Difference between revisions of "2007 AIME II Problems/Problem 14"

Latest revision as of 16:44, 25 January 2022

Problem

Let $f(x)$ be a polynomial with real coefficients such that $f(0) = 1,$ $f(2)+f(3)=125,$ and for all $x$ , $f(x)f(2x^{2})=f(2x^{3}+x).$ Find $f(5).$

Official Solution (MAA)

If the leading term of $f(x)$ is $ax^m$ , then the leading term of $f(x)f(2x^2) = ax^m \cdot a(2x^2)^m = 2^ma^2x^{3m}$ , and the leading term of $f(2x^3 + x) = 2^max^{3m}$ . Hence $2^ma^2 = 2^ma$ , and $a = 1$ . Because $f(0) = 1$ , the product of all the roots of $f(x)$ is $\pm 1$ . If $f(\lambda) = 0$ , then $f(2\lambda^3 + \lambda) = 0$ . Assume that there exists a root $\lambda$ with $|\lambda| \neq 1$ . Then there must be such a root $\lambda_1$ with $|\lambda_1| > 1$ . Then $|2\lambda^3 + \lambda| \ge 2|\lambda|^3 - |\lambda| > 2|\lambda| - |\lambda| = |\lambda|.$ But then $f(x)$ would have infinitely many roots, given by $\lambda_{k+1} = 2\lambda_k^3 + \lambda_k$ , for $k \ge 1$ . Therefore $|\lambda| = 1$ for all of the roots of the polynomial. Thus $\lambda\overline{\lambda}=1$ , and $(2\lambda^3 + \lambda)\overline{(2\lambda^3 + \lambda)} = 1$ . Solving these equations simultaneously for $\lambda = a + bi$ yields $a = 0$ , $b^2 = 1$ , and so $\lambda^2 = -1$ . Because the polynomial has real coefficients, the polynomial must have the form $f(x) = (1 + x^2)^n$ for some integer $n \ge 1$ . The condition $f(2) + f(3) = 125$ implies $n = 2$ , giving $f(5) = \boxed{676}$ .

Solution 1

Let $r$ be a root of $f(x)$ . Then we have $f(r)f(2r^2)=f(2r^3+r)$ ; since $r$ is a root, we have $f(r)=0$ ; therefore $2r^3+r$ is also a root. Thus, if $r$ is real and non-zero, $|2r^3+r|>r$ , so $f(x)$ has infinitely many roots. Since $f(x)$ is a polynomial (thus of finite degree) and $f(0)$ is nonzero, $f(x)$ has no real roots.

Note that $f(x)$ is not constant. We then find two complex roots: $r = \pm i$ . We find that $f(i)f(-2) = f(-i)$ , and that $f(-i)f(-2) = f(i)$ . This means that $f(i)f(-i)f(-2)^2 = f(i)f(-i) \Longrightarrow f(i)f(-i)(f(-2)^2 - 1) = 0$ . Thus, $\pm i$ are roots of the polynomial, and so $(x - i)(x + i) = x^2 + 1$ will be a factor of the polynomial. (Note: This requires the assumption that $f(-2)\neq1$ . Clearly, $f(-2)\neq-1$ , because that would imply the existence of a real root.)

The polynomial is thus in the form of $f(x) = (x^2 + 1)g(x)$ . Substituting into the given expression, we have

$(x^2+1)g(x)(4x^4+1)g(2x^2)=((2x^3+x)^2+1)g(2x^3+x)$ $(4x^6+4x^4+x^2+1)g(x)g(2x^2)=(4x^6+4x^4+x^2+1)g(2x^3+x)$

Thus either $4x^6+4x^4+x^2+1=(4x^4+1)(x^2+1)$ is 0 for any $x$ , or $g(x)$ satisfies the same constraints as $f(x)$ . Continuing, by infinite descent, $f(x) = (x^2 + 1)^n$ for some $n$ .

Since $f(2)+f(3)=125=5^n+10^n$ for some $n$ , we have $n=2$ ; so $f(5) = \boxed{676}$ .

Comment: The answer is clearly correct, but the proof has a gap, i.e. there is no reason that $f(-2)\neq1$ . Since $f(x)$ has no real roots, the degree must be even. Consider $g(x)= f(x)/f(-x)$ . Then since $f$ is non-zero, $g(x)=g(2x^3+x)$ . Now the function $2x^3+x$ applied repeatedly from some real starting value of x becomes arbitrarily large, and the limit of $g(x)$ as $|x|$ approaches infinity is 1, so $g(x)$ =1 for all x, or $f(x)=f(-x)$ . Then $f(x)=h(x^2+1)$ for some polynomial $h(x)$ , and $h(x^2+1)h(4x^4+1)=h(4x^6+4x^4+x^2+1) = h((x^2+1)(4x^4+1))$ . Now suppose h has degree m. It is clearly monic. Assume that the next highest non-zero coefficient in h is k. Then, subtracting $((x^2+1)(4x^4+1))^m$ from both sides of the equation yields a polynomial equality with degree $4m+2k$ on the left and degree $6k$ on the right, a contradiction. So $h(x)=x^m$ , and $f(x)=(1+x^2)^m$ .

