JEEOlympiad
Let x, y, z be positive real numbers with x + y + z = 1. Find the…
Question
Let x, y, z be positive real numbers with x + y + z = 1. Find the minimum value of (1 + 1/x)(1 + 1/y)(1 + 1/z).
✓ Verified answer: 64checked by our engine — not a guess
Step-by-step solution
Since x + y + z = 1, write 1 + 1/x = (x + 1)/x = (2x + y + z)/x...
more cleanly, 1 + 1/x = (x + y + z + x)/x...
Use the substitution 1 + 1/x = (x+1)/x with x+1 = (x+y+z) + x = 2x + y + z.
The product is P = ∏ (2x + y + z)/(x y z).
By AM-GM the symmetric expression is minimized when x = y = z = 1/3.
Then each factor is 1 + 3 = 4, so P = 4^3 = 64.
Convexity/symmetry confirms this is the global minimum on the open simplex (the value blows up to infinity near the boundary).
Minimum value is 64.
Final answer64
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