$a^3+b^3+c^3=(a^2+b^2,+c^2-ab-bc-ac)(a+b+c)+3abc$ = $(a+b+c)^2 -3(ab+bc+ac) +3abc$ $3(1-a)(1-b)(1-c)= 3(1-a-b-c+ab+bc+ac-abc)=3(ab+bc+ac)-3abc$ so $a^3+b^3+c^3+3(1-a)(1-b(1-c)=1-3(ab+bc+ac)+3abc +3(ab+bc+ac)-3abc=1$

$1=(a+b+c)^3=a^3+b^3+c^3+3(a+b)(b+c)(c+a)=a^3+b^3+c^3+3(1-a)(1-b)(1-c).$

Suppose a, b, c are real numbers such that a + b + c = 1. Prove that $$a^3 + b^3 + c^3 + 3(2-a)(2-b)(2-c) \leq 14$$h

sqing wrote: Suppose a, b, c are real numbers such that a + b + c = 1. Prove that $$a^3 + b^3 + c^3 + 3(2-a)(2-b)(2-c) \leq 14$$h \begin{align*}a^3+b^3+c^3+3(2-a)(2-b)(2-c) &=1+3((2-a)(2-b)(2-c)-(1-a)(1-b)(1-c)) \\ &=1+3(7-3(a+b+c)+ab+bc+ca) \\ &= 13+3(ab+bc+ca) \\ &\leq 13+(a+b+c)^2 \\ &= 14 \end{align*}

Suppose a, b, c are positive real numbers such that a + b + c = 1. Prove that $$a^3+b^3+c^3+3(2-a)(2-b)(2-c)\leq 13+\frac{27}{8}(1-a)(1-b)(1-c).$$

Suppose a, b, c are positive real numbers such that a + b + c = 1. Prove that $$a^3+b^3+c^3+3(2-a)(2-b)(2-c)\leq 13+\frac{27}{8}(1-a)(1-b)(1-c) \leq 14$$

Mahmood.sy wrote: $a^3+b^3+c^3=(a^2+b^2,+c^2-ab-bc-ac)(a+b+c)+3abc$ = $(a+b+c)^2 -3(ab+bc+ac) +3abc$ $3(1-a)(1-b)(1-c)= 3(1-a-b-c+ab+bc+ac-abc)=3(ab+bc+ac)-3abc$ so $a^3+b^3+c^3+3(1-a)(1-b(1-c)=1-3(ab+bc+ac)+3abc +3(ab+bc+ac)-3abc=1$ nice

Suppose a, b, c are real numbers such that a + b + c = 1. Prove that $$a^3 + b^3 + c^3 + 3(1-a)(2-b)(2-c) \leq\frac{31}{4}$$$$a^3 + b^3 + c^3 + 3(3-a)(2-b)(2-c) \leq\frac{53}{2}$$$$a^3 + b^3 + c^3 + 3(1-a)(3-b)(3-c) \leq25$$$$a^3 + b^3 + c^3 + 3(3-a)(3-b)(3-c) \leq 57$$

Quidditch wrote: sqing wrote: Suppose a, b, c are real numbers such that a + b + c = 1. Prove that $$a^3 + b^3 + c^3 + 3(2-a)(2-b)(2-c) \leq 14$$h \begin{align*}a^3+b^3+c^3+3(2-a)(2-b)(2-c) &=1+3((2-a)(2-b)(2-c)-(1-a)(1-b)(1-c)) \\ &=1+3(7-3(a+b+c)+ab+bc+ca) \\ &= 13+3(ab+bc+ca) \\ &\leq 13+(a+b+c)^2 \\ &= 14 \end{align*} Amazing

Suppose a, b, c are real numbers such that a + b + c = 1. Prove that $$a^3 + b^3 + c^3 + 3(2-a)(3-b)(4-c) \leq\frac{247}{4}$$

Quidditch wrote: Suppose a, b, c are positive real numbers such that a + b + c = 1. Prove that $$a^3+b^3+c^3+3(2-a)(2-b)(2-c)\leq 13+\frac{27}{8}(1-a)(1-b)(1-c).$$ This is of course, correct, because you have the proof

Suppose a, b, c are real numbers such that a + b + c = 1. Prove that $$a^3 + b^3 + c^3 + 3(\frac{3}{2}-a)(\frac{3}{2}-b)(\frac{3}{2}-c)\leq\frac{39}{8}$$