2018 USAMO Problems/Problem 1

Problem 1

Let $a,b,c$ be positive real numbers such that $a+b+c=4\sqrt[3]{abc}$ . Prove that $2(ab+bc+ca)+4\min(a^2,b^2,c^2)\ge a^2+b^2+c^2.$

Solution

WLOG let $a \leq b \leq c$ . Add $2(ab+bc+ca)$ to both sides of the inequality and factor to get: $4(a(a+b+c)+bc) \geq (a+b+c)^2$ $\frac{4a\sqrt[3]{abc}+bc}{2} \geq 2\sqrt[3]{a^2b^2c^2}$

The last inequality is true by AM-GM. Since all these steps are reversible, the proof is complete.

- It should actually be 4(a)(a+b+c) + 4bc which results in a wrong inequality by AM-GM

Hence, the solution is wrong.

Solution 2

https://wiki-images.artofproblemsolving.com//6/69/IMG_8946.jpg

-srisainandan6

Solution 3

Similarly to Solution 2, we will prove homogeneity but we will use that to solve the problem differently. Let $f(a,b,c)=a+b+c-4\sqrt[3]{abc}$ . Note that $f(a,b,c)=f(ka,kb,kc)$ , thus proving homogeneity.

WLOG, we can scale down all variables such that the lowest one is $1$ . WLOG, let this be $a=1$ . We now have $1+b+c=4\sqrt[3]{bc}$ , and we want to prove $2bc+2b+2c+4\ge 1+b^2+c^2.$ Adding $2bc$ to both sides and subtracting $2b+2c$ gives us $4bc+4\ge 1+ (b+c)(b+c-2)$ , or $4bc+3\ge (b+c)(b+c-2)$ . Let $\sqrt[3]{bc}=x$ . Now, we have $4x^3+3 \ge (4x-1)(4x-3)$ $4x^3 - 16x^2 + 16x \ge 0$ $4x^2 - 16 + 16 \ge 0$ $4(x-2)^2 \ge 0$ By the trivial inequality, this is always true. Since all these steps are reversible, the proof is complete. ~SigmaPiE

Solution 4

WLOG, let $a \le b$ and $a \le c$ and add $2(ab+bc+ca)$ to both sides to make the left side a square. $4(ab+bc+ca)+4a^2 \ge (a+b+c)^2$ $4bc+4a(a+b+c) \ge (a+b+c)^2$ Now we perform the substitution for $a+b+c$ . $4bc+4a(4(abc)^{1/3}) \ge 16(abc)^{2/3}$ Multiply both sides by $a$ because we want all the terms to have a factor of $(abc)^{1/3}$ . $4abc+16a^2(abc)^{1/3} \ge 16a(abc)^{2/3}$ Divide both sides by $4(abc)^{1/3}$ . $(abc)^{2/3}+4a^2 \ge 4a(abc)^{1/3}$ $((abc)^{1/3}-2a)^2 \ge 0$ This is true by the trivial inequality. Lastly, all the steps are reversible so the given inequality has been proved.

-Themathcanadian

2018 USAMO (Problems • Resources)
Preceded by First Question	Followed by Problem 2
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