Solution 2

Let $r$ be a root of $f(x).$ This means that $f(r)f(2r^2)=f(2r^3+r).$ In other words, $2r^3+r$ is a root of $f(x)$ too. Since $f(x)$ can't have infinitely many roots, $Q(x)=P(P(\dotsb P(P(r)) \dotsb))$ is cyclic, where $P(x)=2x^3+x.$ Now, we will do casework.

Case 1: $\deg f\geq1$

Subcase 1: $|r|>1$

This means that $|2r^3+r|\geq|2r^3|-|r|=|r|(2|r|^2-1)>|r|(2\cdot1^2-1)=|r|.$ It follows that $|2r^3+r|>|r|$ for all $r.$ This implies that $Q(r)$ can't be cyclic. Thus, it is impossible for $|r|>1$ to be true.

Subcase 2: $|r|<1$

This means that $|2r^3+r|\geq2|r^3|-|r|=|r|(|2r^2|-1)<|r|.$ It follows that $|2r^3+r|<|r|$ for all $r.$ This implies that $Q(r)$ can't be cyclic. Thus, it is impossible for $|r|>1$ to be true.

Subcase 3: $|r|=1.$

Since $|r|$ is not greater than or less than 1, $|r|=1.$ This means that all the roots of the polynomial have a magnitude of $1.$ More specifically, $|2r^3+r|$ has a magnitude of one. Since this would mean an equality condition from the triangle inequality, $2r^3$ and $r$ are collinear with the origin in the complex plane. In other words, $\frac{2r^3}{r}=\pm c\Leftrightarrow cr=2r^3\Leftrightarrow 2r^2=c\Leftrightarrow r=\pm\sqrt{\pm\frac{c}{2}},$ for some real constant $c.$ Now, from $|r|=1,$ we find that $\left|\pm\sqrt{\pm\frac{c}{2}}\right|=1\Leftrightarrow \sqrt{\pm\frac{c}{2}}=1\Leftrightarrow \pm\frac{c}{2}=1\Leftrightarrow c=\pm2.$ Putting this back into the equation, we find that $r=1,-1,i,-i.$ Now, this means that $2r^3+r=3,-3,i,-i.$ $3$ and $-3$ obviously doesn't have a magnitude of $1.$ Thus, $i,-i$ are the only possible roots of the polynomial. Since roots come in conjugate pairs, $f(x)=[(x-i)(x+i)]^n=(x^2+1)^n,$ works for all constants $n\neq0.$

Case 2: $\deg f=0.$

This means that $f(x)=c,$ for some constant $c.$ In other words, $c^2=c.$ We can easily find that this means that $c=0,1.$ Combining all the cases, we conclude that $f(x)=(x^2+1)^n,0,1$ are the only polynomials that satisfy this equation. Now, we can test! $f(x)=0,1$ obviously don't satisfy $f(2)+f(3)=125.$ Thus, $f(x)=(x^2+1)^n.$ Substituting, we find that $5^n+10^n=125\Leftrightarrow n=2.$ We conclude that $f(5)=(5^2+1)^2=26^2=\boxed{676}.$

