Difference between revisions of "2016 USAMO Problems/Problem 4"

Latest revision as of 23:13, 27 February 2020

Problem

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$ , $(f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2.$

Solution 1

Step 1: Set $x = y = 0$ to obtain $f(0) = 0.$

Step 2: Set $x = 0$ to obtain $f(y)f(-y) = f(y)^2.$

$\indent$ In particular, if $f(y) \ne 0$ then $f(y) = f(-y).$

$\indent$ In addition, replacing $y \to -t$ , it follows that $f(t) = 0 \implies f(-t) = 0$ for all $t \in \mathbb{R}.$

Step 3: Set $x = 3y$ to obtain $\left[f(y) + 3y^2\right]f(8y) = f(4y)^2.$

$\indent$ In particular, replacing $y \to t/8$ , it follows that $f(t) = 0 \implies f(t/2) = 0$ for all $t \in \mathbb{R}.$

Step 4: Set $y = -x$ to obtain $f(4x)\left[f(x) + f(-x) - 2x^2\right] = 0.$

$\indent$ In particular, if $f(x) \ne 0$ , then $f(4x) \ne 0$ by the observation from Step 3, because $f(4x) = 0 \implies f(2x) = 0 \implies f(x) = 0.$ Hence, the above equation implies that $2x^2 = f(x) + f(-x) = 2f(x)$ , where the last step follows from the first observation from Step 2.

$\indent$ Therefore, either $f(x) = 0$ or $f(x) = x^2$ for each $x.$

$\indent$ Looking back on the equation from Step 3, it follows that $f(y) + 3y^2 \ne 0$ for any nonzero $y.$ Therefore, replacing $y \to t/4$ in this equation, it follows that $f(t) = 0 \implies f(2t) = 0.$

Step 5: If $f(a) = f(b) = 0$ , then $f(b - a) = 0.$

$\indent$ This follows by choosing $x, y$ such that $x - 3y = a$ and $3x - y = b.$ Then $x + y = \tfrac{b - a}{2}$ , so plugging $x, y$ into the given equation, we deduce that $f\left(\tfrac{b - a}{2}\right) = 0.$ Therefore, by the third observation from Step 4, we obtain $f(b - a) = 0$ , as desired.

Step 6: If $f \not\equiv 0$ , then $f(t) = 0 \implies t = 0.$

$\indent$ Suppose by way of contradiction that there exists an nonzero $t$ with $f(t) = 0.$ Choose $x, y$ such that $f(x) \ne 0$ and $x + y = t.$ The following three facts are crucial:

$\indent$ 1. $f(y) \ne 0.$ This is because $(x + y) - y = x$ , so by Step 5, $f(y) = 0 \implies f(x) = 0$ , impossible.

$\indent$ 2. $f(x - 3y) \ne 0.$ This is because $(x + y) - (x - 3y) = 4y$ , so by Step 5 and the observation from Step 3, $f(x - 3y) = 0 \implies f(4y) = 0 \implies f(2y) = 0 \implies f(y) = 0$ , impossible.

$\indent$ 3. $f(3x - y) \ne 0.$ This is because by the second observation from Step 2, $f(3x - y) = 0 \implies f(y - 3x) = 0.$ Then because $(x + y) - (y - 3x) = 4x$ , Step 5 together with the observation from Step 3 yield $f(3x - y) = 0 \implies f(4x) = 0 \implies f(2x) = 0 \implies f(x) = 0$ , impossible.

$\indent$ By the second observation from Step 4, these three facts imply that $f(y) = y^2$ and $f(x - 3y) = \left(x - 3y\right)^2$ and $f(3x - y) = \left(3x - y\right)^2.$ By plugging into the given equation, it follows that $\begin{align*} \left(x^2 + xy\right)\left(x - 3y\right)^2 + \left(y^2 + xy\right)\left(3x - y\right)^2 = 0. \end{align*}$ But the above expression miraculously factors into $\left(x + y\right)^4$ ! This is clearly a contradiction, since $t = x + y \ne 0$ by assumption. This completes Step 6.

Step 7: By Step 6 and the second observation from Step 4, the only possible solutions are $f \equiv 0$ and $f(x) = x^2$ for all $x \in \mathbb{R}.$ It's easy to check that both of these work, so we're done.

