User:Cxsmi
Contents
About Me
Hi! I'm just another guy who happens to enjoy math. I often pop onto the AOPS wiki and look for problems to solve, and I sometimes even write solutions for them! I've starred ⭐ a few of my favorite solutions below; please feel free to take a look at any of them. Thanks for visiting my user page, and enjoy your stay!
Solutions
AIME
AMC 8
- 2012 AMC 8 Problem 19 Solution 6 ⭐
- 2002 AMC 8 Problem 17 Solution 3
- 2007 AMC 8 Problem 20 Solution 8
- 2018 AMC 8 Problem 23 Solution 5 ⭐
- 2016 AMC 8 Problem 13 Solution 3
- 2017 AMC 8 Problem 9 Solution 2
- 2012 AMC 8 Problem 20 Solution 7
- 2010 AMC 8 Problem 25 Solution 3
AJHSME
- 1997 AJHSME Problem 22 Solution 1
- 1985 AJHSME Problem 1 Solution 2
- 1985 AJHSME Problem 24 Solution 2 ⭐
- 1985 AJHSME Problem 2 Solution 5
AHSME
- 1950 AHSME Problem 40 Solution 2
- 1950 AHSME Problem 41 Solution 2
- 1972 AHSME Problem 16 Solution 2 ⭐
- 1950 AHSME Problem 45 Solution 3
AMC 12
Significant Problems
Here are some problems that, to me, have been significant on my math journey. This section is mainly for myself, but please please feel free to look at the problems if you're interested.
- 2017 AMC 10A Problem 19 - First AMC 10 Solution of Difficulty 2 or Higher
- 2007 AMC 8 Problem 25 - First AMC 8 Final Five Solution
- 1984 AIME Problem 1 - First AIME Solution
- 2005 AMC 12B Problem 16 - First AMC 12 Solution of Difficulty 2.5 or Higher
- 2016 AMC 10A Problem 21 - First AMC 10 Final Five Solution
- 1987 AIME Problem 11 - First AIME Solution of Difficulty 4 or Higher
- 2017 AMC 12A Problem 23 - First AMC 12 Final Five Solution
Problems
I enjoy writing problems when I see concepts that interest me. I've written a few below; please feel free to solve them! Also, feel free to add solutions or change any mistakes you see. Note: Unless otherwise stated, all questions are in the AIME format -- that is, the answer is an integer between 1 and 999.
Problem 1
Find the least positive integer that satisfies the following. The notation
represents the greatest integer less than or equal to
.
Problem 2
Bob has received a test. He must match a list of words to a list of
definitions such that each word matches to exactly one definition and vice versa. He does not know any of these words, so he will guess randomly. If he does so, he will expect to get
matches. Find the remainder when
is divided by
.
Problem 3
A sequence is defined recursively as
and
for . The value of
can be written in the form
for positive integers
and
such that
is maximized. What is the remainder when
is divided by
?
Solutions
These are solutions for the problems above.
Problem 1 Solutions
Solution 1
We split the condition into two separate conditions, as listed below.
Rearranging the conditions, we find that
Recalling that where
represents the fractional part of
, we rewrite once more.
We now gain some valuable insight. From , we find that
must divide
. From
, we find that
cannot divide both
and
. It is impossible for
to divide only
of
and
, as this would make
false. It must be that
divides neither
nor
. For both this and
to be true simultaneously, we must have that if
, then
. By inspection, this occurs when
.We now test the factors of
to see if we can find a smaller value. As both
and
are congruent to
mod
,
is not a valid solution. However, with
,
, while
. Clearly,
, so our final answer is
.
Problem 2 Solutions
Solution 1
The answer is . I have yet to write a solution.
Problem 3 Solutions
Solution 1
The answer is . I have yet to write a solution.