-
Prove that
Let 
Prove that
Let 
Prove that
,
,
be positive real numbers such that 
.![$$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$](http://latex.artofproblemsolving.com/2/0/7/207934bca53047858a3820f37705171ff0abfd45.png)
with 
.![\[
\frac{a^5}{b^2 + 2c^3} + \frac{2b^5}{3c + a^6} + \frac{c^7}{a + b^4} \geq 2.
\]](http://latex.artofproblemsolving.com/f/f/5/ff5043705025592aeefc55a764a8b7170e13ff4b.png)
Y by
Scientists found a new species of bird called “N-coloured rainbow”. They also found out 3 interesting facts about the bird’s life: 1) every day, N-coloured rainbow is coloured in one of N colors.
2) every day, the color is different from yesterday (not every previous day, just yesterday).
3) there are no four days i, j, k, l in the bird’s life such that i<j<k<l with colours a, b, c, d respectively for which a=c ≠ b=d.
Find the greatest possible age (in days) of this bird as a function of N.
2) every day, the color is different from yesterday (not every previous day, just yesterday).
3) there are no four days i, j, k, l in the bird’s life such that i<j<k<l with colours a, b, c, d respectively for which a=c ≠ b=d.
Find the greatest possible age (in days) of this bird as a function of N.
