9}{8}</math> | ||
− | == Solution 1 ( | + | == Solution 1 (Standard Form)== |
− | Let <math>r_1</math> and <math>r_2</math> be the roots of <math>\tilde{p}(x)</math>. Then, <math>\tilde{p}(x)=(x-r_1)(x-r_2)=x^2-(r_1+r_2)x+r_1r_2</math>. The solutions to <math>\tilde{p}(\tilde{p}(x))=0</math> is the union of the solutions to <cmath> \tilde{p}(x)-r_1=x^2-(r_1+r_2)x+(r_1r_2-r_1)=0 </cmath> and <cmath> \tilde{p}(x)-r_2=x^2-(r_1+r_2)x+(r_1r_2-r_2)=0. </cmath> Note that one of these two quadratics has one solution (a double root) and the other has two as there are exactly three solutions. WLOG, assume that the quadratic with one root is <math>x^2-(r_1+r_2)x+(r_1r_2-r_1)=0</math>. Then, the discriminant is <math>0</math>, so <math>(r_1+r_2)^2 = 4r_1r_2 - 4r_1</math>. Thus, <math>r_1-r_2=\pm 2\sqrt{-r_1}</math>, but for <math>x^2-(r_1+r_2)x+(r_1r_2-r_2)=0</math> to have two solutions, it must be the case that <math>r_1-r_2=- 2\sqrt{-r_1}</math> | + | Let <math>r_1</math> and <math>r_2</math> be the roots of <math>\tilde{p}(x)</math>. Then, <math>\tilde{p}(x)=(x-r_1)(x-r_2)=x^2-(r_1+r_2)x+r_1r_2</math>. The solutions to <math>\tilde{p}(\tilde{p}(x))=0</math> is the union of the solutions to <cmath> \tilde{p}(x)-r_1=x^2-(r_1+r_2)x+(r_1r_2-r_1)=0 </cmath> and <cmath> \tilde{p}(x)-r_2=x^2-(r_1+r_2)x+(r_1r_2-r_2)=0. </cmath> Note that one of these two quadratics has one solution (a double root) and the other has two as there are exactly three solutions. WLOG, assume that the quadratic with one root is <math>x^2-(r_1+r_2)x+(r_1r_2-r_1)=0</math>. Then, the discriminant is <math>0</math>, so <math>(r_1+r_2)^2 = 4r_1r_2 - 4r_1</math>. Thus, <math>r_1-r_2=\pm 2\sqrt{-r_1}</math>, but for <math>x^2-(r_1+r_2)x+(r_1r_2-r_2)=0</math> to have two solutions, it must be the case that <math>r_1-r_2=- 2\sqrt{-r_1} (*)</math>. It follows that the sum of the roots of <math>\tilde{p}(x)</math> is <math>2r_1 + 2\sqrt{-r_1}</math>, whose maximum value occurs when <math>r_1 = - \frac{1}{4} (\star)</math>. Solving for <math>r_2</math> yields <math>r_2 = \frac{3}{4}</math>. Therefore, <math>\tilde{p}(x)=x^2 - \frac{1}{2} x - \frac{3}{16}</math>, so <math>\tilde{p}(1)= \boxed{\textbf{(A) } \frac{5}{16}}</math>. |
− | <u><b> | + | <u><b>Remarks</b></u> |
− | + | * For <math>x^2-(r_1+r_2)x+(r_1r_2-r_2)=0</math> to have two solutions, the discriminant <math>(r_1+r_2)^2-4r_1r_2+4r_2</math> must be positive. From here, we get that <math>(r_1-r_2)^2>-4r_2</math>, so <math>-4r_1>-4r_2 \implies r_1<r_2</math>. Hence, <math>r_1-r_2</math> is negative, so <math>r_1-r_2=-2\sqrt{-r_1}</math>. | |
+ | |||
+ | * Set <math>\sqrt{-r_1}=x</math>. Now <math>r_1+\sqrt{-r_1}=-x^2+x</math>, for which the maximum occurs when <math>x=\frac{1}{2} \rightarrow r_1=-\frac{1}{4}</math>. | ||
~ Leo.Euler | ~ Leo.Euler | ||
Line 18: | Line 20: | ||
Let <math>p(x)=(x-h)^2+k</math> for some real constants <math>h</math> and <math>k.</math> Suppose that <math>p(x)</math> has real roots <math>r</math> and <math>s.</math> | Let <math>p(x)=(x-h)^2+k</math> for some real constants <math>h</math> and <math>k.</math> Suppose that <math>p(x)</math> has real roots <math>r</math> and <math>s.</math> | ||
− | Since <math>p(p(x))=0,</math> we conclude that <math>p(x)=r</math> or <math>p(x)=s.</math> Without | + | Since <math>p(p(x))=0,</math> we conclude that <math>p(x)=r</math> or <math>p(x)=s.</math> Without loss of generality, we assume that <math>p(x)=r</math> has two real solutions and <math>p(x)=s</math> has one real solution. Therefore, we have <math>k=s,</math> from which <math>p(x)=(x-h)^2+s.</math> |
As <math>p(s)=0,</math> we expand the left side to obtain <math>(s-h)^2+s=0,</math> or <cmath>s^2-(2h-1)s+h^2=0. \hspace{15mm}(\bigstar)</cmath> | As <math>p(s)=0,</math> we expand the left side to obtain <math>(s-h)^2+s=0,</math> or <cmath>s^2-(2h-1)s+h^2=0. \hspace{15mm}(\bigstar)</cmath> | ||
Line 28: | Line 30: | ||
~MRENTHUSIASM | ~MRENTHUSIASM | ||
+ | |||
+ | == Solution 2.5 (Vertex Form but Done Differently) == | ||
