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For positive real numbers $a_1, a_2, \cdots, a_{100}$, let $$p=\sum_{i=1}^{100} a_i \text{ and } q=\sum_{1 \leq i < j \leq 100} a_ia_j.$$ Then

 

  1. $q=\frac{p^2}{2}$
  2. $q^2 \geq \frac{p^2}{2}$
  3. $q< \frac{p^2}{2}$
  4. none of the above
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C

Use this formula,

$$\left(\sum_{i==1}^{100} a_i\right)^2=\sum_{i==1}^{100} a_i^2 +2\left(\sum_{1\leq i<j\leq100} a_iaj\right)$$

You can prove this by mathematical induction.

Remember that $a_i>0$ for every $i$.

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