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Question

Ten persons numbered 1,2,⋯,10 play a chess tournament, each player playing against every other exactly one game. It is known that no game ends in a draw. Let w1,w2,....,w10 are the number of games won by players 1,2,3,⋯,10, respectively, and l1,l2,⋯,l10 are the number of games lost by the players 1,2,⋯,10, respectively, then,

A
10i=1wi+10i=1li=90
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B
wi+li=9, where i{1,2,,10}
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C
10i=1w2i=81+10i=1l2i
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D
10i=1w2i=10i=1l2i
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Solution

The correct options are A 10∑i=1wi+10∑i=1li=90 B wi+li=9, where i∈{1,2,⋯,10} D 10∑i=1w2i=10∑i=1l2iClearly, each player will play 9 games. And total number of games is 10C2=45. Clearly, wi+li=9 and 10∑i=1wi=10∑i=1li=45 ⇒ wi=9−li⇒w2i=81+l2i−181i ⇒10∑i=1w2i=81×10+10∑i=1l2i−1810∑i=1li=810+10∑i=1l2i−18×45=10∑i=1l2i

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