2010年AIME II 真题:
Problem 1
Let be the greatest integer multiple of
all of whose digits are even and no two of whose digits are the same. Find the remainder when
is divided by
.
Problem 2
A point is chosen at random in the interior of a unit square
. Let
denote the distance from
to the closest side of
. The probability that
is equal to
, where
and
are relatively prime positive integers. Find
.
Problem 3
Let be the product of all factors
(not necessarily distinct) where
and
are integers satisfying
. Find the greatest positive integer
such that
divides
.
Problem 4
Dave arrives at an airport which has twelve gates arranged in a straight line with exactly feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Dave walks
feet or less to the new gate be a fraction
, where
and
are relatively prime positive integers. Find
.
Problem 5
Positive numbers ,
, and
satisfy
and
. Find
.
Problem 6
Find the smallest positive integer with the property that the polynomial
can be written as a product of two nonconstant polynomials with integer coefficients.
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