Try this beautiful problem from Algebra PRMO 2019 based on smallest positive value
Each of the numbers \(x_1, x_2,……….x_{101}\) is \(±1\). What is the smallest positive value of \(\displaystyle\sum_{1\leq i \leq j \leq {101}} x_i x_j\) ?
Algebra
Integer
sum
Answer:\(10\)
PRMO-2019, Problem 13
Pre College Mathematics
\(S=\displaystyle\sum_{1\leq i \leq j \leq {101}} x_i x_j\) .
we have \((x_1+x_2+x_3+....+x_{101})^2={x_1}^2+{x_2}^2+.....+{x_{101}}^2+2S\)
\(\Rightarrow 2S\)=\((\displaystyle\sum_{i=1}^{101} {x_i})^2-\displaystyle\sum_{j=1}^{101} {x_j}^2\)
Can you now finish the problem ..........
Since we have \(x_i=\pm 1\) so \({x_i}^2=1\)
so \(2S=(\displaystyle\sum_{i=1}^{101} {x_i})^2-{101}\)
Since \(\displaystyle\sum_{i=1}^{101} {x_i}\) will be an integer
so \((\displaystyle\sum_{i=1}^{101} {x_i})^2\) will be a perfect square .
For smalll positive \(S\), \((\displaystyle\sum_{i=1}^{101} {x_i})^2\)must be smallest perfect square greater than \({101}\)
So \((\displaystyle\sum_{i=1}^{101} {x_i})^2={121}\)
\(\Rightarrow (\displaystyle\sum_{i=1}^{101} {x_i})\)=\({11}\) or \({-11}\)
Can you finish the problem........
We can verify that the desired sum can be achieved by putting \(45\) \(x_i\)’s to be –1 and \(56\) \(x_i\)’s to be \(1\) So, \(2S = 121 – 101 = 20\)
\(\Rightarrow s=10\)
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