2018年AIME I 真题:
Problem 1
Let be the number of ordered pairs of integers
with
and
such that the polynomial
can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when
is divided by
.
Problem 2
The number can be written in base
as
, can be written in base
as
, and can be written in base
as
, where
. Find the base-
representation of
.
Problem 3
Kathy has red cards and
green cards. She shuffles the
cards and lays out
of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy, but RRRGR will not. The probability that Kathy will be happy is
, where
and
are relatively prime positive integers. Find
.
Problem 4
In and
. Point
lies strictly between
and
on
and point
lies strictly between
and
on
so that
. Then
can be expressed in the form
, where
and
are relatively prime positive integers. Find
.
Problem 5
For each ordered pair of real numbers satisfying
there is a real number
such that
Find the product of all possible values of
.
Problem 6
Let be the number of complex numbers
with the properties that
and
is a real number. Find the remainder when
is divided by
.
以下是我们为您整理的真题试卷,扫码即可免费领取完整版:
更多AIME 历年真题+真题详解
扫码添加顾问即可免费领取