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Systems of Equations

System of Equations: Level 3 Challenges


There are two containers named AA and BB. Container AA has milk in it, container BB has water (same amount as of milk in container AA) in it. Some amount of milk is transferred from container AA to container BB and then , the same amount is transferred from the container BB to the container AA.

Let container AA has xx amount of water in it and container BB has yy amount of milk in it.

Which one option is correct?

Note : Finally the amount in both the containers is same as it was initially.

Find the number of real solutions (x,y,z)(x,y,z) of

{(x+y)3=z(y+z)3=x(z+x)3=y \begin{cases} (x+y)^{3}=z \\ (y+z)^{3}=x \\ (z+x)^{3}=y \end{cases}

What is the largest integer n1000 n \leq 1000 , such that there exist 2 non-negative integers (a,b)(a, b) satisfying

n=a2+b2ab1? n = \frac{ a^2 + b^2 } { ab - 1 } ?

Hint: (a,b)=(0,0) (a,b) = (0,0) gives us 02+020×01=0 \frac{ 0^2 + 0^2 } { 0 \times 0 - 1 } = 0, so the answer is at least 0. 0 .

{(1+x)(1+x2)(1+x4)=1+y7(1+y)(1+y2)(1+y4)=1+x7\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)(x, y) satisfy the above equations?

{bcdefa=4acdefb=9abdefc=16abcefd=14abcdfe=19abcdef=116\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,fa,b,c,d,e,f satisfy the formulas above. Find the value of (a+c+e)(b+d+f)(a+ c + e) - (b + d + f)


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