+ | Continuing from the previous solution's reasoning, we know that the <math>y</math>-coordinate of the vertex must also be a root of the polynomial. Thus, <math>\tilde{p}(x) = \left(x - \frac{s + r}{2}\right)^2 + s</math>, where <math>r</math> and <math>s</math> are roots of the polynomial. | ||
+ | |||
+ | We also know that in order for <math>\tilde{p}(x) = r</math> to have two solutions, <math>y = r</math> must intersect <math>\tilde{p}(x)</math> above its minimum/vertex. Thus, <math>r > s</math>. Let <math>r = s + a</math>. Then, <math>\tilde{p}(x) = \left(x - \frac {2s + a}{2}\right)^2 + s</math>. | ||
+ | |||
+ | Since <math>s</math> is a root, <math>\tilde{p}(s) = 0</math> so <math>\left(s - \frac{2s + a}{2}\right)^2 +s = 0 \implies \frac{a^2}{4} + s = 0 \implies s = - \frac{1}{4} a^2</math>. We wish to maximize <math>s + s + a = 2s + a = - \frac{1}{2} a^2 + a</math>. | ||
+ | |||
+ | This quadratic is maximized at <math>a = 1</math> (when converting the quadratic to vertex form). Then, since <math>\tilde{p}(s + a) = 0 \implies \tilde{p}(s+1) = 0</math>, <math>\left(s + 1 - s - \frac12\right)^2 + s = 0 \implies \frac{1}{4} + s = 0 \implies s = -\frac{1}{4}</math>. | ||
+ | |||
+ | Thus, <math>\tilde{p}(x) = \left(x - \frac{1}{4}\right)^2 - \frac{1}{4}</math>. So, <math>\tilde{p}(1) = \boxed{\textbf{(A) } \frac{5}{16}}</math>. | ||
+ | |||
+ | ~[https://artofproblemsolving.com/wiki/index.php/User:CrazyVideoGamez CrazyVideoGamez] | ||
== Solution 3 (Symmetry) == | == Solution 3 (Symmetry) == | ||
Line 76: | Line 91: | ||
<cmath> | <cmath> | ||
\[ | \[ | ||
− | p^2 - 4 \left( q - r_1 \right) = 0 . \hspace{1cm} | + | p^2 - 4 \left( q - r_1 \right) = 0 . \hspace{1cm} |
\] | \] | ||
</cmath> | </cmath> | ||
Line 82: | Line 97: | ||
Because <math>r_1 < r_2</math>, <math>r_1 = \frac{- p - \sqrt{p^2 - 4 q}}{2}</math>. | Because <math>r_1 < r_2</math>, <math>r_1 = \frac{- p - \sqrt{p^2 - 4 q}}{2}</math>. | ||
− | Hence, | + | Hence, the above equation can be written as |
<cmath> | <cmath> | ||
\[ | \[ | ||
Line 92: | Line 107: | ||
<cmath> | <cmath> | ||
\[ | \[ | ||
− | \left( p^2 - 4 q \right) - 2 \sqrt{p^2 - 4 q} - 2 p = 0 . \hspace{1cm} ( | + | \left( p^2 - 4 q \right) - 2 \sqrt{p^2 - 4 q} - 2 p = 0 . \hspace{1cm} (1) |
\] | \] | ||
</cmath> | </cmath> | ||
Line 108: | Line 123: | ||
Because <math>p \geq - \frac{1}{2}</math>, the smallest <math>p</math> is <math>-\frac{1}{2}</math>. | Because <math>p \geq - \frac{1}{2}</math>, the smallest <math>p</math> is <math>-\frac{1}{2}</math>. | ||
− | Plugging this value into Equation ( | + | Plugging this value into Equation (1), we get <math>\sqrt{p^2 - 4 q} = 1</math>. |
Hence, <math>q = - \frac{3}{16}</math>. | Hence, <math>q = - \frac{3}{16}</math>. | ||
Line 122: | Line 137: | ||
From here two solutions can progress. | From here two solutions can progress. | ||
+ | |||
+ | ~KingRavi | ||
===Solution 5.1 (Quadratic Formula)=== | ===Solution 5.1 (Quadratic Formula)=== | ||
Line 132: | Line 149: | ||
<cmath>m=2(-q^2+q) \implies m = -2q(q-1).</cmath> | <cmath>m=2(-q^2+q) \implies m = -2q(q-1).</cmath> | ||
To maximize <math>m</math>, we find the vertex of the right-hand side of the equation. The vertex of <math>-2q(q-1)</math> is the average of the roots of the equation which is <math>\frac{0+1}{2} = \frac{1}{2}</math>. This means that <math>r = -q^2 \implies \boldsymbol{r = -\frac{1}{4}}</math> and <math>m = -2q(q-1) \implies m = \frac{1}{2}</math>. Therefore, <math>m-r = s \implies \boldsymbol{s = \frac{3}{4}}</math>. | To maximize <math>m</math>, we find the vertex of the right-hand side of the equation. The vertex of <math>-2q(q-1)</math> is the average of the roots of the equation which is <math>\frac{0+1}{2} = \frac{1}{2}</math>. This means that <math>r = -q^2 \implies \boldsymbol{r = -\frac{1}{4}}</math> and <math>m = -2q(q-1) \implies m = \frac{1}{2}</math>. Therefore, <math>m-r = s \implies \boldsymbol{s = \frac{3}{4}}</math>. | ||
+ | |||
+ | ~KingRavi | ||
