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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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Annoying Probability Math Problem
RYang2   10
N 20 minutes ago by EthanNg6
I was working in my math textbook(not the AoPS one) when I came across this math problem:

Determine if the events are dependent or independent.
1. Drawing a red and a blue marble at the same time from a bag containing 6 red and 4 blue marbles
2.(omitted)

I thought it was independent, since the events happen at the same time, but the textbook answer said dependent.
Can someone help me understand(or prove the textbook wrong)?
10 replies
RYang2
Mar 14, 2018
EthanNg6
20 minutes ago
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Ants and a rope
my_favourite_number_5   4
N 24 minutes ago by EthanNg6
Please solve this problem.

There are n-ants in a rope. The rope is 10 meters long. The velocity of each ants is 1 meter/second. If any 2 ants are touched in the rope, they will inverse their way. If any ants reaches in any corner of the rope, it will fell down. Prove that all ants will fell down in 10 seconds.
4 replies
my_favourite_number_5
Mar 11, 2021
EthanNg6
24 minutes ago
J
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Hello friends
bibidi_skibidi   8
N 38 minutes ago by BS2012
Now unfortunately I don't know the difficulty of the problems posted here but I'll try to replicate:

Bob has 20 apples and 19 oranges. How many ways can he split the fruits between 7 people if each person must have at least 1 apple and 2 oranges?

After looking at the other posters I realized just how bashy this is

Also I can only edit this message for now since new AoPS users can only send 6 messages every day
8 replies
+1 w
bibidi_skibidi
Yesterday at 4:04 AM
BS2012
38 minutes ago
J
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2025 MATHCOUNTS State Hub
SirAppel   549
N 2 hours ago by UberPiggy
Previous Years' "Hubs": (2022) (2023) (2024)Please Read

Now that it's April and we're allowed to discuss ...
[list=disc]
[*] CA: 43 (45 44 43 43 43 42 42 41 41 41)
[*] NJ: 43 (45 44 44 43 39 42 40 40 39 38) *
[*] NY: 42 (43 42 42 42 41 40)
[*] TX: 42 (43 43 43 42 42 40 40 38 38 38)
[*] MA: 41 (45 43 42 41)
[*] WA: 41 (41 45 42 41 41 41 41 41 41 40) *
[*]VA: 40 (41 40 40 40)
[*] FL: 39 (42 41 40 39 38 37 37)
[*] IN: 39 (41 40 40 39 36 35 35 35 34 34)
[*] NC: 39 (42 42 41 39)
[*] IL: 38 (41 40 39 38 38 38)
[*] OR: 38 (44 40? 38 38)
[*] PA: 38 (41 40 40 38 38 37 36 36 34 34) *
[*] MD: 37 (43 39 39 37 37 37)
[*] AZ: 36 (40? 39? 39 36)
[*] CT: 36 (44 38 38 36 35 35 34 34 34 33 33)
[*] MI: 36 (39 41 41 36 37 37 36 36 36 36) *
[*] MN: 36 (40 36 36 36 35 35 35 34)
[*] CO: 35 (41 37 37 35 35 35 ?? 31 31 30) *
[*] GA: 35 (38 37 36 35 34 34 34 34 34 33)
[*] OH: 35 (41 37 36 35)
[*] AR: 34 (46 45 35 34 33 31 31 31 29 29)
[*] NV: 34 (41 38 ?? 34)
[*] TN: 34 (38 ?? ?? 34)
[*] WI: 34 (40 37 37 34 35 30 28 29 29 29) *
[*] HI: 32 (35 34 32 32)
[*] NH: 31 (42 35 33 31 30)
[*] DE: 30 (34 33 32 30 30 29 28 27 26? 24)
[*] SC: 30 (33 33 31 30)
[*] IA: 29 (33 30 31 29 29 29 29 29 29 29 29 29) *
[*] NE: 28 (34 30 28 28 27 27 26 26 25 25)
[*] SD: 22 (30 29 24 22 22 22 21 21 20 20)
[/list]
Cutoffs Unknown

* means that CDR is official in that state.

