Plan ahead for the next school year. Schedule your class today!

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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
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Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

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0 replies
jwelsh
Jul 1, 2025
0 replies
1999 AMC 8 #24
Mrdavid445   17
N an hour ago by thdwlgh1229
When $1999{}^2{}^0{}^0{}^0$ is divided by $5$ , the remainder is

$ \text{(A)}\ 4\qquad\text{(B)}\ 1\qquad\text{(C)}\ 2\qquad\text{(D)}\ 3\qquad\text{(E)}\ 0 $
17 replies
Mrdavid445
Jun 17, 2011
thdwlgh1229
an hour ago
Grandi Series is weird??
sadas123   22
N an hour ago by Not__Infinity
I was just watching this video and I kinda understood what they were saying they said:

$1-1+1-1+1-1+1-1 .... = \frac{1}{2}$

so at first I thought that = 0

but then I realized that if you give an variable such that:

$x= 1-1+1-1+1-1 .....$
then you get
$x = 1-x$
isolating x you get that
$2x = 1$
so
$x= \frac{1}{2}$

is there a way to get the definite answer because I am still confused.
22 replies
sadas123
Yesterday at 12:52 AM
Not__Infinity
an hour ago
Modular Equality with Coprime Integers
pandev3   5
N 3 hours ago by pandev3
Let $(a)_n = a - n \left\lfloor \frac{a}{n} \right\rfloor$, which is equivalent to $a \mod n$ for any integer $a$.

Let $a_1, a_2, a_3, a_4$ and $n$ be positive integers such that $a_i$ is coprime with $n$ for $i = 1, 2, 3, 4$.

It holds that $(k a_1)_n + (k a_2)_n + (k a_3)_n + (k a_4)_n = 2n$ for $k = 1, 2, \dots, n - 1$.

Prove that $(a_1)_n + (a_j)_n = n$ for some $j$ where $2 \leq j \leq 4$.
5 replies
pandev3
Nov 21, 2024
pandev3
3 hours ago
Concurrency of Lines Involving Altitudes and Circumcenters in a Triangle
JackMinhHieu   0
3 hours ago
Hi everyone,
I recently came across an interesting geometry problem that I'd like to share. It involves a triangle inscribed in a circle, altitudes, points on arcs, and a surprising concurrency involving circumcenters. Here's the problem:
Problem:
Let ABC be an acute triangle inscribed in a circle (O). Let the altitudes AD, BE, CF intersect at the orthocenter H.
Points M and N lie on the minor arcs AB and AC of circle (O), respectively, such that MN // AB.
Let I be the circumcenter of triangle NEC, and J be the circumcenter of triangle MFB.
Prove that the lines OD, BI, and CJ are concurrent.

I find the configuration quite elegant, and I'm looking for different ways to approach the problem — whether it's synthetic, coordinate, vector-based, or inversion.
Any ideas, hints, or full solutions are appreciated. Thank you!
0 replies
JackMinhHieu
3 hours ago
0 replies
geo series question
Spacepandamath13   5
N 3 hours ago by laer
$1+1/3+1/9+1/27+1/81+1/243...=?$
5 replies
Spacepandamath13
Today at 1:17 AM
laer
3 hours ago
question
duckShroom   8
N 3 hours ago by littleduckysteve
Hi everyone,

So I'll give you the short version:

- My school's math program is nearly nonexistent

- The highest freshman course is a 2 year long course combining geometry, alg 2, and precalc. (Most people are struggling and drop out, which is because of the horrible orginaization and the fact they moved middle school back a year so the highest middle school course is alg 1)

- In the first year, we essentially skipped geometry and they'll most likely butcher precalc too

- I'm either the best or second best math student in my year and the year above me, and the school knows. Even so, they won't let me skip the 2nd year of the course I'm in to do AP Calc BC

- I know bits and pieces of precalc from comp stuff, but I need to learn standardized precalc on a deep level out of school without spending money. I know alg 2 well (besides some trig stuff) and geo, well, I haven't actually formally learnt it but I basically know it from the school's 'teachings', khan, and comp stuff.

- I also am reading a calc textbook to learn overall concepts and some basic derivates and all. (https://calculusmadeeasy.org/)

So basically, I need a good free online precalc course or textbook. Thanks for listening to my half rant and half question post thing. All help will be appreciated!

P.S: pm me if you want to be in a AIME study forum
8 replies
duckShroom
Jan 7, 2025
littleduckysteve
3 hours ago
2018 preRMO p22, k=sum of no in partition of {1, 2, ..., 20}, good integer
parmenides51   8
N 4 hours ago by cortex_classes
A positive integer $k$ is said to be good if there exists a partition of $ \{1, 2, 3,..., 20\}$ into disjoint proper subsets such that the sum of the numbers in each subset of the partition is $k$. How many good numbers are there?
8 replies
parmenides51
Aug 8, 2019
cortex_classes
4 hours ago
2018 preRMO p20, product of whose digits equals n^2 -15n -27
parmenides51   11
N 4 hours ago by cortex_classes
Determine the sum of all possible positive integers $n, $ the product of whose digits equals $n^2 -15n -27$.
11 replies
parmenides51
Aug 8, 2019
cortex_classes
4 hours ago
Infinite 6's
BlackPanther007   23
N 4 hours ago by cortex_classes
Let $N=6+66+666+....+666..66$, where there are hundred $6's$ in the last term in the sum. How many times does the digit $7$ occur in the number $N$
23 replies
BlackPanther007
Aug 19, 2018
cortex_classes
4 hours ago
Question about Notation
Quaratinium   4
N 4 hours ago by aidan0626
How are we suppose to know if something like $f^5(x)$ wants the 5th derivative, raise the function to the 5th power, or plug the thing back in the function five times?
4 replies
Quaratinium
Jun 28, 2018
aidan0626
4 hours ago
P-Adic MathDash Problem
LilKirb   2
N 4 hours ago by cheltstudent
Let $N = 2^{23} - 36.$ Given that $2^{21} - 9$ is a prime, find the number of nonnegative integers $0 \leq x \leq N$ such that $N$ is a divisor of $\binom{N}{x}.$

