Systems of Equations
Let container \(A\) has \(x\) amount of water in it and container \(B\) has \(y\) amount of milk in it.
Which one option is correct?
Note : Finally the amount in both the containers is same as it was initially.
by
Sandeep Bhardwaj
Find the number of real solutions \((x,y,z)\) of
\[ \begin{cases} (x+y)^{3}=z \\ (y+z)^{3}=x \\ (z+x)^{3}=y \end{cases} \]
by
Souryajit Roy
What is the largest integer \( n \leq 1000 \), such that there exist 2 non-negative integers \((a, b)\) satisfying
\[ n = \frac{ a^2 + b^2 } { ab - 1 } ? \]
Hint: \( (a,b) = (0,0) \) gives us \( \frac{ 0^2 + 0^2 } { 0 \times 0 - 1 } = 0\), so the answer is at least \( 0 .\)
by
Calvin Lin
\[\begin{cases} (1+x)(1+x^2)(1+x^4) = 1+y^7 \\ (1+y)(1+y^2)(1+y^4) = 1+x^7 \end{cases} \]
How many ordered pairs of real numbers \((x, y)\) satisfy the above equations?
by
Sharky Kesa
\[\left\{ \begin{array}{l} \frac{{bcdef}}{a} = 4\\ \frac{{acdef}}{b} = 9\\ \frac{{abdef}}{c} = 16\\ \frac{{abcef}}{d} = \frac{1}{4}\\ \frac{{abcdf}}{e} = \frac{1}{9}\\ \frac{{abcde}}{f} = \frac{1}{{16}} \end{array} \right.\] Positive numbers \[a,b,c,d,e,f\] satisfy the formulas above. Find the value of \[(a+ c + e) - (b + d + f)\]
by
Jason Snow