TV competition

Ten contestants in the competition answer five questions, one per round. Anyone who answers correctly will receive as many points as the number of competitors who answered incorrectly in that round.

After the contest, one of the contestants said: We got 116 points in total, of which I was 30.

a. In how many rounds did all competitors respond correctly?
b. In how many rounds did at least five competitors respond correctly?
c. In how many rounds did three competitors respond correctly?

Correct answer:

O₁ =  0
O₂ =  2
O₃ =  2

Step-by-step explanation:

$$\begin{gathered} 116=(10-a)\cdot a+(10-b)\cdot b+(10-c)\cdot c+(10-d)\cdot d+(10-e)\cdot e \\ 30=(10-a)\cdot x+(10-b)\cdot y+(10-c)\cdot z+(10-d)\cdot w+(10-e)\cdot q \\ a\text{geqb},b\text{geqc},c\text{geqd},d\text{geqe} \\ a,b,c,d,e\in N \\ {a}_1=5,b_1=5,c_1=4,d_1=3,e_1=3 \\ x=y=z=w=q=1 \\ x,y,z,w,q\in N \\ 116=(10-5)\cdot 5+(10-5)\cdot 5+(10-4)\cdot 4+(10-3)\cdot 3+(10-3)\cdot 3 \\ 30=(10-5)\cdot 1+(10-5)\cdot 1+(10-4)\cdot 1+(10-3)\cdot 1+(10-3)\cdot 1 \\ {O}_1=0 \end{gathered}$$

$O_2=2$

$O_3=2$

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