Algebra

Systems of Equations

System of Equations: Level 4 Challenges

         

x1x2+y1y2+z1z2=kxyz(1x2)(1y2)(1z2)\dfrac{x}{1-x^2}+\dfrac{y}{1-y^2}+\dfrac{z}{1-z^2} \\ =\dfrac{kxyz}{(1-x^2)(1-y^2)(1-z^2)}

If x,y,x,y, and zz are real numbers satisfying xy+yz+xz=1xy+yz+xz=1, find the value of kk for which the equation above holds true.

If a,ba, b and cc are non-zero real numbers, that satisfy the equations a2+b2+c2=1 a^2 + b^2 + c^2 = 1 and a(1b+1c)+b(1c+1a)+c(1a+1b)=3a\left(\frac {1}{b} + \frac {1}{c}\right) + b\left(\frac {1}{c} + \frac {1}{a}\right) + c\left( \frac {1}{a} + \frac {1}{b}\right) = -3 , how many possible values are there for a+b+ca + b + c?

How many unordered triplets (x,y,z)(x,y,z) are in the range [0,2][0,2] and satisfy the system of equations: {2x24x+2=y2y24y+2=z2z24z+2=x\left\{\begin{array}{l}2x^2-4x+2=y\\ 2y^2-4y+2=z\\ 2z^2-4z+2=x\end{array}\right.

Details and Assumptions:

An unordered triplet means that (1,2,3)(1,2,3) is the same as (3,2,1)(3,2,1) or (1,3,2)(1,3,2).

The positive reals x,yx,y and zz satisfy the given equations:

x2+xy+y23=25x^{2}+xy+\frac{y^{2}}{3}=25

z2+y23=9z^{2}+\frac{y^{2}}{3}=9

z2+zx+x2=16z^{2}+zx+x^{2}=16

Find the value of (xy+2yz+3xz)2(xy+2yz+3xz)^{2}.

xx satisfies the equation x+xxx=199100xx+x\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}=\dfrac{199}{100}\sqrt{\dfrac{x}{x+\sqrt{x}}}

Also, its value can be expressed in the form mn\frac{m}{n}, where mm and nn are coprime positive integers, find m+nm+n.

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