问题描述：
关于北美数学竞赛题..因为实在做不出来,甚至题目的意思也没搞懂,
1.A betting company ofers the possibility to bet on any of the n contestants in a race,with odds a 1,...,a n,respectively.
This means that if you bet X dollars on contestant j (where X is any positive real number),then:
• If contestant j loses,you lose your X dollars.
• If contestant j wins,you get your X dollars back,together with a profit of a jX dollars.
If the company does not choose the odds wisely,it may be possible to bet on all of the contestants in a manner that
guarantees a profit no matter what.For instance,if there are 3 contestants with odds 2,2,and 10,you could bet $1 on
each of the first two and $0.50 on the the third one,guaranteeing a profit no matter who wins the race.
(a) What relation must the real numbers a 1,...,a n satisfy so that it is impossible to guarantee a profit in this way?
(b) What is the largest profit that you can guarantee on a total bet of $1 if the relation is NOT satisfied?
2.The function f is defined on the positive integers by the formulas
f (1) = 1,
f (2n) = 2 f (n) + 2n,
nf (2n + 1) = (2n + 1) f (n)
for all n 1.
(a) Prove that f (n) is always an integer.
(b) For what values of n does the equality f (n) = n hold?(Be sure to prove that no other value works.)
3.(Proposed by Michael Wu,a student at Mathcamp 2009 & 2010.) There are n children equally spaced around a merry-
go-round with n seats,waiting to get on.The children climb onto the merry-go-round one by one (but not necessarily
going in order around the circle),always using the seat in front of them and only taking a seat if it is empty.After one
child climbs on and takes a seat,the merry-go-round rotates 360/n degrees counterclockwise so that each remaining
child is again lined up with a seat.For what values of n is it possible for the children to climb on,in some order,so that
everyone gets a seat?(Do remember to prove both that it's possible for the values you claim and that it's impossible for
all other values.)
1.A betting company ofers the possibility to bet on any of the n contestants in a race,with odds a 1,...,a n,respectively.
This means that if you bet X dollars on contestant j (where X is any positive real number),then:
• If contestant j loses,you lose your X dollars.
• If contestant j wins,you get your X dollars back,together with a profit of a jX dollars.
If the company does not choose the odds wisely,it may be possible to bet on all of the contestants in a manner that
guarantees a profit no matter what.For instance,if there are 3 contestants with odds 2,2,and 10,you could bet $1 on
each of the first two and $0.50 on the the third one,guaranteeing a profit no matter who wins the race.
(a) What relation must the real numbers a 1,...,a n satisfy so that it is impossible to guarantee a profit in this way?
(b) What is the largest profit that you can guarantee on a total bet of $1 if the relation is NOT satisfied?
2.The function f is defined on the positive integers by the formulas
f (1) = 1,
f (2n) = 2 f (n) + 2n,
nf (2n + 1) = (2n + 1) f (n)
for all n 1.
(a) Prove that f (n) is always an integer.
(b) For what values of n does the equality f (n) = n hold?(Be sure to prove that no other value works.)
3.(Proposed by Michael Wu,a student at Mathcamp 2009 & 2010.) There are n children equally spaced around a merry-
go-round with n seats,waiting to get on.The children climb onto the merry-go-round one by one (but not necessarily
going in order around the circle),always using the seat in front of them and only taking a seat if it is empty.After one
child climbs on and takes a seat,the merry-go-round rotates 360/n degrees counterclockwise so that each remaining
child is again lined up with a seat.For what values of n is it possible for the children to climb on,in some order,so that
everyone gets a seat?(Do remember to prove both that it's possible for the values you claim and that it's impossible for
all other values.)
问题解答：
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