Algebra

Symmetric Polynomials

Symmetric Polynomials: Level 4 Challenges

         

Let a=20142014a=2014^{2014}, b=20152015b=2015^{2015},c=20162016c=2016^{2016} . Find the value of 12015aa+2015ab+2015ac+12015ba+2015bb+2015bc+12015ca+2015cb+2015cc \frac{1}{2015^{a-a}+2015^{a-b}+2015^{a-c}} \\ +\frac{1}{2015^{b-a}+2015^{b-b}+2015^{b-c}} \\ +\frac{1}{2015^{c-a}+2015^{c-b}+2015^{c-c}}

{a1+b1+c1=λa2+b2+c2=λa3+b3+c3=λ\displaystyle \begin{cases} { a }^{ 1 }+{ b }^{ 1 }+{ c }^{ 1 }=\lambda \\ { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }=\lambda \\ { a }^{ 3 }+{ b }^{ 3 }+{ c }^{ 3 }=\lambda \end{cases}

If λZ+\lambda \in \mathbb{{Z}^{+}} satisfying the system of equations above with abc=5!abc=5!, determine λ\lambda.

Note: "!!" represents factorial, not exclamatory sign.

x,yx, y and zz are complex numbers satisfying

{x1+y1+z1=1x2+y2+z2=2x3+y3+z3=3 \begin{cases} x^1+y^1+z^1 & = 1\\ x^2 + y^2 + z^2 & = 2 \\ x^3 + y^3 + z^3 & = 3 \\ \end{cases}

The value of x4+y4+z4 x^4 + y^4 + z^4 can be expressed as ab \frac {a}{b} , where a a and bb are positive coprime integers. What is the value of a+b a +b ?

This problem is proposed by Harshit.

α+β+γ=6α3+β3+γ3=87(α+1)(β+1)(γ+1)=33\large \begin{aligned}\alpha+\beta+\gamma&=6 \\\alpha^3+\beta^3+\gamma^3&=87\\ (\alpha+1)(\beta+1)(\gamma+1)&=33 \end{aligned}

Suppose α\alpha, β\beta, and γ\gamma are complex numbers that satisfy the system of equations above.

If 1α+1β+1γ=mn\frac1\alpha+\frac1\beta+\frac1\gamma=\tfrac mn for positive coprime integers mm and nn, find m+nm+n.

If the roots of p(x)=x3+3x2+4x8p(x) = x^3 + 3x^2 + 4x - 8 are a\color{#D61F06}{a}, b\color{#3D99F6}{b} and c\color{#69047E}{c}, what is the value of

a2(1+a2)+b2(1+b2)+c2(1+c2)?\color{#D61F06}{a}^2 \big(1 + \color{#D61F06}{a}^2\big) + \color{#3D99F6}{b}^2 \big(1 + \color{#3D99F6}{b}^2\big) + \color{#69047E}{c}^2 \big(1 + \color{#69047E}{c}^2\big)?

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