Alternative Solution 1

From steps 1 and 2 of Solution 1 we have that $f(0)=0$ , and $f(-y) \cdot f(y)=(f(y))^2$ . Therefore, if $f(y) \ne 0$ , then $f(y)=f(-y)$ . Furthermore, setting $y=-x$ gives us $(f(x)-x^2) \cdot f(4x) + (f(-x)-x^2) \cdot f(4x) = f(0) = 0$ . The LHS can be factored as $(f(x)+f(-x)-2x^2) \cdot f(4x) = 0$ . In particular, if $f(4x) \ne 0$ , then we have $f(x)+f(-x)=2x^2$ . However, since we have from step 2 that $f(x)=f(-x)$ , assuming $f(x) \ne 0$ , the equation becomes $2f(x)=2x^2$ , so for every $x$ , $f(x)$ is equivalent to either $0$ or $x^2$ . From step 6 of Solution 1, we can prove that $f(x)=0$ , and $f(x)=x^2$ are the only possible solutions.

Solution 2

Step 1: $x=y=0 \implies f(0)=0$

Step 2: $x=0 \implies f(y)f(-y)=f(y)^{2}$ . Now, assume $y \not = 0$ . Then, if $f(y)=0$ , we substitute in $-y$ to get $f(y)f(-y)=f(-y)^{2}$ , or $f(y)=f(-y)=0$ . Otherwise, we divide both sides by $f(y)$ to get $f(y)=f(-y)$ . If $y=0$ , we obviously have $f(0)=f(0)$ . Thus, the function is even. . Step 3: $y=-x \implies 2f(4x)(f(x)-x^{2})=0$ . Thus, $\forall x$ , we have $f(4x)=0$ or $f(x)=x^{2}$ .

Step 4: We now assume $f(x) \not = 0$ , $x\not = 0$ . We have $f(\frac{x}{4})=\frac{x^{2}}{16}$ . Now, setting $x=y=\frac{x}{4}$ , we have $f(\frac{x}{2})=\frac{x^{2}}{4}$ or $f(\frac{x}{2})=0$ . The former implies that $f(x)=0$ or $x^{2}$ . The latter implies that $f(x)=0$ or $f(x)=\frac{x^{2}}{2}$ . Assume the latter. $y=-2x \implies -\frac{3y^{2}}{2}f(7y)-{2y^{2}}f(5y)=\frac{x^{2}}{2}$ . Clearly, this implies that $f(x)$ is negative for some $m$ . Now, we have $f(\frac{m}{4})=\frac{m^{2}}{16} \implies f(\frac{m}{2})=0,\frac{m^{2}}{4} \implies f(m) \geq 0$ , which is a contradiction. Thus, $\forall x$ $f(x)=0$ or $f(x)=x^{2}$ .

Step 5: We now assume $f(x)=0$ , $f(y)=y^{2}$ for some $x,y \not = 0$ . Let $m$ be sufficiently large integer, let $z=|4^{m}x|$ and take the absolute value of $y$ (since the function is even). Choose $c$ such that $3z-c=y$ . Note that we have $\frac{c}{z}$ ~ $3$ and $\frac{y}{z}$ ~ $0$ . Note that $f(z)=0$ . Now, $x=z, y=c \implies$ LHS is positive, as the second term is positive and the first term is nonnegative and thus the right term is equal to $(z+c)^{4}$ ~ $256z^{4}$ . Now if $f(z-3c)=0$ , the second term of the LHS/RHS clearly ~0 as $m \to \infty$ . if $f(z-3c)=0$ , then we have LHS/RHS ~ $0$ , otherwise, we have LHS/RHS~ $\frac{8^{2}\cdot 3z^{4}}{256z^{4}}$ ~ $\frac{3}{4}$ , a contradiction, as we're clearly not dividing by $0$ , and we should have LHS/RHS=1.

