Complaining 9611

Ondra, Matěj, and Kuba are returning from collecting nuts. They have a total of 120. Matěj complains that Ondra has the most as always. The father orders Ondra to sprinkle it on his Matěj so that the number of nuts doubles. Now Cuba is complaining that he has the most.
At his father's command, Matěj sprinkles Kuba by doubling the number of nuts. Ondra is angry that he has the least of them now. So Cuba sprinkles Ondra by doubling the number of nuts. Now everyone has the same, and finally, they are calm.

How many nuts did each of the boys initially have?

Correct answer:

o =  55
m =  35
k =  30

Step-by-step explanation:

$$\begin{gathered} o+m+k=120 \\ {o}_1=o-m \\ {m}_1=m+m=2m \\ {k}_1=k \\ {m}_2=m_1-k_1=2m-k \\ {k}_2=k_1+k_1=2k_1=2k \\ {o}_2=o_1 \\ {o}_3=o_2+o_2=o_1+o_1=(o-m)+(o-m) \\ {k}_3=k_2-o_2=2k-o_1=2k-(o-m) \\ {m}_3=m_2 \\ o+m+k=120 \\ 2k-(o-m)=120/3 \\ (o-m)+(o-m)=120/3 \\ o+m+k=120 \\ 2\cdot k-(o-m)=120/3 \\ (o-m)+(o-m)=120/3 \\ k+m+o=120 \\ 6k+3m-3o=120 \\ 6m-6o=-120 \\ Pivot:Row1\leftrightarrow Row2 \\ 6k+3m-3o=120 \\ k+m+o=120 \\ 6m-6o=-120 \\ Row2-\frac{1}{6}\cdot Row1\to Row2 \\ 6k+3m-3o=120 \\ 0.5m+1.5o=100 \\ 6m-6o=-120 \\ Pivot:Row2\leftrightarrow Row3 \\ 6k+3m-3o=120 \\ 6m-6o=-120 \\ 0.5m+1.5o=100 \\ Row3-\frac{0.5}{6}\cdot Row2\to Row3 \\ 6k+3m-3o=120 \\ 6m-6o=-120 \\ 2o=110 \\ o=\frac{110}{2}=55 \\ m=\frac{-120+6o}{6}=\frac{-120+6\cdot 55}{6}=35 \\ k=\frac{120-3m+3o}{6}=\frac{120-3\cdot 35+3\cdot 55}{6}=30 \\ k=30 \\ m=35 \\ o=55 \end{gathered}$$

$m=35$

$$\begin{gathered} k=30 \\ skuska/zkouska: \\ {o}_1=o-m=55-35=20 \\ {m}_1=m+m=35+35=70 \\ {k}_1=k=30 \\ {o}_2=o_1=20 \\ {m}_2=m_1-k_1=70-30=40 \\ {k}_2=k_1+k_1=30+30=60 \\ {o}_3=o_2+o_2=20+20=40 \\ {m}_3=m_2=40 \\ {k}_3=k_2-o_2=60-20=40 \end{gathered}$$

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