equals
of relatively prime integers 
and 
such that 
is divisible by 
.
is a solution to the system![\[ xy=6 \qquad \text{and} \qquad x^2y+xy^2+x+y=63, \]](http://latex.artofproblemsolving.com/5/1/1/511f1b14f971d1ca56dda72179d9a26b2a578833.png)
is a square and 
and 
are the midpoints of 
and 
respectively. Then 
,
be a constant. The simultaneous equations
times around a cardboard tube of diameter 
cm, forming a roll 
cm in diameter. Approximate the length of the paper in meters. (Pretend the paper forms 
.
to 
is
be an integer. Carl has 
books arranged on a bookshelf. Each book has a height and a width. No two books have the same height, and no two books have the same width. Initially, the books are arranged in increasing order of height from left to right. In a move, Carl picks any two adjacent books where the left book is wider and shorter than the right book, and swaps their locations. Carl does this repeatedly until no further moves are possible. Prove that regardless of how Carl makes his moves, he must stop after a finite number of moves, and when he does stop, the books are sorted in increasing order of width from left to right.
of non-zero real numbers have the property that each number is the product of the other two?
Y by
Algebra
It is known that
Given this fact, determine the exact value of 
(Source: Mandelbrot)
Counting and Probability
A
wooden cube is painted on all six faces, then cut into 27 unit cubes. One unit cube is randomly selected and rolled. After it is rolled, 
out of the 
faces are visible. What is the probability that exactly one of the five visible faces is painted? (Source: MATHCOUNTS)
Number Theory(This technically isn't the last problem but the last chapter doesn't have challenge problems)
The integer p is a 50-digit prime number. When its square is divided by 120, the remainder is not 1. What is the remainder?
I didn't include geometry because I haven't taken it yet, feel free to post it
Answer these problems and post what you think is the order of difficulty
It is known that
Given this fact, determine the exact value of 
(Source: Mandelbrot)Counting and Probability
A
wooden cube is painted on all six faces, then cut into 27 unit cubes. One unit cube is randomly selected and rolled. After it is rolled, 
out of the 
faces are visible. What is the probability that exactly one of the five visible faces is painted? (Source: MATHCOUNTS)Number Theory(This technically isn't the last problem but the last chapter doesn't have challenge problems)
The integer p is a 50-digit prime number. When its square is divided by 120, the remainder is not 1. What is the remainder?
I didn't include geometry because I haven't taken it yet, feel free to post it
Answer these problems and post what you think is the order of difficulty
This post has been edited 1 time. Last edited by ysn613, May 26, 2025, 7:06 PM
Reason: h
Reason: h
Therefore, the desired expression is equal to 
,
(since its multiplying by 
)
gives us 
,
of them, and there are 
scenarios (one at the bottom and one at the side) for a total of 
,
always gives 
gives 
(if and only if 
,
or 
,
gives 
digits condition is arbitrary, and only prevents 
from being equal to 
or 
as 
and consider 
.
so 
and 
so 
so there are only two possibilities for 
.
leads to 
so 
and 
.
is on 
such that 
and 
Find 
and 
of regular heptagon 
meet at 
.
to 
given 
: A spinner has 5 equals sections. 2 are colored red, 2 are colored blue, and the other is colored yellow. What is the probability of spinning and landing on a blue section, then spinning again and landing on a yellow section?
has side lengths of 
and 
Find the perimeter of the triangle (pretty easy lol)