===Solution 5.2 (Derivation-Rotated Conics)=== | ===Solution 5.2 (Derivation-Rotated Conics)=== | ||
Line 177: | Line 196: | ||
Therefore, we have <math>\boldsymbol{r=-\frac{1}{4}}</math> and <math>\boldsymbol{s=\frac{3}{4}}</math>. | Therefore, we have <math>\boldsymbol{r=-\frac{1}{4}}</math> and <math>\boldsymbol{s=\frac{3}{4}}</math>. | ||
+ | |||
+ | ~KingRavi | ||
===Solutions 5.1 and 5.2 Rejoined=== | ===Solutions 5.1 and 5.2 Rejoined=== | ||
Line 184: | Line 205: | ||
==Solution 6 (Factored Form)== | ==Solution 6 (Factored Form)== | ||
− | Let <math>p(x)=(x-p)(x-q).</math> Then, < | + | Let <math>p(x)=(x-p)(x-q).</math> Then, <cmath> |
+ | p(p(x))=((x-p)(x-q)-p)((x-p)(x-q)-q)=0,</cmath> | ||
+ | which means that either <math>(x-p)(x-q)-p=0</math> or <math>(x-p)(x-q)-q=0</math>. Both of these equations are quadratics, so <math>p(p(x))</math> has four roots, unless there's a double root. | ||
− | Without loss of generality, let <math>(x-p)(x-q)-p</math> be the expression that produces the double root, so its discriminant is zero. When expanded, <cmath>(x-p)(x-q)-p=x^2-(p+q)x+pq-p=0.</cmath> | + | Without loss of generality, let <math>(x-p)(x-q)-p</math> be the expression that produces the double root, so its discriminant is zero. When expanded, |
+ | <cmath>(x-p)(x-q)-p=x^2-(p+q)x+pq-p=0.</cmath> | ||
+ | The value of <math>x</math> is irrelevant to the discriminant, which is <math>(-(p+q))^2-4(pq-p).</math> Setting this equal to zero and simplifing, this equation becomes <math>p^2-2pq+q^2+4p=0,</math> which is a quadratic in <math>p</math>. | ||
− | |||
Now, we seek the value of <math>p</math>. The previous quadratic is equivalent to <math>p^2-(2q-4)p+q^2=0</math>. Using the quadratic formula by having <math>a=1, b=-(2q-4)</math>, and <math>c=q^2,</math> we have | Now, we seek the value of <math>p</math>. The previous quadratic is equivalent to <math>p^2-(2q-4)p+q^2=0</math>. Using the quadratic formula by having <math>a=1, b=-(2q-4)</math>, and <math>c=q^2,</math> we have | ||
<cmath>p = \frac{2q-4\pm\sqrt{(-(2q-4))^2-4(1(q^2))}}{2(1)} = \frac{2q-4\pm4\sqrt{1-q}}{2}=q-2+2\sqrt{1-q}.</cmath> | <cmath>p = \frac{2q-4\pm\sqrt{(-(2q-4))^2-4(1(q^2))}}{2(1)} = \frac{2q-4\pm4\sqrt{1-q}}{2}=q-2+2\sqrt{1-q}.</cmath> | ||
Our focus is on maximizing <math>p+q</math>, so we need the maximum values of <math>p</math> and <math>q</math> respectively (by taking the positive square root). Adding <math>q,</math> we see that | Our focus is on maximizing <math>p+q</math>, so we need the maximum values of <math>p</math> and <math>q</math> respectively (by taking the positive square root). Adding <math>q,</math> we see that | ||
− | <cmath>p+q=2(q-1+\sqrt{1-q}).</cmath> | + | <cmath>p+q=2\left(q-1+\sqrt{1-q}\right).</cmath> |
Let <math>k=\sqrt{1-q}.</math> Then, <math>-k^2=q-1,</math> so <math>p+q=2(-k^2+k).</math> | Let <math>k=\sqrt{1-q}.</math> Then, <math>-k^2=q-1,</math> so <math>p+q=2(-k^2+k).</math> | ||
+ | |||
The graph of <math>-k^2+k</math> is a parabola that opens up downwards, and has its maximum value at <math>y</math>-value of the vertex. The <math>x</math> coordinate of the vertex is the average of the two roots, which in this case, are <math>0</math> and <math>1</math>, so the average of these two is <math>\frac{1}{2}</math>. This means that the maximum value of <math>-k^2+k</math> occurs at <math>k=\frac{1}{2}.</math> | The graph of <math>-k^2+k</math> is a parabola that opens up downwards, and has its maximum value at <math>y</math>-value of the vertex. The <math>x</math> coordinate of the vertex is the average of the two roots, which in this case, are <math>0</math> and <math>1</math>, so the average of these two is <math>\frac{1}{2}</math>. This means that the maximum value of <math>-k^2+k</math> occurs at <math>k=\frac{1}{2}.</math> | ||
− | |||
− | Since <math>p=q-2+2\sqrt{1-q},</math> we can plug <math>q=\frac{3}{4}</math> in, to see that <math>p=\frac{ | + | Substituting <math>k=\sqrt{1-q},</math> we see that <math>q=\frac{3}{4}.</math> Since <math>p=q-2+2\sqrt{1-q},</math> we can plug <math>q=\frac{3}{4}</math> in, to see that <math>p=-\frac{1}{4}.</math> |
