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what number have you memorized perfect squares
ellenssim 18
N
2 hours ago
by lavender_cloud
Up to what number have you memorized perfect squares, and how often does it help you in solving problems?
18 replies
1234th Post!
PikaPika999 264
N
2 hours ago
by PikaPika999
I hit my 1234th post! (I think I missed it, I'm kinda late, :oops_sign:)
But here's a puzzle for you all! Try to create the numbers 1 through 25 using the numbers 1, 2, 3, and 4! You are only allowed to use addition, subtraction, multiplication, division, and parenthesis. If you're post #1, try to make 1. If you're post #2, try to make 2. If you're post #3, try to make 3, and so on. If you're a post after 25, then I guess you can try to make numbers greater than 25 but you can use factorials, square roots, and that stuff. Have fun!
1:
But here's a puzzle for you all! Try to create the numbers 1 through 25 using the numbers 1, 2, 3, and 4! You are only allowed to use addition, subtraction, multiplication, division, and parenthesis. If you're post #1, try to make 1. If you're post #2, try to make 2. If you're post #3, try to make 3, and so on. If you're a post after 25, then I guess you can try to make numbers greater than 25 but you can use factorials, square roots, and that stuff. Have fun!
1:

264 replies
Find the amount of possible values from the expression
Darealzolt 2
N
2 hours ago
by Elephant12
Find the amount of possible values from
For all non zero integers
![\[
\frac{|a|}{a}+\frac{|b|}{b}+\frac{|c|}{c}
\]](http://latex.artofproblemsolving.com/8/4/a/84a1a990f8bbde7670e9154dd9a46bb18c14d547.png)

2 replies
Standard Factoring/Manipulation
Darealzolt 1
N
2 hours ago
by Andyluo
Find the value of the following expression
![\[
\frac{4000^2+4050^2}{(2000+2025)^2+(2000-2025)^2}
\]](http://latex.artofproblemsolving.com/b/9/9/b991bda876c059439578dd1eeb54b1f8010df755.png)
1 reply
Worst math problems
LXC007 7
N
Today at 5:10 AM
by buddyram
What is the most egregiously bad problem or solution you have encountered in school?
7 replies

Challenge: Make every number to 100 using 4 fours
CJB19 267
N
Today at 3:56 AM
by Marshall_Huang
I've seen this attempted a lot but I want to see if the AoPS community can actually do it. Using ONLY 4 fours and math operations, make as many numbers as you can. Try to go in order. I'll start:

















267 replies
Trivial Factoring MathCounts Question? 2025 MathCounts National Sprint Round #29
ilikemath247365 35
N
Today at 2:56 AM
by Andyluo
What is the value of the expression below?
.

35 replies
The daily problem!
Leeoz 214
N
Today at 1:51 AM
by maxamc
Every day, I will try to post a new problem for you all to solve! If you want to post a daily problem, you can! :)
Please hide solutions and answers, hints are fine though! :)
Problems usually get harder throughout the week, so Sunday is the easiest and Saturday is the hardest!
Past Problems!
Please hide solutions and answers, hints are fine though! :)
Problems usually get harder throughout the week, so Sunday is the easiest and Saturday is the hardest!
Past Problems!
[quote=March 21st Problem]Alice flips a fair coin until she gets 2 heads in a row, or a tail and then a head. What is the probability that she stopped after 2 heads in a row? Express your answer as a common fraction.[/quote]
Answer
[quote=March 22nd Problem]In a best out of 5 math tournament, 2 teams compete to solve math problems, with each of the teams having a 50% chance of winning each round. The tournament ends when one team wins 3 rounds. What is the probability that the tournament will end before the fifth round? Express your answer as a common fraction.[/quote]
Answer
[quote=March 23rd Problem]The equations of
and
intersect at the point
. What is the value of
?[/quote]
Answer
[quote=March 24th Problem]Anthony rolls two fair six sided dice. What is the sum of all the different possible products of his rolls?[/quote]
Answer
[quote=March 25th Problem]If
, find the value of
.[/quote]
Answer
[quote=March 26th Problem]There is a group of 6 friends standing in line. However, 3 of them don't want to stand next to each other. In how many ways can they stand in line?[/quote]
Answer
[quote=March 27th Problem]Two real numbers,
and
are chosen from 0 to 1. What is the probability that their positive difference is more than
?[/quote]
Answer
[quote=March 28th Problem]What is the least possible value of the expression
?[/quote]
Answer
[quote=March 29th Problem]How many integers from 1 to 2025, inclusive, contain the digit “1”?[/quote]
Answer
[quote=April 3rd Problem]In
families, there are
children respectively. If a random child from any of the families is chosen, what is the probability that the child has
siblings? Express your answer as a common fraction.[/quote]
Answer
[quote=April 5th Problem]A circle with a radius of 3 units is centered at the point (0,0) on the coordinate plane. How many lattice points, points which both of the coordinates are integers, are strictly inside the circle?[/quote]
Answer
[quote=April 6th Problem]If the probability that someone asks for a problem is
, find the probability that out of
people, exactly
of them ask for a problem.[/quote]
Answer
[quote=April 8th Problem]Find the value of
such that
.[/quote]
Answer
[quote=April 9th Problem]In unit square
, point
lies on diagonal
such that
. Find the area of quadrilateral
.[/quote]
Answer
[quote=April 10th Problem]An function in the form
has
,
, and
. Find the value of
.[/quote]
Answer
Answer

