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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jun 2, 2025
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!

[list][*]MATHCOUNTS/AMC 8 Basics
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For those interested in Olympiad level training in math, computer science, physics, and chemistry, be sure to enroll in our WOOT courses before August 19th to take advantage of early bird pricing!

Summer camps are starting this month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have a transformative summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

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0 replies
jlacosta
Jun 2, 2025
0 replies
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Divisors of number
RagvaloD   6
N 7 minutes ago by Assassino9931
Source: All Russian Olympiad 2017,Day2,grade 11,P7
There is number $N$ on the board. Every minute Ivan makes next operation: takes any number $a$ written on the board,
erases it, then writes all divisors of $a$ except $a$( Can be same numbers on the board). After some time on the board there are $N^2$ numbers.
For which $N$ is it possible?
6 replies
RagvaloD
May 1, 2017
Assassino9931
7 minutes ago
J
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Beware the degeneracies!
Rijul saini   6
N 38 minutes ago by sansgankrsngupta
Source: India IMOTC 2025 Day 1 Problem 1
Let $a,b,c$ be real numbers satisfying $$\max \{a(b^2+c^2),b(c^2+a^2),c(a^2+b^2) \} \leqslant 2abc+1$$Prove that $$a(b^2+c^2)+b(c^2+a^2)+c(a^2+b^2) \leqslant 6abc+2$$and determine all cases of equality.

Proposed by Shantanu Nene
6 replies
Rijul saini
Yesterday at 6:30 PM
sansgankrsngupta
38 minutes ago
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IMO 2009, Problem 1
orl   141
N an hour ago by ND_
Source: IMO 2009, Problem 1
Let $ n$ be a positive integer and let $ a_1,a_2,a_3,\ldots,a_k$ $ ( k\ge 2)$ be distinct integers in the set $ { 1,2,\ldots,n}$ such that $ n$ divides $ a_i(a_{i + 1} - 1)$ for $ i = 1,2,\ldots,k - 1$. Prove that $ n$ does not divide $ a_k(a_1 - 1).$

Proposed by Ross Atkins, Australia
141 replies
orl
Jul 15, 2009
ND_
an hour ago
J
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Minimization without derivative
Butterfly   2
N 2 hours ago by Butterfly

Find the minimum value of $f(x)=\frac{(3x+2)^4}{x(x+3)}$ on $(0,+\infty)$ without derivative.
2 replies
Butterfly
2 hours ago
Butterfly
2 hours ago
J
No more topics!
H
J
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Nice inequality
sqing   3
N Apr 24, 2025 by Oksutok
Source: WYX
Let $a_1,a_2,\cdots,a_n (n\ge 2)$ be real numbers . Prove that : There exist positive integer $k\in \{1,2,\cdots,n\}$ such that $$\sum_{i=1}^{n}\{kx_i\}(1-\{kx_i\})<\frac{n-1}{6}.$$Where $\{x\}=x-\left \lfloor x \right \rfloor.$
3 replies
sqing
Apr 24, 2019
Oksutok
Apr 24, 2025
J
Nice inequality
G H J
Reveal topic tags
inequalities
China
L
G H BBookmark kLocked kLocked NReply
Source: WYX
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sqing
42590 posts
sqing
#1 h  Apr 24, 2019, 1:01 PM • 3 Y
Y by Semcio, Adventure10, Mango247
Let $a_1,a_2,\cdots,a_n (n\ge 2)$ be real numbers . Prove that : There exist positive integer $k\in \{1,2,\cdots,n\}$ such that $$\sum_{i=1}^{n}\{kx_i\}(1-\{kx_i\})<\frac{n-1}{6}.$$Where $\{x\}=x-\left \lfloor x \right \rfloor.$
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Oksutok
212 posts
Oksutok
#2 h  Apr 23, 2025, 3:04 PM
Y by
Any solution?
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Seungjun_Lee
526 posts
Seungjun_Lee
#3 h  Apr 23, 2025, 3:23 PM
Y by
This is a problem that EthanWYX2009 has posted on his/her blog here. Very nice problem.
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Oksutok
212 posts
Oksutok
#4 h  Apr 24, 2025, 1:17 PM
Y by
No is the same problem
This post has been edited 1 time. Last edited by Oksutok, Apr 24, 2025, 1:17 PM
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