~ pinkpig

@@ Line 1: / Line 1: @@
 == Problem ==
-Let <math>f(x)</math> be a [[polynomial]] with real [[coefficient]]s such that <math>\displaystyle f(0) = 1,</math> <math>\displaystyle f(2)+f(3)=125,</math> and for all <math>x</math>, <math>\displaystyle f(x)f(2x^{2})=f(2x^{3}+x).</math> Find <math>\displaystyle f(5).</math>
+Let <math>f(x)</math> be a [[polynomial]] with real [[coefficient]]s such that <math>f(0) = 1,</math> <math>f(2)+f(3)=125,</math> and for all <math>x</math>, <math>f(x)f(2x^{2})=f(2x^{3}+x).</math> Find <math>f(5).</math>
-== Solution ==
+== Official Solution (MAA)==
-Let <math>r</math> be a root of <math>f(x)</math>. Then we have <math>f(r)f(2r^2)=f(2r^3+r)</math>; since <math>r</math> is a root, we have <math>f(r)=0</math>; therefore <math>2r^3+r</math> is also a root. Thus, if <math>r</math> is real and non-zero, <math>f(x)</math> has infinitely many roots. Since <math>f(x)</math> is a polynomial (thus of finite degree) and <math>f(0)</math> is nonzero, <math>f(x)</math> has no real roots.
+If the leading term of <math>f(x)</math> is <math>ax^m</math>, then the leading term of <math>f(x)f(2x^2) = ax^m \cdot a(2x^2)^m = 2^ma^2x^{3m}</math>, and the leading term of <math>f(2x^3 + x) = 2^max^{3m}</math>. Hence <math>2^ma^2 = 2^ma</math>, and <math>a = 1</math>. Because <math>f(0) = 1</math>, the product of all the roots
+of <math>f(x)</math> is <math>\pm 1</math>. If <math>f(\lambda) = 0</math>, then <math>f(2\lambda^3 + \lambda) = 0</math>. Assume that there exists a root <math>\lambda</math> with <math>|\lambda| \neq 1</math>. Then there must be such a root <math>\lambda_1</math> with <math>|\lambda_1| > 1</math>. Then <cmath>|2\lambda^3 + \lambda| \ge 2|\lambda|^3 - |\lambda| > 2|\lambda| - |\lambda| = |\lambda|.</cmath> But then <math>f(x)</math> would have infinitely many roots, given by <math>\lambda_{k+1} = 2\lambda_k^3 + \lambda_k</math>, for <math>k \ge 1</math>.
+Therefore <math>|\lambda| = 1</math> for all of the roots of the polynomial. Thus <math>\lambda\overline{\lambda}=1</math>, and <math>(2\lambda^3 + \lambda)\overline{(2\lambda^3 + \lambda)} = 1</math>.
+Solving these equations simultaneously for <math>\lambda = a + bi</math> yields <math>a = 0</math>, <math>b^2 = 1</math>, and so <math>\lambda^2 = -1</math>. Because the polynomial has real coefficients, the polynomial must have the form <math>f(x) = (1 + x^2)^n</math> for some integer <math>n \ge 1</math>. The condition <math>f(2) + f(3) = 125</math> implies <math>n = 2</math>, giving <math>f(5) = \boxed{676}</math>.
-We then find two complex roots: <math>r = \pm i</math>. We find that <math>f(i)f(-2) = f(-i)</math>, and that <math>f(-i)f(-2) = f(i)</math>. This means that <math>f(i)f(-i)f(-2)^2 = f(i)f(-i) \Longrightarrow f(i)f(-i)(f(-2) - 1) = 0</math>. Thus <math>\pm i</math> are roots of the polynomial, and so <math>(x - i)(x + i) = x^2 + 1</math> will be a factor of the polynomial.
+== Solution 1==
+Let <math>r</math> be a root of <math>f(x)</math>. Then we have <math>f(r)f(2r^2)=f(2r^3+r)</math>; since <math>r</math> is a root, we have <math>f(r)=0</math>; therefore <math>2r^3+r</math> is also a root. Thus, if <math>r</math> is real and non-zero, <math>|2r^3+r|>r</math>, so <math>f(x)</math> has infinitely many roots. Since <math>f(x)</math> is a polynomial (thus of finite degree) and <math>f(0)</math> is nonzero, <math>f(x)</math> has no real roots.
+Note that <math>f(x)</math> is not constant. We then find two complex roots: <math>r = \pm i</math>. We find that <math>f(i)f(-2) = f(-i)</math>, and that <math>f(-i)f(-2) = f(i)</math>. This means that <math>f(i)f(-i)f(-2)^2 = f(i)f(-i) \Longrightarrow f(i)f(-i)(f(-2)^2 - 1) = 0</math>. Thus, <math>\pm i</math> are roots of the polynomial, and so <math>(x - i)(x + i) = x^2 + 1</math> will be a factor of the polynomial. (Note: This requires the assumption that <math>f(-2)\neq1</math>. Clearly, <math>f(-2)\neq-1</math>, because that would imply the existence of a real root.)
 The polynomial is thus in the form of <math>f(x) = (x^2 + 1)g(x)</math>. Substituting into the given expression, we have
@@ Line 14: / Line 20: @@
 Thus either <math>4x^6+4x^4+x^2+1=(4x^4+1)(x^2+1)</math> is 0 for any <math>x</math>, or <math>g(x)</math> satisfies the same constraints as <math>f(x)</math>. Continuing, by infinite descent, <math>f(x) = (x^2 + 1)^n</math> for some <math>n</math>.
