Systems of Equations

System of Equations: Level 5 Challenges


{4x2+1x=5y2+1y=6z2+1zxyz=x+y+z\begin{cases} 4\dfrac{\sqrt{x^{2} + 1}}{x} = 5\dfrac{\sqrt{y^{2} + 1}}{y}=6\dfrac{\sqrt{z^{2} + 1}}{z} \\ xyz = x + y + z\end{cases}

If x=ab,y=cde,z=fgx = \dfrac{\sqrt{a}}{b} , y = c\dfrac{\sqrt{d}}{e} , z = f\sqrt{g} satisfy the above system of equations, then find (a+b++g) (a + b + \ldots + g ).

  • The expression is in simplest terms, meaning that all of the variables are positive integers, a,d,g a, d, g are square free and c,ec, e are coprime.

Let aa and bb be positive integers such that a ba\ge\ b anda+1b+b+1a\frac{a+1}{b}+\frac{b+1}{a}is also a positive integer. Find sum of all possible values of aa which are less than 1000.

The system of equations 4a4+36a2b2+9b4=20a2+30b21 4a^4+36a^2b^2+9b^4=20a^2+30b^2-1 2a3b+3ab3=5ab2a^3b+3ab^3=5ab has nn nonnegative (meaning both aa and bb are nonnegative) real solutions. Let these solutions be (a1,b1),(a2,b2),(an,bn)(a_1,b_1),(a_2,b_2),\cdots (a_n,b_n). The value of the sum i=1nai+bi\sum_{i=1}^n a_i+b_i can be expressed in the form a+bcd\dfrac{a+b\sqrt{c}}{d} where gcd(a,b,d)=1\gcd (a,b,d)=1 and cc is squarefree. Find a+b+c+da+b+c+d.

A positive real number is given. In each move, we can either add 33 to it, subtract 33 to it, multiply it by 33 and divide it by 33. The sum of all numbers such that after exactly 33 moves, the original number comes back is ab\frac{a}{b}, where aa and bb are relatively prime positive integers. Find a+ba + b.

Find the largest possible value of x3+y3+z3x^3+y^3+z^3 for real x, y, z,x,\ y,\ z, such that {xyz2=64y128xx2yz=32y32z3xy2z=128x64z \begin{cases} xyz^2=-64y-128x\\ x^2yz=-32y-32z\\ 3xy^2z=128x-64z \end{cases}


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