The following is known:
$$\sum\limits_{i=1}^{50} x_i = 1250, \sum\limits_{i=1}^{50} x_i^2 = 45000, \sum\limits_{i=1}^{50} y_i = 3100, \sum\limits_{i=1}^{50} y_i^2 = 250000$$
It is also known that there is a strong positive correlation between $x$ and $y$. Does this mean that $\sum\limits_{i=1}^{50} x_i y_i$ must necessarily be greater than 45000?
My reasoning is that, since $\overline{x}=25$ and $\overline{y}=62$, because of a strong correlation, $\sum\limits_{i=1}^{50} x_i y_i \ge 77500$. However, we cannot deduce the exact sum of $\sum\limits_{i=1}^{50} x_i y_i,$ can we?
The product moment correlation coefficient $r$ is defined by:
$r=\frac{\Sigma xy-n\bar x\bar y}{\sqrt{\left (\Sigma x^2-n\bar x^2\right )\left (\Sigma y^2-n\bar y^2\right)}}$
To have 'strong positive correlation' we can say that $r$ must be greater than some critical value $r_c$. We also know that $r\le 1$
So $r_c \le \frac{\Sigma xy-n\bar x\bar y}{\sqrt{\left (\Sigma x^2-n\bar x^2\right )\left (\Sigma y^2-n\bar y^2\right)}}\le 1$
Thus $r_c \sqrt{\left (\Sigma x^2-n\bar x^2\right )\left (\Sigma y^2-n\bar y^2\right)}\le \Sigma xy-n\bar x\bar y\le \sqrt{\left (\Sigma x^2-n\bar x^2\right )\left (\Sigma y^2-n\bar y^2\right)}$
or $n\bar x\bar y +r_c \sqrt{\left (\Sigma x^2-n\bar x^2\right )\left (\Sigma y^2-n\bar y^2\right)}\le \Sigma xy \le n\bar x\bar y + \sqrt{\left (\Sigma x^2-n\bar x^2\right )\left (\Sigma y^2-n\bar y^2\right)}$
You are right that we can't say exactly what $\Sigma xy$ is, but we can give it bounds.