+ | |||
− | Because the roots of this quadratic are <math>\frac{3}{4}</math> and <math>\frac{ | + | Because the roots of this quadratic are <math>\frac{3}{4}</math> and <math>-\frac{1}{4},</math> our quadratic is <math>\left(x-\frac{3}{4}\right)\left(x+\frac{1}{4}\right).</math> <math>\tilde{p}(1)=\frac{5}{16}</math>, so the answer is <math>\boxed{\textbf{(A) } \frac{5}{16}}</math>. |
~Benedict T (countmath1) | ~Benedict T (countmath1) | ||
+ | |||
+ | ==Solution 7 (Vieta’s Formulas)== | ||
+ | |||
+ | Since <math>\tilde{p}(x)</math> is a disrespectful polynomial, it must satisfy <math>\tilde{p}(\tilde{p}(x))</math>. Let <math>\tilde{p}(x)</math> have roots <math>x_1, x_2</math>. So, the equation <math>\tilde{p}(\tilde{p}(x))</math> becomes two separate equations: | ||
+ | <cmath>\begin{align*} | ||
+ | x^2+bx+c=x_1, \\ | ||
+ | x^2+bx+c=x_2. | ||
+ | \end{align*}</cmath> | ||
+ | Let <math>r_1,r_2,r_3,r_4</math> be the roots of <math>\tilde{p}(\tilde{p}(x))</math>. Since there are <math>3</math> real roots, then either one of them is complex, or two of them are the same. It is known that if complex number <math>w</math> is a root to a certain polynomial, then so is <math>\overline{w}</math>. So, if one root to <math>\tilde{p}(\tilde{p}(x))</math> is complex, then another root should also be complex — we can't have that. So, two roots are equal; WLOG, assume it is <math>r_3=r_4</math>. That means the roots to <math>\tilde{p}(\tilde{p}(x))</math> are <math>r_1,r_2,r_3,r_3</math>. | ||
+ | |||
+ | Now, if the roots of <math>x^2+bx+c=x_1</math> are <math>r_1</math> and <math>r_3</math> and the roots of <math>x^2+bx+c=x_2</math> are <math>r_2</math> and <math>r_3</math>, then we have <math>-b = r_1+r_3 = r_2+r_3</math> giving <math>r_1=r_2</math> by [[Vieta's Formula]]. That can't happen either. So, <math>x^2+bx+c=x_1</math> must have roots <math>r_1, r_2</math> and <math>x^2+bx+c=x_2</math> must have a double root <math>r_3</math>. Again by Vieta's, we have | ||
+ | <cmath>\begin{align*} | ||
+ | r_1+r_2&=-b, \\ | ||
+ | 2r_3&=-b. | ||
+ | \end{align*}</cmath> | ||
+ | That implies <math>r_2-r_3=r_3-r_1</math>. So, let <math>r_1=r_3-k</math> and <math>r_2=r_3+k</math>. Also, from <math>2r_3=-b</math>, we have <math>r_3=-\frac{b}{2}</math>. By Vieta's again, we have | ||
+ | <cmath>\begin{align*} | ||
+ | r_1\cdot r_2 &= c-x_1 = r_1\cdot r_2 = (r_3-k)(r_3-k) = r_3^2-k^2, \\ | ||
+ | r_3^2&=c-x_2. | ||
+ | \end{align*}</cmath> | ||
+ | Subtracting yields <math>x_1-x_2=k^2</math>. Adding that with <math>x_1+x_2=-b</math> brings us to | ||
+ | <cmath>\begin{align*} | ||
+ | x_1 &= \frac{k^2-b}{2}, \\ | ||
+ | x_2 &= \frac{-k^2-b}{2}. | ||
+ | \end{align*}</cmath> | ||
+ | Thus, <math>c=x_1\cdot x_2=\frac{b^2-k^4}{4}</math>. Now, <math>c-x_2 = r_3^2</math> by Vieta's. Plugging in <math>c, x_2, r_3=-\frac{b}{2}</math> gives | ||
+ | <cmath>\frac{b^2-k^4}{4} + \frac{k^2+b}{2} = \frac{b^2-k^4+2k^2-2b}{4} = \frac{b^2}{4}.</cmath> | ||
+ | Simplifying yields <math>2b=k^4-2k^2=(k^2-1)^2-1 \ge -1</math> by [[Trivial Inequality]]. Thus, <math>\text{min}(b) = -\frac{1}{2}</math>. Since <math>\tilde{p}(x)</math> have the maximum value to the sum of the roots, which is <math>-b</math>, we want to minimize <math>b</math>. That means that <math>b</math> for <math>\tilde{p}(x)</math> is <math>\frac{1}{2}</math>. That means <math>k^2=1</math> and <math>c=-\frac{3}{16}</math>, and thus giving us <math>\tilde{p}(1) = 1+b+c = 1-\frac{1}{2}-\frac{3}{16} = \boxed{\textbf{(A) } \frac{5}{16}}</math>. | ||
+ | |||
+ | ~sml1809 | ||
+ | |||
+ | ==Solution 8 (Calculus)== | ||
+ | |||
+ | Let <math>p(x)=(x-r_1)(x-r_2)</math>. Then | ||
+ | <cmath> | ||
+ | p(p(x)) = ((x-r_1)(x-r_2)-r_1)((x-r_1)(x-r_2)-r_2) | ||
+ | </cmath> | ||
+ | As noted in other solutions, in order for <math>p(p(x))</math> to have three (distinct) real roots, one of the roots must be repeated twice. | ||
