An Inequality Made Up Of Greek Alphabets
Let 0 < α , β , γ < 2 1 be real numbers with α + β + γ = 1 . The minimum value of δ which satisfies the inequality
α 3 + β 3 + γ 3 + 4 α β γ ≤ δ
can be expressed in the form b a , where a and b are positive coprime integers. Find a + b .
The answer is 41.
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2 solutions
x = y is due to the fact that x and y can be interchanged or symmetrical in equations. If calculus were to be applied to determine, then it shall agree.
α 3 + β 3 + γ 3 + 4 α β γ = α 2 + β 2 + γ 2 − ( α β + β γ + γ α ) + 7 α β γ = α 2 + β 2 + γ 2 − 2 1 − ( α 2 + β 2 + γ 2 ) + 7 α β γ = 2 1 + ( α 2 + β 2 + γ 2 ) + 7 α β γ
Moderator note:
This solution is currently incomplete.
Your solution is incomplete. Alternatively, If α, β, γ are each 1/3, they meet the condition α +β+γ = 1, and so α3+ β3+ γ3 +4 αβ γ = 7/27 7 + 27 = 34
Impressive problem! The alternative solution which I submitted is incorrect, and the solution indeed seems to be 41as noted below. If α, β, γ ≤ 1/2, then choose α ≈ 1/2, and β, γ (by symmetry) each 1/4, they meet the condition α +β+γ = 1, and so α3+ β3+ γ3 +4 αβ γ = 9/32 9+32 = 41
from caushy-shawzez inequality we get 7/27
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Note that you are asked for the minimum value.
2 7 7 is an upper bound, but it is not the least upper bound. For example, we could also say that
α 3 + β 3 + γ 3 + 4 α β γ ≤ 2 3 1 + 2 3 1 + 2 3 1 + 4 2 3 1 = 8 7 .
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@Calvin Lin But 1/2+1/2+1/2 is 3/2 and doesn't satisfy the condition a+b+c=1
same ans from cauchy inequality 7\27
@cccc dddd Isn't the solution incomplete?
@cccc dddd Can you complete your solution? Thanks.
a^3 + b^3 + g^3 = S^3 - 3 P I S + 3 P allows simplification of a^3 + b^3 + g^3 + 4 a b g into
(3 a + b)(a + 3 b) - (a + b)(7 a b + 3) + 1 OR
-7 a^2 b - 7 a b^2 + 3 a^2 + 10 a b + 3 b^2 - 3 a - 3 b + 1
Partial differentiation gives stationary points for maximum, minimum and saddle point; (1/3, 1/3, 1/3) for 7/27 and (1/7, 3/7, 3/7) for 13/49 failed to be the correct answers. Stationary points are not always the extremes within certain coverage. Basically, a + b = 4/7 xor a = b are first found.
Minimum extreme of d or minimum of maximum of R.H.S. to L.H.S. gives a guaranteed coverage.
a^3 + b^3 + g^3 + 4 a b g <= 9/32 = 0.28125 = d
Means d is not f(a, b, g) and therefore can equal to the extreme for exact simple fraction.
Applying computer to search for a = 0 to 0.5 and b = 0 to 0.5 as for d which take 0.499999999999999999999 of L.H.S. as 0.5 of R.H.S., modify into a = b = 0 to 0.5 with steps of 0.0001 for example, set memory for maximum (and also minimum) as the similar idea to thermometer for extremes of temperature, maximum of 0.28125 and minimum of 0.25 are obtainable. With z for d or R.H.S.:
z = (3 x + y)(x + 3 y) - (x + y)(7 x y + 3) + 1
First, found that x = y;
Second, found that maximum of z = 0.28125 at x = 0.25 and minimum of z = 0.25 at x = 0.5 where steps of 0.0001 can be adjusted to guess for logical exact maximum and exact minimum. Concerning this question, no proper mathematics is available for obtaining the answer because boundary at edges which are not stationary points are decided by g = 0.5 and g = 0 respectively as being specified artificially.
Answer: d = 0.28125 = 9/32 (exact) for 9 + 32 = 41.