Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Quadratic Equation

Olympiad workbook · IOQM · Quadratic Equation

Let x, y, z be complex numbers such that aligned &...

Let x,y,zx, y, z be complex numbers such that xy+z+yz+x+zx+y=9x2y+z+y2z+x+z2x+y=64x3y+z+y3z+x+z3x+y=488\begin{aligned} & \frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=9 \\ & \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}=64 \\ & \frac{x^3}{y+z}+\frac{y^3}{z+x}+\frac{z^3}{x+y}=488 \end{aligned} If xyz+yzx+zxy=mn\frac{x}{y z}+\frac{y}{z x}+\frac{z}{x y}=\frac{m}{n} where m,nm, n are positive integers with GCD⁡(m,n)=1\operatorname{GCD}(m, n)=1, find m+nm+n.

Correct answer

16

Step-by-step solution

xy+z+yz+x+zx+y=9(x+y+z)(1y+z+1z+x+1x+y)=12\begin{aligned} & \frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=9 \\ & (x+y+z)\left(\frac{1}{y+z}+\frac{1}{z+x}+\frac{1}{x+y}\right)=12 \end{aligned} Let x+y+z=S1x+y+z=S_1 1y+z+1z+x+1x+y=12S1\frac{1}{y+z}+\frac{1}{z+x}+\frac{1}{x+y}=\frac{12}{S_1} Now, (x+y+z)(xy+z+yz+x+zx+y)=9S1(x+y+z)\left(\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}\right)=9 S_1 64+S1=9S1⇒S1=8(x+y+z)(x2y+z+y2z+x+z2x+y)=64S1488+(x2+y2+z2)=64×8x2+y2+z2=24xy+yz+zx=64−242=20\begin{aligned} & 64+S_1=9 S_1 \\ & \Rightarrow \quad S_1=8 \\ & (x+y+z)\left(\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}\right)=64 S_1 \\ & 488+\left(x^2+y^2+z^2\right)=64 \times 8 \\ & x^2+y^2+z^2=24 \\ & x y+y z+z x=\frac{64-24}{2}=20 \end{aligned}  Now, 1x+y+1y+z+1z+x=128=3218−x+18−y+18−z=32(8−x)(8−y)(8−z)=56⇒xyz=104\begin{aligned} & \text { Now, } \frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{z+x}=\frac{12}{8}=\frac{3}{2} \\ & \frac{1}{8-x}+\frac{1}{8-y}+\frac{1}{8-z}=\frac{3}{2} \\ & \quad(8-x)(8-y)(8-z)=56 \\ & \Rightarrow \quad x y z=104 \end{aligned} So, xyz+yzx+zxy=24104=313=mn\frac{x}{y z}+\frac{y}{z x}+\frac{z}{x y}=\frac{24}{104}=\frac{3}{13}=\frac{m}{n} ⇒m+n=16\Rightarrow \quad m+n=16
Practice Quadratic Equation on Quantrex Academy →

More from Quadratic Equation

Suppose the prime numbers p and q satisfy q^2+3 p=197 p^2+q . Write (q)/(p) as l+(m)/(n)… Olympiad workbookLet X=\ -5,-4,-3,-2,-1,0,1,2,3,4,5\ and S=\ (a, b) ∈ X × X: x^2+a x+b. and x^3+b x+a have… Olympiad workbookIf x= 2 + 3 + 6 is a root of x^4+a x^3+b x^2+c x+d=0 where a, b, c, d are integers, what… Olympiad workbookSuppose that P is the polynomial of least degree with integer coefficients such that P( 7… Olympiad workbookLet A=\ m: m. an integer and the roots of x^2+m x + 2020 = 0 are positive integers) and… Olympiad workbookLet x, y, z be complex numbers such that aligned & (x)/(y+z)+(y)/(z+x)+(z)/(x+y)=9 &… Olympiad workbookSuppose a, b are positive real numbers such that a a +b b =183, a b +b a =182 . Find… Olympiad workbookLet f(x)=x^2+a x+b . If for all nonzero real x f(x+(1)/(x))=f(x)+f((1)/(x)) and the roots… Olympiad workbook Full Quadratic Equation list All Olympiad workbook PYQs