+ | |||
+ | For a general polynomial <math>q(x)</math> with an <math>n</math>-times repeated root <math>s</math>, write <math>q(x)=(x-s)^n u(x)</math>, where <math>u(x)</math> is another polynomial. Taking the derivative, | ||
+ | <cmath> | ||
+ | \begin{align*} | ||
+ | q'(x)&= n(x-s)^{n-1} u(x) + (x-s)^n u'(x) \\ | ||
+ | &= (x-s)^{n-1} \left[nu(x) + (x-s)u'(x)\right] | ||
+ | \end{align*} | ||
+ | </cmath> | ||
+ | This shows that a repeated root / linear factor will appear in the derivative as well. | ||
+ | |||
+ | Returning to the problem, consider the derivative of <math>p(p(x))</math>, which is | ||
+ | <cmath> | ||
+ | [p(p(x))]' = p'(p(x))p'(x) = \left[2(x-r_1)(x-r_2)-r_1-r_2\right] \left[2x-r_1-r_2\right] | ||
+ | </cmath> | ||
+ | By the observation above, the repeated root of <math>p(p(x))</math> must be a root of this derivative. Therefore, a search through its roots will eventually give information to resolve <math>r_1</math> and <math>r_2</math>. Setting the derivative equal to zero yields the candidate equations | ||
+ | <cmath> | ||
+ | 2(x-r_1)(x-r_2)-r_1-r_2 = 0 \hspace{.5cm}\text{or}\hspace{.5cm} 2x-r_1-r_2 = 0 | ||
+ | </cmath> | ||
+ | Case 1: Starting with the first equation, if <math>x_0</math> is a solution, then rearrange to obtain | ||
+ | <cmath> | ||
+ | (x_0-r_1)(x_0-r_2) = \frac{r_1+r_2}{2} | ||
+ | </cmath> | ||
+ | Then | ||
+ | <cmath> | ||
+ | \begin{align*} | ||
+ | p(p(x_0)) &= ((x_0-r_1)(x_0-r_2)-r_1)((x_0-r_1)(x_0-r_2)-r_2) \\ | ||
+ | &= \left(\frac{r_1+r_2}{2} - r_1\right)\left(\frac{r_1+r_2}{2} - r_2\right) \\ | ||
+ | &= \frac{1}{4}(r_1-r_2)^2 = 0 | ||
+ | \end{align*} | ||
+ | </cmath> | ||
+ | This shows that <math>r_1=r_2</math>. However, this gives <math>p(x) = (x-r_1)^2</math> and <math>p(p(x)) = \left[(x-r_1)^2-r_1\right]^2</math>, which only has at most two distinct real roots. This contradicts the three-root condition given by the problem. | ||
+ | |||
+ | Case 2: The failure of Case 1 shows that the second equation must contain the repeated root. The solution for the second equation is readily <math>x_0 = (r_1+r_2)/2</math>. Then, | ||
+ | <cmath> | ||
+ | p(p(x_0)) = \left[\left(\frac{r_1+r_1}{2}-r_1\right)\left(\frac{r_1+r_2}{2}-r_2\right)-r_1\right]\left[\left(\frac{r_1+r_2}{2}-r_1\right)\left(\frac{r_1+r_2}{2}-r_2\right)-r_2\right] = 0 | ||
+ | </cmath> | ||
+ | Solving for <math>r_1</math> yields four candidate equations: | ||
+ | <cmath> | ||
+ | r_1 = r_2\pm 2\sqrt{-r_2} \hspace{.5cm}\text{or}\hspace{.5cm} r_1 = r_2\pm 2\sqrt{1-r_2} - 2 | ||
+ | </cmath> | ||
+ | Since it is desired to maximise the sum of roots, add <math>r_2</math> to both sides to consider | ||
+ | <cmath> | ||
+ | r_1+r_2 = 2r_2\pm 2\sqrt{-r_2} \hspace{.5cm}\text{or}\hspace{.5cm} r_1+r_2 = 2r_2\pm 2\sqrt{1-r_2} - 2 | ||
+ | </cmath> | ||
+ | For the first family of solutions, the "plus" version is immediately larger, so maximise | ||
+ | <cmath> | ||
+ | f(r_2) = 2r_2 + 2\sqrt{-r_2} | ||
+ | </cmath> | ||
+ | Taking the derivative and setting equal to zero, | ||
+ | <cmath> | ||
+ | f'(r_2) = 2 - \frac{1}{\sqrt{-r_2}} = 0 | ||
+ | </cmath> | ||
+ | yielding <math>r_2=-1/4</math> and <math>f(-1/4)=1/2</math>, which can be verified to be a maximum. Since <math>f(r_2)=r_1+r_2</math>, <math>r_1=3/4</math>. A similar calculation on the second family of solutions yields the complementary version <math>r_1=-1/4</math> and <math>r_2=3/4</math>. | ||
+ | |||
+ | In summary, this calculation shows that <math>\tilde{p}(x)=\left(x+\frac{1}{4}\right)\left(x-\frac{3}{4}\right)</math>. Finally, <math>\tilde{p}(1)=\left(1+\frac{1}{4}\right)\left(1-\frac{3}{4}\right)=\boxed{\textbf{(A) } \frac{5}{16}}</math>. | ||
+ | |||
+ | ~BinaryField | ||
+ | |||
+ | ==Solution 9 (Lagrange Multipliers)== | ||
