2015年AIME II 真题:
Problem 1
Let be the least positive integer that is both
percent less than one integer and
percent greater than another integer. Find the remainder when
is divided by
.
Problem 2
In a new school, percent of the students are freshmen,
percent are sophomores,
percent are juniors, and
percent are seniors. All freshmen are required to take Latin, and
percent of sophomores,
percent of the juniors, and
percent of the seniors elect to take Latin. The probability that a randomly chosen Latin student is a sophomore is
, where
and
are relatively prime positive integers. Find
.
Problem 3
Let be the least positive integer divisible by
whose digits sum to
. Find
.
Problem 4
In an isosceles trapezoid, the parallel bases have lengths and
, and the altitude to these bases has length
. The perimeter of the trapezoid can be written in the form
, where
and
are positive integers. Find
.
Problem 5
Two unit squares are selected at random without replacement from an grid of unit squares. Find the least positive integer
such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than
.
Problem 6
Steve says to Jon, "I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form for some positive integers
and
. Can you tell me the values of
and
?"
After some calculations, Jon says, "There is more than one such polynomial."
Steve says, "You're right. Here is the value of ." He writes down a positive integer and asks, "Can you tell me the value of
?"
Jon says, "There are still two possible values of ."
Find the sum of the two possible values of .
Problem 7
Triangle has side lengths
,
, and
. Rectangle
has vertex
on
, vertex
on
, and vertices
and
on
. In terms of the side length
, the area of
can be expressed as the quadratic polynomial
Then the coefficient , where
and
are relatively prime positive integers. Find
.
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