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anemone
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Prove that if $\displaystyle \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=1$, then $\displaystyle \frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}=0$
Proving the sum equals 0 means showing that the result of adding all the numbers in the sum together is equal to 0.
This means that the sum of all the numbers in the given sum is equal to 1.
Proving that the sum equals 0 helps to show that the two sums are inverses of each other. This means that the numbers in one sum can be used to cancel out and result in the other sum, making them equal.
One strategy is to use algebraic manipulation, where you rearrange the terms in the sum to show that they add up to 0. Another strategy is to use the properties of addition, such as the commutative and associative properties, to rearrange the terms and show that they equal 0.
In most cases, proving the sum equals 0 is sufficient to show that it is the inverse of a sum that equals 1. However, in certain mathematical systems, such as modular arithmetic, there may be exceptions where the sums do not equal 0 and 1, but rather their equivalents in that system.