+ | First note that <math>p(x)</math> must have two distinct roots, else <math>p(p(x))</math> will have fewer than <math>3</math> roots. Let these be <math>a,b.</math> Since <math>p(x)</math> has <math>3</math> real roots, and complex roots come in pairs, there must be a double root. So there must be <math>a_1, a_2</math> s.t <math>p(a_1) = p(a_2) = a</math> and <math>b_1, b_2</math> s.t <math>p(b_1) = p(b_2) = b</math> but two of <math>a_1, a_2, b_1, b_2</math> must be the same. WLOG either <math>a_1 = a_2</math> or <math>a_1 = b_1</math>. In the latter, case, then, | ||
+ | <cmath>\begin{align*} | ||
+ | p(x) &= a + (x - a_1)(x - a_2) \\ | ||
+ | p(x) &= b + (x - b_1)(x-b_2) \\ | ||
+ | &= b + (x - a_1)(x - b_2) \\ | ||
+ | \implies -(a_1 + a_2)x &= -(a_1 + b_2)x \\ | ||
+ | \implies a_2 &= b_2 | ||
+ | \end{align*} </cmath> | ||
+ | which implies <math>a = b</math> which is not possible. Hence assume WLOG <math>a_1 = a_2.</math> Then note that | ||
+ | <cmath>\begin{align*} | ||
+ | p(x) &= (x-a)(x - b) \\ | ||
+ | &= a + (x- a_1)^2 \\ | ||
+ | -(a + b)x &= -2a_1x, | ||
+ | \end{align*} </cmath> | ||
+ | so | ||
+ | <cmath>\begin{align*} | ||
+ | p(x) &= a + \left(x - \frac{a + b}{2}\right)^2 \\ | ||
+ | x^2 - (a + b)x + ab &= x^2 - (a + b)x + \left(a + \frac{1}{4}(a^2 + 2ab + b^2)\right) \\ | ||
+ | 0 &= a^2 - 2ab + b^2 + 4a. | ||
+ | \end{align*} </cmath> | ||
+ | The problem wants us to maximize the sum of the roots, <math>a + b.</math> Thus define <math>f(a,b) = a + b, g(a,b) = a^2 - 2ab + b^2 + 4a.</math> By the method of Lagrange Multipliers, we need to solve, | ||
+ | <cmath>\begin{align*} | ||
+ | \vec{\nabla} f &= \lambda \vec{\nabla} g \\ | ||
+ | g &= 0. | ||
+ | \end{align*} </cmath> | ||
+ | In the first equation, note, | ||
+ | <cmath>\begin{align*} | ||
+ | \vec{\nabla} f &= \lambda \vec{\nabla} g \\ | ||
+ | \langle 1, 1 \rangle &= \lambda \langle 2a - 2b + 4, -2a + 2b \rangle \\ | ||
+ | 2a - 2b + 4 &= -2a + 2b \\ | ||
+ | a - b &= -1. | ||
+ | \end{align*} </cmath> | ||
+ | Substituting into the second equation, | ||
+ | <cmath>\begin{align*} | ||
+ | g(a,b) &= (a-b)^2 + 4a \\ | ||
+ | 0 &= 1 + 4a \\ | ||
+ | a &= \frac{-1}{4}, b = \frac{3}{4} \\ | ||
+ | \tilde{p}(1) &= \left(1 + \frac{1}{4}\right)\left(1 - \frac{3}{4}\right) = \boxed{\textbf{(A) } \dfrac{5}{16}}. | ||
+ | \end{align*} </cmath> | ||
+ | ~Aaryabhatta1 | ||
+ | |||
+ | ==Solution 10 (Solution 8 but Simpler)== | ||
+ | |||
+ | Let <math>p(x) = (x-r)(x-s)</math>, having roots <math>r, s</math>. WLOG, set <math>r<s</math>. The graph of <math>p(x)</math> will have a vertex at <math>(\frac{r+s}{2}, p(\frac{r+s}{2}))</math>. Since <math>p(p(x)) = 0 \iff p(x) = r </math> or <math>p(x) = s</math> has exactly 3 distinct solutions, we know that <math>r = p(\frac{r+s}{2})</math> and <math>s > p(\frac{r+s}{2})</math> providing 1 and 2 solutions respectively for a total of 3 (if both <math>r,s > p(\frac{r+s}{2})</math> there will be 4 solutions and if either <math>r,s<p(\frac{r+s}{2})</math> there will be at most 2 solutions). Then, | ||
+ | <cmath>r = p(\frac{r+s}{2}) = \left(\frac{s-r}{2}\right)\left(\frac{r-s}{2}\right) = -\frac{(s-r)^2}{4}</cmath> | ||
+ | <cmath>(s-r)^2 = -4r</cmath> | ||
+ | <cmath>s+r = 2r\pm\sqrt{-4r}</cmath> | ||
+ | The negative case clearly doesn't work (giving max <math>s+r=0</math> where <math>r,s=0</math>), leaving us with <math>s+r = 2r+\sqrt{-4r}</math>. Finding the derivative we see that <math>s+r</math> is maximized when <math>r = -\frac{1}{4}</math>, which gives <math>s = \frac{3}{4}</math>. Plugging these back into <math>p(x)</math> we see that <math>\tilde{p}(x)=(x+\frac{1}{4})(x-\frac{3}{4})</math>. Then <math>\tilde{p}(1) = \boxed{\textbf{(A) } \dfrac{5}{16}}</math>. | ||
+ | |||
+ | ~Chupdogs | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/byo2vnCi6tk | ||
+ | |||
+ | ~MathProblemSolvingSkills.com | ||
+ | |||
+ | ==Video Solution by SpreadTheMathLove== | ||
+ | https://www.youtube.com/watch?v=fUcBjeP4XQ0 | ||
==See Also== | ==See Also== |
Latest revision as of 10:23, 16 July 2024
- The following problem is from both the 2021 Fall AMC 10A #25 and 2021 Fall AMC 12A #23, so both problems redirect to this page.
