Let $n_1,n_2,\cdots ,n_{1998}$ be positive integers such that
$$\ n_1^2 + n_2^2 + \cdots + n_{1997}^2 = n_{1998}^2 $$
Show that at least two of the numbers are even.
An old jr Balkan problem
- Thamim Zahin
- Posts:98
- Joined:Wed Aug 03, 2016 5:42 pm
I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.
- Thamim Zahin
- Posts:98
- Joined:Wed Aug 03, 2016 5:42 pm
Re: An old jr Balkan problem
It is trivial that the condition is true when RHS is even. And if the RHS is odd, then LHS have to contain even number of even number. So, only hard case is to proof that it is impossible to have all integer odd in LHS.
We know that every odd number can be written as $2k+1$ form. So every square of a odd number can be written as $(2k+1)^2=4k^2+4k+1$.
Now, we can write LHS as
$$ \displaystyle\sum_{i=1}^{1997} 4k_i^2 +\sum_{i=1}^{1997}4k_i+1997=4k^2_{1998}+4k_{1998}+1 $$
$$ \Rightarrow \displaystyle\sum_{i=1}^{1997} 4k_i^2 +\sum_{i=1}^{1997}4k_i+1996=4k^2_{1998}+4k_{1998} $$
Dividing by $4$ we get,
$$ \displaystyle\sum_{i=1}^{1997} k_i^2 +\sum_{i=1}^{1997}k_i+499=k^2_{1998}+k_{1998} $$
Now,$\displaystyle\sum_{i=1}^{1997} k_i^2$ and $\displaystyle\sum_{i=1}^{1997} k_i$ has same parity and $499$ is odd. So LHS is odd. But $4k^2_{1998}+4k_{1998}$ is even. $Contradiction$.
So there has to be more even numbers of even numbers in LHS.
$[Done]$
We know that every odd number can be written as $2k+1$ form. So every square of a odd number can be written as $(2k+1)^2=4k^2+4k+1$.
Now, we can write LHS as
$$ \displaystyle\sum_{i=1}^{1997} 4k_i^2 +\sum_{i=1}^{1997}4k_i+1997=4k^2_{1998}+4k_{1998}+1 $$
$$ \Rightarrow \displaystyle\sum_{i=1}^{1997} 4k_i^2 +\sum_{i=1}^{1997}4k_i+1996=4k^2_{1998}+4k_{1998} $$
Dividing by $4$ we get,
$$ \displaystyle\sum_{i=1}^{1997} k_i^2 +\sum_{i=1}^{1997}k_i+499=k^2_{1998}+k_{1998} $$
Now,$\displaystyle\sum_{i=1}^{1997} k_i^2$ and $\displaystyle\sum_{i=1}^{1997} k_i$ has same parity and $499$ is odd. So LHS is odd. But $4k^2_{1998}+4k_{1998}$ is even. $Contradiction$.
So there has to be more even numbers of even numbers in LHS.
$[Done]$
I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.