数学竞赛题与逻辑优化示例.pdfVIP

TeamRound

1.[2]Findthenumberofpositiveintegersxsthan100forwhich

xxxxxxx

3+5+7+11+13+17+19

isprime.

Answer:0Weclaimthatourintegerisdivisibleby3forallpositiveintegersx.Indeed,wehave

xxxxxxxxxxxxxx

3+5+7+11+13+17+19≡(0)+(−1)+(1)+(−1)+(1)+(−1)+(1)

x

x

≡3[(1)+(−1)]

≡0(mod3).

Itisclearthatforallx≥1,ourintegerisstrictlygreaterthan3,soitwillalwaysbecomposite,

makingouranswer0.

32

2.[4]DeterminethesetofallrealnumberspforwhichthepolynomialQ(x)=x+px−px−1has

threedistinctrealroots.

Answer:First,wenotethat

322

x+px−px−1=(x−1)(x+(p+1)x+1).

Hence,x2+(p+1)x+1hastwodistinctroots.Consequently,thediscriminantofthisequationmust

bepositive,so(2

p+1)40,soeitherp1orp3.However,theproblemspecifiesthatthequadratic

−−

musthavedistinctro