Contents
- 1 Problem
- 2 Solution 1 (Standard Form)
- 3 Solution 2 (Vertex Form)
- 4 Solution 2.5 (Vertex Form but Done Differently)
- 5 Solution 3 (Symmetry)
- 6 Solution 4 (Discriminant)
- 7 Solution 5 (Factored Form)
- 8 Solution 6 (Factored Form)
- 9 Solution 7 (Vieta’s Formulas)
- 10 Solution 8 (Calculus)
- 11 Solution 9 (Lagrange Multipliers)
- 12 Solution 10 (Solution 8 but Simpler)
- 13 Video Solution
- 14 Video Solution by SpreadTheMathLove
- 15 See Also
Problem
A quadratic polynomial with real coefficients and leading coefficient is called
if the equation
is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial
for which the sum of the roots is maximized. What is
?
Solution 1 (Standard Form)
Let and
be the roots of
. Then,
. The solutions to
is the union of the solutions to
and
Note that one of these two quadratics has one solution (a double root) and the other has two as there are exactly three solutions. WLOG, assume that the quadratic with one root is
. Then, the discriminant is
, so
. Thus,
, but for
to have two solutions, it must be the case that
. It follows that the sum of the roots of
is
, whose maximum value occurs when
. Solving for
yields
. Therefore,
, so
.
Remarks
- For
to have two solutions, the discriminant
must be positive. From here, we get that
, so
. Hence,
is negative, so
.
- Set
. Now
, for which the maximum occurs when
.
~ Leo.Euler
Solution 2 (Vertex Form)
Let for some real constants
and
Suppose that
has real roots
and
Since we conclude that
or
Without loss of generality, we assume that
has two real solutions and
has one real solution. Therefore, we have
from which
As we expand the left side to obtain
or
Since
has real solutions for
the discriminant is nonnegative:
We solve this inequality to get
Either by the axis of symmetry or Vieta's Formulas, note that As we wish to maximize
we maximize
Substituting
into
we obtain
We factor the left side to get
or
Finally, the unique such polynomial is from which
~MRENTHUSIASM
Solution 2.5 (Vertex Form but Done Differently)
Continuing from the previous solution's reasoning, we know that the -coordinate of the vertex must also be a root of the polynomial. Thus,
, where
and
are roots of the polynomial.
We also know that in order for to have two solutions,
must intersect
above its minimum/vertex. Thus,
. Let
. Then,
.
Since is a root,
so
. We wish to maximize
.
This quadratic is maximized at (when converting the quadratic to vertex form). Then, since
,
.
Thus, . So,
.
Solution 3 (Symmetry)
Let . We seek to maximize the sum of the roots
, so we maximize
.
Let . Note that
is symmetric about
.
has
real solutions, Due to the complex conjugate theorem,
) must have
real roots. Therefore,
must have exactly
double root.
This root cannot be to the left or to the right of , as the symmetry of the function would mean that there would be another double root reflected across the
. It follows that the double root could only be situated at
.
. Expanding and writing this out in terms of k,
.
In order for this to have a solution, the discriminant has to be non-negative. In other words, .
This simplifies to , or
.
As we seek to maximize , we set
and see that
.
Therefore, , and
.
~ConcaveTriangle
Solution 4 (Discriminant)
Equation is equivalent to the following system of equations:
Denote .
Denote by and
two roots of
.
Because has three real roots, we must have the properties that
and
are real with
.
Without loss of generality, we assume
.
We notice that all roots of
are the collection of all roots of
and all roots of
.
Because each of these two equations is quadratic, it has two roots (may be identical). To get a total number of three roots, one equation must have two identical roots.
Because , equation
has two identical roots.
Hence, the discriminant of this equation satisfies
Because ,
.
Hence, the above equation can be written as
This can be reorganized as
Define .
Hence, the value of
should ensure that equation
has at least one real nonnegative root.
This condition can be satisfied if the discriminant of this equation is nonnegative: .
Hence,
.
Now, we are ready to find .
Following from Vieta's formula, .
Hence, to get
maximized, we need to find smallest
.
Because , the smallest
is
.
Plugging this value into Equation (1), we get
.
Hence,
.
Thus, .
Therefore,
.
~Steven Chen (www.professorchenedu.com)
Solution 5 (Factored Form)
The disrespectful function has leading coefficient
, so it can be written in factored form as
. Now the problem states that all
must satisfy
. Plugging our form in, we get
The roots of this equation are
and
. By the fundamental theorem of algebra, each root must have two roots for a total of four possible values of
yet the problem states that this equation is satisfied by three values of
. Therefore one equation must give a double root. Without loss of generality, let the equation
be the equation that produces the double root. Expanding gives
. We know that if there is a double root to this equation, the discriminant must be equal to zero, so
.
From here two solutions can progress.
~KingRavi
Solution 5.1 (Quadratic Formula)
We can rewrite as
. Let's keep our eyes on the ball; we want to find the disrespectful quadratic that maximizes the sum of the roots, which is
. Let this be equal to a new variable,
, so that our problem is reduced to maximizing this variable.
We can rewrite our equation in terms of
as
.
This is a quadratic in , so we can use the quadratic formula:
It will be easier to think without the square root, so let
. We can rewrite the equation as
. We want to maximize
, so we take the plus value of the right-hand-side of the equation. Then,
To maximize
, we find the vertex of the right-hand side of the equation. The vertex of
is the average of the roots of the equation which is
. This means that
and
. Therefore,
.
~KingRavi
Solution 5.2 (Derivation-Rotated Conics)
We see that the equation is in the form of the general equation of a rotated conic
Because
, this rotated conic is a parabola.
The definition of a parabola is the locus of all points that are equidistant from a point (focus) and line (directrix). Let the focus and directrix of this particular parabola be and
. Then we can try to find the general form of a rotated parabola in terms of
and
.
The distance between two points and
is
. Therefore this is the distance from any point on the parabola to the focus.
The distance from a point to a line
is
.
