Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2008 iTest Problems/Problem 90

Revision as of 09:30, 20 April 2022 by Samrocksnature (talk | contribs) (→‎Solution)

Problem

For $a,b,c$ positive reals, let $N=\dfrac{a^2+b^2}{c^2+ab}+\dfrac{b^2+c^2}{a^2+bc}+\dfrac{c^2+a^2}{b^2+ca}$. Find the minimum value of $\lfloor 2008N\rfloor$.

Solution 1

By the Trivial Inequality (with equality happening if $a = b$), \begin{align*} a^2 - 2ab + b^2 &\ge 0 \\ a^2 + b^2 &\ge 2ab. \end{align*} Add $2c^2$ to both sides and use the reciprocal property to get \begin{align*} a^2 + b^2 + 2c^2 &\ge 2ab + 2c^2 \\ \frac{1}{2ab+2c^2} &\ge \frac{1}{a^2 + b^2 + 2c^2}. \end{align*} Since $2a^2 + 2b^2 \ge 0$, multiplying both sides by this value would not change the inequality sign, and doing so results in \begin{align*} \frac{a^2+b^2}{ab+c^2} &\ge \frac{2a^2 + 2b^2}{a^2+b^2+2c^2} \\ &\ge 2 \left( \frac{a^2+b^2}{a^2+c^2+b^2+c^2} \right). \end{align*} By using similar steps, we find that $N \ge 2 \left( \frac{a^2+b^2}{a^2+c^2+b^2+c^2} + \frac{b^2+c^2}{b^2+a^2+c^2+a^2} + \frac{a^2+c^2}{a^2+b^2+c^2+b^2}\right).$


Let $x = a^2+b^2$, $y = b^2+c^2$, and $z = a^2+c^2$, making $N \ge 2\left(\frac{x}{y+z} + \frac{y}{x+z} + \frac{z}{x+y}\right)$. Note that $\frac{x}{y+z} + \frac{y}{x+z} + \frac{z}{x+y} = (x+y+z)(\frac{1}{y+z} + \frac{1}{x+z} + \frac{1}{x+y}) - 3$. By the Cauchy-Schwarz Inequality, $(y+z+x+z+x+y)(\frac{1}{y+z} + \frac{1}{x+z} + \frac{1}{x+y}) \ge 9$, so $\frac{x}{y+z} + \frac{y}{x+z} + \frac{z}{x+y} \ge \frac32$. Equality happens if $(y+z)^2 = (x+z)^2 = (x+y)^2$, which is possible if $x = y = z$. If $x = y = z$, then $a = b = c$.


Therefore, the minimum value of $N$ (which happens if $a = b = c$) is $2 \cdot \frac32 = 3$, so the minimum value of $\lfloor 2008N\rfloor$ is $\boxed{6024}$.

Solution 2 (Cauchy-Schwarz Spam)

Note that By C-S, \[N=\sum \frac{a^2}{c^2+ab} \geq \frac{(2a+2b+2c)^2}{2c^2+2b^2+2a^2+2ab+2bc+2ac}=\frac{4(a+b+c)^2}{(a+b+c)^2+a^2+b^2+c^2}.\] Thus, reciprocating and C-S gives \[\frac{1}{N} \leq \frac{(a+b+c)^2+a^2+b^2+c^2}{4(a+b+c)^2}=\frac{1}{4}+\sum_{\text{cyc}} \frac{a^2}{4(a+b+c)^2} \geq \frac{1}{4}+\frac{(a+b+c)^2}{3 \cdot 4(a+b+c)^2}=\frac{1}{4}+\frac{1}{12}=\frac{1}{3}.\] Reciprocating once again gives $N \geq 3,$ with $N=3$ clearly obtained at $a=b=c,$ therefore the minimum is \[\lfloor 2008N\rfloor=\boxed{6024}.\]

See Also

2008 iTest (Problems)
Preceded by:
Problem 89 		Followed by:
Problem 91
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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • 61 • 62 • 63 • 64 • 65 • 66 • 67 • 68 • 69 • 70 • 71 • 72 • 73 • 74 • 75 • 76 • 77 • 78 • 79 • 80 • 81 • 82 • 83 • 84 • 85 • 86 • 87 • 88 • 89 • 90 • 91 • 92 • 93 • 94 • 95 • 96 • 97 • 98 • 99 • 100
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2008_iTest_Problems/Problem_90&oldid=173501"