Express the answer in the form $a \cdot b^c - d,$ where $a,b,c,d$ are positive integers, $a$ is not divisible by $b,$ and $b$ is as small as possible with $b\neq1$
2 replies
LilKirb
Yesterday at 12:24 PM
cheltstudent
4 hours ago
Just for fun
Alexyuan2017   5
N 4 hours ago by Quadratic_rush
What is the derivative of y=x. see if you can figure it out without knowing calculus :D
5 replies
Alexyuan2017
May 6, 2023
Quadratic_rush
4 hours ago
9 science bowl
valenbb   13
N 4 hours ago by Andrew2019
i would like to know if the math kids on AoPS did science bowl

for me my team made it to nationals but we didn't do that good there, it was still a really fun experience tho :)
13 replies
valenbb
Yesterday at 6:27 PM
Andrew2019
4 hours ago
Calculus-flavored sum
NamelyOrange   4
N 5 hours ago by Quadratic_rush
Evaluate $\sum_{n = 0}^{\infty}\frac{1}{(4n)!}$.

(Source: AOPS user @SomeDumbChild)
4 replies
NamelyOrange
Jun 24, 2025
Quadratic_rush
5 hours ago
Interesting Combinatorics Problem
Ro.Is.Te.   14
N May 29, 2025 by Wolfpierce
Amanda has $1000$ red marbles, $2000$ yellow marbles, $3000$ green marbles, and $4000$ blue marbles. If Amanda takes the marbles one by one without replacing them until the $3999th$ marble. Then the probability that the $4000th$ marble is red is?
14 replies
Ro.Is.Te.
May 23, 2025
Wolfpierce
May 29, 2025
Interesting Combinatorics Problem
G H J
G H BBookmark kLocked kLocked NReply
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Ro.Is.Te.
19 posts
#1
Y by
Amanda has $1000$ red marbles, $2000$ yellow marbles, $3000$ green marbles, and $4000$ blue marbles. If Amanda takes the marbles one by one without replacing them until the $3999th$ marble. Then the probability that the $4000th$ marble is red is?
This post has been edited 2 times. Last edited by Ro.Is.Te., May 23, 2025, 11:51 AM
Reason: LaTeX
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CJB19
178 posts
#2 • 4 Y
Y by Demetri, sbarrack, Pengu14, aidan0626
?
This post has been edited 1 time. Last edited by CJB19, May 23, 2025, 1:01 PM
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sultanine
902 posts
#3
Y by
CJB19 wrote:
?

yeah I kinda agree with u
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SpeedCuber7
1895 posts
#5
Y by
CJB19 wrote:
?

nope

there are 10000 marbles in the bin, not 4000
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CJB19
178 posts
#6
Y by
Oh wait I'm silly
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aidan0626
2068 posts
#7 • 2 Y
Y by Pengu14, BAM10
i'm pretty sure the answer is still correct tho lol
the chance of getting a red marble is the same regardless of how many you've already taken out
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SpeedCuber7
1895 posts
#8
Y by
aidan0626 wrote:
i'm pretty sure the answer is still correct tho lol
the chance of getting a red marble is the same regardless of how many you've already taken out

no it isn't??

if you take out 3999 blues, then the probability is 1000/6001
Z K Y
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aidan0626
2068 posts
#9
Y by
aidan0626 wrote:
i'm pretty sure the answer is still correct tho lol
the chance of getting a red marble is the same regardless of how many you've already taken out

no it isn't??

if you take out 3999 blues, then the probability is 1000/6001

do you understand how probability works
where in the problem did it say 3999 blues are taken out
well i don't want to actually write the whole thing up noooo :(
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SpeedCuber7
1895 posts
#10
Y by
aidan0626 wrote:
aidan0626 wrote:
i'm pretty sure the answer is still correct tho lol
the chance of getting a red marble is the same regardless of how many you've already taken out

no it isn't??

if you take out 3999 blues, then the probability is 1000/6001

do you understand how probability works
where in the problem did it say 3999 blues are taken out
well i don't want to actually write the whole thing up noooo :(

what if they take a red out in one of the 3999 marbles? then the probability must be different
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Aaronjudgeisgoat
935 posts
#11
Y by
im pretty sure post 2 is right.

it doesnt matter what the first 3999 numbers are, just initially choose what the 4000th marble is and nothing else matters.
for example, consider this. what if instead the problem was as follows: Amanda has $1000$ red marbles, $2000$ yellow marbles, $3000$ green marbles, and $4000$ blue marbles. IAmanda takes the marbles one by one, from the last marble to the $6002th$ marble. Then the probability that the $1st$ marble is red is?

same problem, ur removing 3999 marbles, and u wanna find the probability that the first one is red. you can see how this is the same as the previosu problem right? and the answer is still 1/10

ur argument is that it depends on what the 3999 marbles are. and of course, thats true. but in the end, it all averages out because both solutions are the same, except ur looking at the problem in a different way
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ZMB038
444 posts
#12
Y by
I think it’s just $\frac{1}{10}$
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LXC007
176 posts
#13
Y by
sol
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RedChameleon
1257 posts
#14
Y by
The confusion this is giving me is the same confusion I get when contemplating the monty hall problem
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Juran2014
1 post
#15
Y by
.............
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Wolfpierce
37 posts
#16
Y by
Yeah just 1/10 I think I don't know I'm not even in 5th grade
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