TV competition

In the competition, 10 contestants answer five questions, one question per round. Anyone who answers correctly will receive as many points as the number of competitors answered incorrectly in that round.

One of the contestants after the contest 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 5 competitors respond correctly?
c. In how many rounds did 3 competitors respond correctly?

Result

O1 =  0
O2 =  2
O3 =  2

Solution:

$116=(10-a) \cdot \ a+(10-b) \cdot \ b+(10-c) \cdot \ c+(10-d) \cdot \ d+ (10-e) \cdot \ e \ \\ 30=(10-a) \cdot \ x+(10-b) \cdot \ y+(10-c) \cdot \ z+(10-d) \cdot \ w +(10-e) \cdot \ q \ \\ a>=b,b>=c, c>=d, d>=e \ \\ a,b,c,d,e \in N \ \\ \ \\ \ \\ a_{1}=5, b_{1}=5, c_{1}=4, d_{1}=3, e_{1}=3 \ \\ x=y=z=w=q=1 \ \\ x,y,z,w,q \in N \ \\ \ \\ 116=(10-5) \cdot \ 5+(10-5) \cdot \ 5+(10-4) \cdot \ 4+(10-3) \cdot \ 3+ (10-3) \cdot \ 3 \ \\ 30=(10-5) \cdot \ 1+(10-5) \cdot \ 1+(10-4) \cdot \ 1+(10-3) \cdot \ 1 +(10-3) \cdot \ 1 \ \\ \ \\ O_{1}=0$
$O_{2}=2$
$O_{3}=2$

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