We can set these two equal to each other and we get
Squaring both sides of the equation, we get
Expanding both sides of the equation gives
Multiplying both sides of the equation by
and rearranging gives
Now we can compare to our rotated parabola,
. From this,
or
. From here we have a system of three equations:
Plugging in
we get
Solving for the first equation,
.
Subtracting the first two equations, .
Plugging into the third equation, .
Substituting in, we get
.
Now and
.
This means that the focus of the parabola is and the directrix is
. The maximum value of
would lie at the vertex of the parabola, which is the midpoint of the focus and the foot of the focus at the directrix. The line that the vertex and focus lie on is perpendicular to the directrix, so it has slope
. It can be written as
and must go through
so
. This perpendicular line intersects the directrix, so to find the point at which this foot occurs, we set the equation of the lines equal to each other:
Adding, we get
or
and
. The vertex of the parabola is now at the midpoint of
and
which is
.
Therefore, we have and
.
~KingRavi
Solutions 5.1 and 5.2 Rejoined
Now that we know the roots of , we can plug in our equation:
~KingRavi
Solution 6 (Factored Form)
Let Then,
which means that either
or
. Both of these equations are quadratics, so
has four roots, unless there's a double root.
Without loss of generality, let be the expression that produces the double root, so its discriminant is zero. When expanded,
The value of
is irrelevant to the discriminant, which is
Setting this equal to zero and simplifing, this equation becomes
which is a quadratic in
.
Now, we seek the value of . The previous quadratic is equivalent to
. Using the quadratic formula by having
, and
we have
Our focus is on maximizing
, so we need the maximum values of
and
respectively (by taking the positive square root). Adding
we see that
Let
Then,
so
The graph of is a parabola that opens up downwards, and has its maximum value at
-value of the vertex. The
coordinate of the vertex is the average of the two roots, which in this case, are
and
, so the average of these two is
. This means that the maximum value of
occurs at
Substituting we see that
Since
we can plug
in, to see that
Because the roots of this quadratic are and
our quadratic is
, so the answer is
.
~Benedict T (countmath1)
Solution 7 (Vieta’s Formulas)
Since is a disrespectful polynomial, it must satisfy
. Let
have roots
. So, the equation
becomes two separate equations:
Let
be the roots of
. Since there are
real roots, then either one of them is complex, or two of them are the same. It is known that if complex number
is a root to a certain polynomial, then so is
. So, if one root to
is complex, then another root should also be complex — we can't have that. So, two roots are equal; WLOG, assume it is
. That means the roots to
are
.
Now, if the roots of are
and
and the roots of
are
and
, then we have
giving
by Vieta's Formula. That can't happen either. So,
must have roots
and
must have a double root
. Again by Vieta's, we have
That implies
. So, let
and
. Also, from
, we have
. By Vieta's again, we have
Subtracting yields
. Adding that with
brings us to
Thus,
. Now,
by Vieta's. Plugging in
gives
Simplifying yields
by Trivial Inequality. Thus,
. Since
have the maximum value to the sum of the roots, which is
, we want to minimize
. That means that
for
is
. That means
and
, and thus giving us
.
~sml1809
Solution 8 (Calculus)
Let . Then
As noted in other solutions, in order for
to have three (distinct) real roots, one of the roots must be repeated twice.
For a general polynomial with an
-times repeated root
, write
, where
is another polynomial. Taking the derivative,
This shows that a repeated root / linear factor will appear in the derivative as well.
Returning to the problem, consider the derivative of , which is
By the observation above, the repeated root of
must be a root of this derivative. Therefore, a search through its roots will eventually give information to resolve
and
. Setting the derivative equal to zero yields the candidate equations
Case 1: Starting with the first equation, if
is a solution, then rearrange to obtain
Then
This shows that
. However, this gives
and
, which only has at most two distinct real roots. This contradicts the three-root condition given by the problem.
Case 2: The failure of Case 1 shows that the second equation must contain the repeated root. The solution for the second equation is readily . Then,
Solving for
yields four candidate equations:
Since it is desired to maximise the sum of roots, add
to both sides to consider
For the first family of solutions, the "plus" version is immediately larger, so maximise
Taking the derivative and setting equal to zero,
yielding
and
, which can be verified to be a maximum. Since
,
. A similar calculation on the second family of solutions yields the complementary version
and
.
In summary, this calculation shows that . Finally,
.
~BinaryField
Solution 9 (Lagrange Multipliers)
First note that must have two distinct roots, else
will have fewer than
roots. Let these be
Since
has
real roots, and complex roots come in pairs, there must be a double root. So there must be
s.t
and
s.t
but two of
must be the same. WLOG either
or
. In the latter, case, then,
which implies
which is not possible. Hence assume WLOG
Then note that
so
The problem wants us to maximize the sum of the roots,
Thus define
By the method of Lagrange Multipliers, we need to solve,
In the first equation, note,
Substituting into the second equation,
~Aaryabhatta1
Solution 10 (Solution 8 but Simpler)
Let , having roots
. WLOG, set
. The graph of
will have a vertex at
. Since
or
has exactly 3 distinct solutions, we know that
and
providing 1 and 2 solutions respectively for a total of 3 (if both
there will be 4 solutions and if either
there will be at most 2 solutions). Then,
The negative case clearly doesn't work (giving max
where
), leaving us with
. Finding the derivative we see that
is maximized when
, which gives
. Plugging these back into
we see that
. Then
.
~Chupdogs
Video Solution
~MathProblemSolvingSkills.com
Video Solution by SpreadTheMathLove
https://www.youtube.com/watch?v=fUcBjeP4XQ0
See Also
2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
2021 Fall AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Last Problem | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.