## March 2023

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In the Math Corner of my March 2023 Diary I offered the following brainteaser.

Leta,b, andcbe positive real numbers such thata² +b² +c² ≤ 3. Prove that

√(1 +a) + √(1 +b) + √(1 +c) ≥ (√2)(a+b+c)² / 3

When does equality occur?

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**• Solution**

Difficult! Obviously equality occurs when *a* = *b* = *c* = 1; but the general proof eluded me.

Two readers with better math than mine *did* prove the inequality, but they used advanced techniques.

Reader A used Lagrange multipliers, a neat trick in calculus. (However, several users of Twitter, where Reader A posted his proof, found a flaw in it.)

Reader B plunged deeper, resorting to Schur minimization.

At mid-July the sense of the meeting is that a tweeter identifying only as "Hedi H," Twitter handle "@HHHHdi," has come up with the nearest yet to a solution by elementary algebra, although "elementary" here translates as "undergraduate math level."

Hedi H's result depends on two standard inequalities, those of Jensen and Cauchy-Schwarz. Both are highly serviceable in any problem involving algebraic inequalities; the trick is to find how to apply them in some particular case.

I'm curious to see what approach will be taken by the person who posed the problem, when *Math Horizons* publishes the proof later
this year (?).