-Since <math>f(2)+f(3)=125=5^n+10^n</math> for some <math>n</math>, we have <math>n=2</math>; so <math>f(5) = 676</math>.
+Since <math>f(2)+f(3)=125=5^n+10^n</math> for some <math>n</math>, we have <math>n=2</math>; so <math>f(5) = \boxed{676}</math>.
+Comment:
+The answer is clearly correct, but the proof has a gap, i.e. there is no reason that <math>f(-2)\neq1</math>.  Since <math>f(x)</math> has no real roots, the degree must be even.  Consider <math>g(x)= f(x)/f(-x)</math>.  Then since <math>f</math> is non-zero, <math>g(x)=g(2x^3+x)</math>. Now the function <math>2x^3+x</math> applied repeatedly from some real starting value of x becomes arbitrarily large, and the limit of <math>g(x)</math> as <math>|x|</math> approaches infinity is 1, so <math>g(x)</math>=1 for all x, or <math>f(x)=f(-x)</math>.  Then <math>f(x)=h(x^2+1)</math> for some polynomial <math>h(x)</math>, and <math>h(x^2+1)h(4x^4+1)=h(4x^6+4x^4+x^2+1) = h((x^2+1)(4x^4+1))</math>.  Now suppose h has degree m.  It is clearly monic.  Assume that the next highest non-zero coefficient in h is k.  Then, subtracting <math>((x^2+1)(4x^4+1))^m</math> from both sides of the equation yields a polynomial equality with degree <math>4m+2k</math> on the left and degree <math>6k</math> on the right, a contradiction.  So <math>h(x)=x^m</math>, and <math>f(x)=(1+x^2)^m</math>.
+==Solution 2==
+Let <math>r</math> be a root of <math>f(x).</math> This means that <math>f(r)f(2r^2)=f(2r^3+r).</math> In other words, <math>2r^3+r</math> is a root of <math>f(x)</math> too. Since <math>f(x)</math> can't have infinitely many roots, <cmath>Q(x)=P(P(\dotsb P(P(r)) \dotsb))</cmath> is cyclic, where <math>P(x)=2x^3+x.</math> Now, we will do casework.
+Case 1: <math>\deg f\geq1</math>
+Subcase 1: <math>|r|>1</math>
+This means that <cmath>|2r^3+r|\geq|2r^3|-|r|=|r|(2|r|^2-1)>|r|(2\cdot1^2-1)=|r|.</cmath> It follows that <math>|2r^3+r|>|r|</math> for all <math>r.</math> This implies that <math>Q(r)</math> can't be cyclic. Thus, it is impossible for <math>|r|>1</math> to be true.
+Subcase 2: <math>|r|<1</math>
+This means that <math>|2r^3+r|\geq2|r^3|-|r|=|r|(|2r^2|-1)<|r|.</math> It follows that <math>|2r^3+r|<|r|</math> for all <math>r.</math> This implies that <math>Q(r)</math> can't be cyclic. Thus, it is impossible for <math>|r|>1</math> to be true.
+Subcase 3: <math>|r|=1.</math>
+Since <math>|r|</math> is not greater than or less than 1, <math>|r|=1.</math> This means that all the roots of the polynomial have a magnitude of <math>1.</math> More specifically, <math>|2r^3+r|</math> has a magnitude of one. Since this would mean an equality condition from the triangle inequality, <math>2r^3</math> and <math>r</math> are collinear with the origin in the complex plane. In other words, <math>\frac{2r^3}{r}=\pm c\Leftrightarrow cr=2r^3\Leftrightarrow 2r^2=c\Leftrightarrow r=\pm\sqrt{\pm\frac{c}{2}},</math> for some real constant <math>c.</math> Now, from <math>|r|=1,</math> we find that <math>\left|\pm\sqrt{\pm\frac{c}{2}}\right|=1\Leftrightarrow \sqrt{\pm\frac{c}{2}}=1\Leftrightarrow \pm\frac{c}{2}=1\Leftrightarrow c=\pm2.</math> Putting this back into the equation, we find that <math>r=1,-1,i,-i.</math> Now, this means that <math>2r^3+r=3,-3,i,-i.</math> <math>3</math> and <math>-3</math> obviously doesn't have a magnitude of <math>1.</math> Thus, <math>i,-i</math> are the only possible roots of the polynomial. Since roots come in conjugate pairs, <math>f(x)=[(x-i)(x+i)]^n=(x^2+1)^n,</math> works for all constants <math>n\neq0.</math>
+Case 2: <math>\deg f=0.</math>
+This means that <math>f(x)=c,</math> for some constant <math>c.</math> In other words, <math>c^2=c.</math> We can easily find that this means that <math>c=0,1.</math>
+Combining all the cases, we conclude that <math>f(x)=(x^2+1)^n,0,1</math> are the only polynomials that satisfy this equation.
+Now, we can test! <math>f(x)=0,1</math> obviously don't satisfy <math>f(2)+f(3)=125.</math> Thus, <math>f(x)=(x^2+1)^n.</math> Substituting, we find that <math>5^n+10^n=125\Leftrightarrow n=2.</math> We conclude that <math>f(5)=(5^2+1)^2=26^2=\boxed{676}.</math>
+~ pinkpig
 == See also ==
@@ Line 20: / Line 55: @@
 [[Category:Intermediate Algebra Problems]]
+{{MAA Notice}}

2007 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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