[quote=March 22nd Problem]In a best out of 5 math tournament, 2 teams compete to solve math problems, with each of the teams having a 50% chance of winning each round. The tournament ends when one team wins 3 rounds. What is the probability that the tournament will end before the fifth round? Express your answer as a common fraction.[/quote]
Answer

[quote=March 23rd Problem]The equations of




Answer

[quote=March 24th Problem]Anthony rolls two fair six sided dice. What is the sum of all the different possible products of his rolls?[/quote]
Answer

[quote=March 25th Problem]If


Answer

[quote=March 26th Problem]There is a group of 6 friends standing in line. However, 3 of them don't want to stand next to each other. In how many ways can they stand in line?[/quote]
Answer

[quote=March 27th Problem]Two real numbers,



Answer

[quote=March 28th Problem]What is the least possible value of the expression

Answer

[quote=March 29th Problem]How many integers from 1 to 2025, inclusive, contain the digit “1”?[/quote]
Answer

[quote=April 3rd Problem]In



Answer

[quote=April 5th Problem]A circle with a radius of 3 units is centered at the point (0,0) on the coordinate plane. How many lattice points, points which both of the coordinates are integers, are strictly inside the circle?[/quote]
Answer

[quote=April 6th Problem]If the probability that someone asks for a problem is



Answer

[quote=April 8th Problem]Find the value of


Answer

[quote=April 9th Problem]In unit square





Answer

[quote=April 10th Problem]An function in the form





Answer

214 replies
Last challenge problems in the books
ysn613 18
N
Today at 12:24 AM
by valenbb
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


Counting and Probability
A



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
18 replies
Math with Connect4 Boards
Math-lover1 8
N
Today at 12:16 AM
by Math-lover1
Hi! So I was playing Connect4 with my friends the other day and I wondered: how many "legal" arrangements of Connect4 can be reached at the ending position?
We assume that we do not stop the game when there is a four in a row, and we have 21 red pieces and 21 yellow pieces. We also drop the pieces one by one into a standard 7 by 6 board. We can start the game with any color piece.
https://en.wikipedia.org/wiki/Connect_Four
Initial Thoughts
Attempt to use one-to-one correspondences
We assume that we do not stop the game when there is a four in a row, and we have 21 red pieces and 21 yellow pieces. We also drop the pieces one by one into a standard 7 by 6 board. We can start the game with any color piece.
https://en.wikipedia.org/wiki/Connect_Four
Initial Thoughts
This problem seems easy at first; the number of arrangments is simply
However, I quickly saw that some boards

OOOOOOO
OOOOOOO
OOOOOOO
OOOOOOO
OOOOOOO
OOOOOOO
were impossible to construct by just dropping pieces one by one like a normal game.OOOOOOO
OOOOOOO
OOOOOOO
OOOOOOO
OOOOOOO
Attempt to use one-to-one correspondences
After I realized that my Initial Thoughts weren't going to work, I tried to use one-to-one correspondences. I represented the columns as ABCDEFG from left to right and represented dropping the red/yellow pieces as a string of length 21 of these letters. This seemed to solve my problem, but new roadblocks popped up.
Roadblock 1 There is more than one way to represent a certain configuration using this correspondence. A quick example
Roadblock 2 Even if we didn't overcount, we still need to account for the fact that the total number of A, B, C... over both of the strings have to each equal 7. The amount of cases (1 A goes to the red pieces, 6 As go to the yellow pieces,...) would be very difficult to calculate, even using a computer.
Roadblock 1 There is more than one way to represent a certain configuration using this correspondence. A quick example
red pieces fill all the left 3 columns, yellow pieces fill all the right 3 columns
shows that we overcount some configurations by using this method.Roadblock 2 Even if we didn't overcount, we still need to account for the fact that the total number of A, B, C... over both of the strings have to each equal 7. The amount of cases (1 A goes to the red pieces, 6 As go to the yellow pieces,...) would be very difficult to calculate, even using a computer.
8 replies
