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MC III Algebra help
4.20:
I tried using geometry and graphing but that didn't work. I also couldn't figure out how to assign variables for Cauchy-Schwarz.
4.25:
I couldn't figure out what to add/subtract to turn it into Cauchy-Schwarz.
4.20: Try trig substitution: $x=r\cos\theta, y=r\sin\theta$.
4.25: Try the following:
\[ \frac{a}{b+3c}+\frac{b}{8c+4a}+\frac{9c}{3a+2b} = \frac{8a}{8b+24c}+\frac{3b}{24c+12a}+\frac{36c}{12a+8b} . \]
I still can't figure out how to use any of the inequalities to make progress on 4.25 and 4.17.
For 4.17, follow the example of 4.8; instead of adding 1 to each fraction, you can add 1 to the first fraction, add 4 to the second fraction, and add 5 to the third one, and get a factor of $(a+b+c)$.
For 4.25, after the transformation given above, use change of variables:
\[ x = 12a, y=8b, z=24c, \] then the expression becomes \[ \frac{\dfrac{2}{3}x}{y+z} + \frac{\dfrac{3}{8}y}{z+x} + \frac{\dfrac{3}{2}z}{x+y}, \] and follow the same technique as 4.17 and 4.8.These are usually Math Olympiad level questions. However, in recent years more and more questions using these inequalities are getting into the AIME level contests. You don't have to get all of them at once, but it is good to know about these techniques, and try to use them.
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