Notes

For those asking about the removal of the tiers, I'd like to quote Jason himself:
[quote=peace09]
learn from my mistakes
[/quote]

Help contribute by sharing your state's cutoffs!
549 replies
SirAppel
Apr 1, 2025
UberPiggy
2 hours ago
J
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MVT question
mqoi_KOLA   11
N Yesterday at 5:03 PM by Rohit-2006
Let \( f \) be a function which is continuous on \( [0,1] \) and differentiable on \( (0,1) \), with \( f(0) = f(1) = 0 \). Assume that there is some \( c \in (0,1) \) such that \( f(c) = 1 \). Prove that there exists some \( x_0 \in (0,1) \) such that \( |f'(x_0)| > 2 \).
11 replies
mqoi_KOLA
Apr 10, 2025
Rohit-2006
Yesterday at 5:03 PM
J
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polynomial with real coefficients
Peter   9
N Yesterday at 4:58 PM by Rohit-2006
Source: IMC 1998 day 1 problem 5
Let $P$ be a polynomial of degree $n$ with only real zeros and real coefficients.
Prove that for every real $x$ we have $(n-1)(P'(x))^2\ge nP(x)P''(x)$. When does equality occur?
9 replies
Peter
Nov 1, 2005
Rohit-2006
Yesterday at 4:58 PM
J
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Integral Inequality with better bound than usual cauchy
StarLex1   5
N Yesterday at 4:47 PM by RobertRogo
Source: a friend of mine
Suppose that $f:[0,1]\rightarrow\mathbb{R}$ ,is a convex function and $f(0) = 0 $
Prove that
\[\left(\int^{1}_{0}f(x)dx\right)^2\leq \dfrac{3}{4}\int^1_{0}f^2(x)dx\]
Note
5 replies
StarLex1
Yesterday at 7:58 AM
RobertRogo
Yesterday at 4:47 PM
J
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Kyiv Taras Shevchenko University Mechmat Competition 1974-92 book
rogue   1
N Yesterday at 4:34 PM by rogue
Source: Kyiv Taras Shevchenko University Mechmat Competition
Kyiv Taras Shevchenko University Mechmat Competition 1974-92 book [in Ukrainian]
https://mechmat.knu.ua/wp-content/uploads/2024/03/mechmat1974-92.pdf
1 reply
rogue
Sep 2, 2023
rogue
Yesterday at 4:34 PM
J
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Romanian National Olympiad 1996 – Grade 11 – Problem 4
Filipjack   2
N Yesterday at 2:28 PM by loup blanc
Source: Romanian National Olympiad 1996 – Grade 11 – Problem 4
Let $A,B,C,D \in \mathcal{M}_n(\mathbb{C}),$ $A$ and $C$ invertible. Prove that if $A^k B = C^k D$ for any positive integer $k,$ then $B=D.$
2 replies
Filipjack
Apr 13, 2025
loup blanc
Yesterday at 2:28 PM
J
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Simple limit with standard recurrence
AndreiVila   3
N Yesterday at 1:50 PM by ZeroAlephZeta
Source: Romanian District Olympiad 2025 11.1
Consider the sequence $(a_n)_{n\geq 1}$ given by $a_1=1$ and $a_{n+1}=\frac{a_n}{1+\sqrt{1+a_n}}$, for all $n\geq 1$. Show that $$\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n} = \lim_{n\rightarrow\infty}\sum_{k=1}^n \log_2(1+a_k)=2.$$Mathematical Gazette
3 replies
AndreiVila
Mar 8, 2025
ZeroAlephZeta
Yesterday at 1:50 PM
J
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Putnam 1999 A6
djmathman   3
N Yesterday at 1:38 PM by zhoujef000
The sequence $(a_n)_{n\geq 1}$ is defined by $a_1=1,a_2=2,a_3=24,$ and, for $n\geq 4,$ \[a_n=\dfrac{6a_{n-1}^2a_{n-3}-8a_{n-1}a_{n-2}^2}{a_{n-2}a_{n-3}}.\] Show that, for all $n$, $a_n$ is an integer multiple of $n$.
3 replies
djmathman
Dec 22, 2012
zhoujef000
Yesterday at 1:38 PM
J
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Convex geometry
ILOVEMYFAMILY   2
N Yesterday at 1:02 PM by alexheinis
1) Find all closed convex sets with nonempty interior that have exactly one supporting hyperplane in the plane.

2) Find all closed convex sets with nonempty interior that have exactly two supporting hyperplane in the plane.

2 replies
ILOVEMYFAMILY
Yesterday at 4:12 AM
alexheinis
Yesterday at 1:02 PM
J
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Pyramid packing in sphere
smartvong   1
N Yesterday at 12:04 PM by smartvong
Source: own
Let $A_1$ and $B$ be two points that are diametrically opposite to each other on a unit sphere. $n$ right square pyramids are fitted along the line segment $\overline{A_1B}$, such that the apex and altitude of each pyramid $i$, where $1\le i\le n$, are $A_i$ and $\overline{A_iA_{i+1}}$ respectively, and the points $A_1, A_2, \dots, A_n, A_{n+1}, B$ are collinear.

(a) Find the maximum total volume of $n$ pyramids, with altitudes of equal length, that can be fitted in the sphere, in terms of $n$.