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 45: / Line 45: @@
 '''Step 7:''' By Step 6 and the second observation from Step 4, the only possible solutions are <math>f \equiv 0</math> and <math>f(x) = x^2</math> for all <math>x \in \mathbb{R}.</math> It's easy to check that both of these work, so we're done.
-==Alternate Solution 1==
+==Alternative Solution 1==
-From steps 1 and 2 we have that, setting <math>x=y=0</math>, <math>f(0)=0</math>, and, setting <math>x=0</math>, <math>f(-y)f(y)=(f(y))^2</math>, so if <math>f(y) \ne 0</math>, then <math>f(y)=f(-y)</math>. Furthermore, setting <math>y=-x</math> gives us <math>(f(x)-x^2) \cdot f(4x) + (f(-x)-x^2) \cdot f(4x) = f(0) = 0</math>. The LHS can be factored as <math>(f(x)+f(-x)-2x^2) \cdot f(4x) = 0</math>. Therefore, if <math>f(4x) \ne 0</math>, then we have <math>f(x)+f(-x)-2x^2=0</math>, or <math>f(x)+f(-x)=2x^2</math>. However, since from step 2 we have that <math>f(x)=f(-x)</math> assuming f(x) \ne 0<math>, the equation becomes </math>2f(x)=2x^2<math>, so </math>f(x)=0, x^2<math> are the only solutions.
+From steps 1 and 2 of Solution 1 we have that <math>f(0)=0</math>, and <math>f(-y) \cdot f(y)=(f(y))^2</math>. Therefore, if <math>f(y) \ne 0</math>, then <math>f(y)=f(-y)</math>. Furthermore, setting <math>y=-x</math> gives us <math>(f(x)-x^2) \cdot f(4x) + (f(-x)-x^2) \cdot f(4x) = f(0) = 0</math>. The LHS can be factored as <math>(f(x)+f(-x)-2x^2) \cdot f(4x) = 0</math>. In particular, if <math>f(4x) \ne 0</math>, then we have <math>f(x)+f(-x)=2x^2</math>. However, since we have from step 2 that <math>f(x)=f(-x)</math>, assuming <math>f(x) \ne 0</math>, the equation becomes <math>2f(x)=2x^2</math>, so for every <math>x</math>, <math>f(x)</math> is equivalent to either <math>0</math> or <math>x^2</math>. From step 6 of Solution 1, we can prove that <math>f(x)=0</math>, and <math>f(x)=x^2</math> are the only possible solutions.
 ==Solution 2==
-Step 1: </math>x=y=0 \implies f(0)=0<math>
+Step 1: <math>x=y=0 \implies f(0)=0</math>
-Step 2: </math>x=0 \implies f(y)f(-y)=f(y)^{2}<math>. Now, assume </math>y \not = 0<math>. Then, if </math>f(y)=0<math>, we substitute in </math>-y<math> to get </math>f(y)f(-y)=f(-y)^{2}<math>, or </math>f(y)=f(-y)=0<math>. Otherwise, we divide both sides by </math>f(y)<math> to get </math>f(y)=f(-y)<math>. If </math>y=0<math>, we obviously have </math>f(0)=f(0)<math>. Thus, the function is even.
+Step 2: <math>x=0 \implies f(y)f(-y)=f(y)^{2}</math>. Now, assume <math>y \not = 0</math>. Then, if <math>f(y)=0</math>, we substitute in <math>-y</math> to get <math>f(y)f(-y)=f(-y)^{2}</math>, or <math>f(y)=f(-y)=0</math>. Otherwise, we divide both sides by <math>f(y)</math> to get <math>f(y)=f(-y)</math>. If <math>y=0</math>, we obviously have <math>f(0)=f(0)</math>. Thus, the function is even.
 .
-Step 3: </math>y=-x \implies 2f(4x)(f(x)-x^{2})=0<math>. Thus, </math>\forall x<math>, we have </math>f(4x)=0<math> or </math>f(x)=x^{2}<math>.
+Step 3: <math>y=-x \implies 2f(4x)(f(x)-x^{2})=0</math>. Thus, <math>\forall x</math>, we have <math>f(4x)=0</math> or <math>f(x)=x^{2}</math>.
-Step 4: We now assume </math>f(x) \not = 0<math>, </math>x\not = 0<math>. We have </math>f(\frac{x}{4})=\frac{x^{2}}{16}<math>. Now, setting </math>x=y=\frac{x}{4}<math>, we have </math>f(\frac{x}{2}=\frac{x^{2}}{4}<math> or </math>f(\frac{x}{2})=0<math>. The former implies that </math>f(x)=0<math> or </math>x^{2}<math>. The latter implies that </math>f(x)=0<math> or </math>f(x)=\frac{x^{2}}{2}<math>. Assume the latter. </math>y=-2x \implies -\frac{3y^{2}}{2}f(7y)-{2y^{2}}f(5y)=\frac{x^{2}}{2}<math>. Clearly, this implies that </math>f(x)<math> is negative for some </math>m<math>. Now, we have </math>f(\frac{m}{4})=\frac{m^{2}}{16} \implies f(\frac{m}{2})=0,\frac{m^{2}}{4} \implies f(m) \geq 0<math>, which is a contradiction. Thus, </math>\forall x<math></math>f(x)=0<math> or </math>f(x)=x^{2}<math>.
