2010年AIME I 真题:
Problem 1
Maya lists all the positive divisors of . She then randomly selects two distinct divisors from this list. Let
be the probability that exactly one of the selected divisors is a perfect square. The probability
can be expressed in the form
, where
and
are relatively prime positive integers. Find
.
Problem 2
Find the remainder when is divided by
.
Problem 3
Suppose that and
. The quantity
can be expressed as a rational number
, where
and
are relatively prime positive integers. Find
.
Problem 4
Jackie and Phil have two fair coins and a third coin that comes up heads with probability . Jackie flips the three coins, and then Phil flips the three coins. Let
be the probability that Jackie gets the same number of heads as Phil, where
and
are relatively prime positive integers. Find
.
Problem 5
Positive integers ,
,
, and
satisfy
,
, and
. Find the number of possible values of
.
Problem 6
Let be a quadratic polynomial with real coefficients satisfying
for all real numbers
, and suppose
. Find
.
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