2013年AIME II 真题:
Problem 1
Suppose that the measurement of time during the day is converted to the metric system so that each day has metric hours, and each metric hour has
metric minutes. Digital clocks would then be produced that would read
just before midnight,
at midnight,
at the former
AM, and
at the former
PM. After the conversion, a person who wanted to wake up at the equivalent of the former
AM would set his new digital alarm clock for
, where
,
, and
are digits. Find
.
Problem 2
Positive integers and
satisfy the condition
Find the sum of all possible values of
.
Problem 3
A large candle is centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes
seconds to burn down the first centimeter from the top,
seconds to burn down the second centimeter, and
seconds to burn down the
-th centimeter. Suppose it takes
seconds for the candle to burn down completely. Then
seconds after it is lit, the candle's height in centimeters will be
. Find
.
Problem 4
In the Cartesian plane let and
. Equilateral triangle
is constructed so that
lies in the first quadrant. Let
be the center of
. Then
can be written as
, where
and
are relatively prime positive integers and
is an integer that is not divisible by the square of any prime. Find
.
Problem 5
In equilateral let points
and
trisect
. Then
can be expressed in the form
, where
and
are relatively prime positive integers, and
is an integer that is not divisible by the square of any prime. Find
.
Problem 6
Find the least positive integer such that the set of
consecutive integers beginning with
contains no square of an integer.
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