+Step 4: We now assume <math>f(x) \not = 0</math>, <math>x\not = 0</math>. We have <math>f(\frac{x}{4})=\frac{x^{2}}{16}</math>. Now, setting <math>x=y=\frac{x}{4}</math>, we have <math>f(\frac{x}{2})=\frac{x^{2}}{4}</math> or <math>f(\frac{x}{2})=0</math>. The former implies that <math>f(x)=0</math> or <math>x^{2}</math>. The latter implies that <math>f(x)=0</math> or <math>f(x)=\frac{x^{2}}{2}</math>. Assume the latter. <math>y=-2x \implies -\frac{3y^{2}}{2}f(7y)-{2y^{2}}f(5y)=\frac{x^{2}}{2}</math>. Clearly, this implies that <math>f(x)</math> is negative for some <math>m</math>. Now, we have <math>f(\frac{m}{4})=\frac{m^{2}}{16} \implies f(\frac{m}{2})=0,\frac{m^{2}}{4} \implies f(m) \geq 0</math>, which is a contradiction. Thus, <math>\forall x</math><math>f(x)=0</math> or <math>f(x)=x^{2}</math>.
-Step 5: We now assume </math>f(x)=0<math>, </math>f(y)=y^{2}<math> for some </math>x,y \not = 0<math>. Let </math>m<math> be sufficiently large integer, let </math>z=|4^{m}x|<math> and take the absolute value of </math>y<math>(since the function is even). Choose </math>c<math> such that </math>3z-c=y<math>. Note that we have </math>\frac{c}{z}<math>~</math>3<math> and </math>\frac{y}{z}<math>~</math>0<math>. Note that </math>f(z)=0<math>. Now, </math>x=z, y=c \implies<math> LHS is positive, as the second term is positive and the first term is nonnegative and thus the right term is equal to </math>(z+c)^{4}<math>~</math>256z^{4}<math>. Now if </math>f(z-3c)=0<math>, the second term of the LHS/RHS clearly ~0 as </math>m \to \infty<math>. if </math>f(z-3c)=0<math>, then we have LHS/RHS ~ </math>0<math>, otherwise, we have LHS/RHS~</math>\frac{8^{2}\cdot 3z^{4}}{256z^{4}}<math>~</math>\frac{3}{4}<math>, a contradiction, as we're clearly not dividing by </math>0$, and we should have LHS/RHS=1.
+Step 5: We now assume <math>f(x)=0</math>, <math>f(y)=y^{2}</math> for some <math>x,y \not = 0</math>. Let <math>m</math> be sufficiently large integer, let <math>z=|4^{m}x|</math> and take the absolute value of <math>y</math>(since the function is even). Choose <math>c</math> such that <math>3z-c=y</math>. Note that we have <math>\frac{c}{z}</math>~<math>3</math> and <math>\frac{y}{z}</math>~<math>0</math>. Note that <math>f(z)=0</math>. Now, <math>x=z, y=c \implies</math> LHS is positive, as the second term is positive and the first term is nonnegative and thus the right term is equal to <math>(z+c)^{4}</math>~<math>256z^{4}</math>. Now if <math>f(z-3c)=0</math>, the second term of the LHS/RHS clearly ~0 as <math>m \to \infty</math>. if <math>f(z-3c)=0</math>, then we have LHS/RHS ~ <math>0</math>, otherwise, we have LHS/RHS~<math>\frac{8^{2}\cdot 3z^{4}}{256z^{4}}</math>~<math>\frac{3}{4}</math>, a contradiction, as we're clearly not dividing by <math>0</math>, and we should have LHS/RHS=1.

2016 USAMO (Problems • Resources)
Preceded by Problem 3	Followed by Problem 5
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions

2016 USAJMO (Problems • Resources)
Preceded by Problem 5	Last Problem
1 • 2 • 3 • 4 • 5 • 6
All USAJMO Problems and Solutions

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