(b) Find the maximum total volume of $n$ pyramids that can be fitted in the sphere, in terms of $n$.

(c) Find the maximum total volume of the pyramids that can be fitted in the sphere as $n$ tends to infinity.

Note: The altitudes of the pyramids are not necessarily equal in length for (b) and (c).
1 reply
smartvong
Apr 13, 2025
smartvong
Yesterday at 12:04 PM
J
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Putnam 2021 B3
awesomemathlete   6
N Yesterday at 11:38 AM by HacheB2031
Let $h(x,y)$ be a real-valued function that is twice continuously differentiable throughout $\mathbb{R}^2$, and define
\[ \rho (x,y)=yh_x -xh_y . \]Prove or disprove: For any positive constants $d$ and $r$ with $d>r$, there is a circle $S$ of radius $r$ whose center is a distance $d$ away from the origin such that the integral of $\rho$ over the interior of $S$ is zero.
6 replies
awesomemathlete
Dec 5, 2021
HacheB2031
Yesterday at 11:38 AM
J
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J
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Nice Combinatorics Problem
RabtejKalra   10
N Apr 12, 2025 by sadas123
A number is considered happy if it contains at least one digit exactly twice. For instance, the numbers 2020 and 2024 are happy, but the numbers 2019 and 2022 are not. How many happy counting numbers are there that are less than 10,000?
10 replies
RabtejKalra
Apr 11, 2025
sadas123
Apr 12, 2025
J
Nice Combinatorics Problem
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RabtejKalra
63 posts
RabtejKalra
#1 h  Apr 11, 2025, 10:16 PM
Y by
A number is considered happy if it contains at least one digit exactly twice. For instance, the numbers 2020 and 2024 are happy, but the numbers 2019 and 2022 are not. How many happy counting numbers are there that are less than 10,000?
Z K Y
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HacheB2031
367 posts
HacheB2031
#2 h  Apr 11, 2025, 10:32 PM
Y by
ez sol
This post has been edited 1 time. Last edited by HacheB2031, Apr 11, 2025, 10:36 PM
Reason: imagine having a computational error wouldnt be me
Z K Y
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Bummer12345
132 posts
Bummer12345
#3 h  Apr 11, 2025, 10:38 PM
Y by
@above there can also be numbers with a digit that repeats more than twice
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HacheB2031
367 posts
HacheB2031
#4 h  Apr 11, 2025, 10:42 PM
Y by
Bummer12345 wrote:
@above there can also be numbers with a digit that repeats more than twice

oops yeah i didn't count $0043$ and stuff let me fix that
HacheB2031 wrote:
idoit sol
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RabtejKalra
63 posts
RabtejKalra
#5 h  Apr 11, 2025, 11:03 PM
Y by
No, 0043 and stuf like that would just be considered 43, so they're sad numbers
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maxamc
540 posts
maxamc
#6 h  Apr 11, 2025, 11:05 PM
Y by
sol
This post has been edited 1 time. Last edited by maxamc, Apr 12, 2025, 12:19 AM
Reason: mistake
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iwastedmyusername
78 posts
iwastedmyusername
#7 h  Apr 12, 2025, 12:00 AM
Y by
maxamc wrote:
sol

You overcounted 243 4 digit numbers (aabb is the same as a permutation of bbaa). So I think the actual answer should be 4626-243=4383
Z K Y
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maxamc
540 posts
maxamc
#8 h  Apr 12, 2025, 12:18 AM
Y by
iwastedmyusername wrote:
maxamc wrote:
sol

You overcounted 243 4 digit numbers (aabb is the same as a permutation of bbaa). So I think the actual answer should be 4626-243=4383

oops, edited my post
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sadas123
1214 posts
sadas123
#9 h  Apr 12, 2025, 2:08 AM
Y by
Bashy 3 minute solution

I took a bit of inspiration from you @maxamc
This post has been edited 2 times. Last edited by sadas123, Apr 12, 2025, 1:23 PM
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maxamc
540 posts
maxamc
#10 h  Apr 12, 2025, 3:21 AM
Y by
sadas123 wrote:
Bashy 3 minute solution

I took a bit of inspiration from you @maxamc

You overcounted 243 4 digit numbers (aabb is the same as a permutation of bbaa). So I think the actual answer should be 4626-243=4383
Z K Y
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sadas123
1214 posts
sadas123
#11 h  Apr 12, 2025, 1:22 PM
Y by
maxamc wrote:
sadas123 wrote:
Bashy 3 minute solution

I took a bit of inspiration from you @maxamc

You overcounted 243 4 digit numbers (aabb is the same as a permutation of bbaa). So I think the actual answer should be 4626-243=4383

oh woops wait I am dumb :( I should have saw that updating my thingie
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