<s>[INST] Minimum and maximum There are five real numbers $a,b,c,d,e$ such that
$$a + b + c + d + e = 7$$$$a^2+b^2+c^2+d^2+e^2 = 10$$
How can we find the maximum and minimum possible values of any one of the numbers ?
Source
 [/INST] Maximum value that can be taken by one of the variables is $\frac{9}{5}$ and the minimum value is $1$.
This can be argued based on symmetry as follows (and I hope I have not overlooked anything in my symmetry argument).
Say we want to find the extreme values of $a$. By symmetry when $a$ reaches the extremum the remaining variables have to be equal. So when $a$ takes the extremum $b=c=d=e$.
Hence, $a + 4b = 7$ and $a^2 + 4b^2 = 10$ i.e. $a^2 + 4 (\frac{7-a}{4})^2 = 10$ 
i.e. $a^2 + \frac{(a-7)^2}{4} = 10$ i.e. $5a^2 -14a + 49 = 40 $ i.e. $5a^2 - 14a + 9 =0$.
Hence, $(5a-9)(a-1)=0$.
So, minimum is $a=1$ and maximum is $a = \frac{9}{5}$.
The round about way would be to use Lagrange multiplier technique and set up 7 equations in 7 unknowns and solve it.
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