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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
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC 10 Problem Series[/list]
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)!

Be sure to mark your calendars for the following upcoming events:
[list][*]June 5th, Thursday, 7:30pm ET: Open Discussion with Ben Kornell and Andrew Sutherland, Art of Problem Solving's incoming CEO Ben Kornell and CPO Andrew Sutherland host an Ask Me Anything-style chat. Come ask your questions and get to know our incoming CEO & CPO!
[*]June 9th, Monday, 7:30pm ET, Game Jam: Operation Shuffle!, Come join us to play our second round of Operation Shuffle! If you enjoy number sense, logic, and a healthy dose of luck, this is the game for you. No specific math background is required; all are welcome.[/list]
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0 replies
jlacosta
Jun 2, 2025
0 replies
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Easy Number Theory
math_comb01   42
N 10 minutes ago by heheman
Source: INMO 2024/3
Let $p$ be an odd prime and $a,b,c$ be integers so that the integers $$a^{2023}+b^{2023},\quad b^{2024}+c^{2024},\quad a^{2025}+c^{2025}$$are divisible by $p$.
Prove that $p$ divides each of $a,b,c$.
$\quad$
Proposed by Navilarekallu Tejaswi
42 replies
1 viewing
math_comb01
Jan 21, 2024
heheman
10 minutes ago
J
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A function on a 2D grid
Rijul saini   3
N an hour ago by alexheinis
Source: India IMOTC 2025 Day 4 Problem 2
Does there exist a function $f:\{1,2,...,2025\}^2 \rightarrow \{1,2,...,2025\}$ such that:

$\bullet$ for any positive integer $i \leqslant 2025$, the numbers $f(i,1),f(i,2),...,f(i,2025)$ are all distinct, and
$\bullet$ for any positive integers $i \leqslant 2025$ and $j \leqslant 2024$, $f(f(i,j),f(i,j+1))=i$?

Proposed by Shantanu Nene
3 replies
Rijul saini
Jun 4, 2025
alexheinis
an hour ago
J
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3 lines concurrent with 1 circle
parmenides51   4
N an hour ago by LeYohan
Source: 2021 Irish Mathematical Olympiad P8
A point $C$ lies on a line segment $AB$ between $A$ and $B$ and circles are drawn having $AC$ and $CB$ as diameters. A common tangent to both circles touches the circle with $AC$ as diameter at $P \ne C$ and the circle with $CB$ as diameter at $Q \ne C$.
Prove that $AP, BQ$ and the common tangent to both circles at $C$ all meet at a single point which lies on the circumference of the circle with $AB$ as diameter.
4 replies
parmenides51
May 30, 2021
LeYohan
an hour ago
J
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Integer Polynomial with P(a)=b, P(b)=c, and P(c)=a
Brut3Forc3   30
N an hour ago by megahertz13
Source: 1974 USAMO Problem 1
Let $ a,b,$ and $ c$ denote three distinct integers, and let $ P$ denote a polynomial having integer coefficients. Show that it is impossible that $ P(a) = b, P(b) = c,$ and $ P(c) = a$.
30 replies
Brut3Forc3
Mar 13, 2010
megahertz13
an hour ago
J
No more topics!
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4 var inequality
sqing   1
N Apr 26, 2025 by sqing
Source: https://bbs.emath.ac.cn/thread-39778-1-1.html
Let $ a,b,c,d>0 $ and $ a+b+c+d=4. $ Prove that$$a\sqrt{bc}+b\sqrt{cd}+c\sqrt{da}+d\sqrt{ab}\leq 2(1+\sqrt{abcd})$$Let $ a,b,c,d\geq -1 $ and $ a+b+c+d=0. $ Prove that$$ab+bc+cd\leq \frac{5}{4}$$
1 reply
sqing
Apr 26, 2025
sqing
Apr 26, 2025
J
4 var inequality
G H J
Reveal topic tags
inequalities proposed
inequalities
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G H BBookmark kLocked kLocked NReply
Source: https://bbs.emath.ac.cn/thread-39778-1-1.html
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sqing
42617 posts
sqing
#1 h  Apr 26, 2025, 9:06 AM
Y by
Let $ a,b,c,d>0 $ and $ a+b+c+d=4. $ Prove that$$a\sqrt{bc}+b\sqrt{cd}+c\sqrt{da}+d\sqrt{ab}\leq 2(1+\sqrt{abcd})$$Let $ a,b,c,d\geq -1 $ and $ a+b+c+d=0. $ Prove that$$ab+bc+cd\leq \frac{5}{4}$$
This post has been edited 1 time. Last edited by sqing, Apr 26, 2025, 9:21 AM
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sqing
42617 posts
sqing
#2 h  Apr 26, 2025, 9:13 AM
Y by
Let $ a,b,c\geq 0 $ and $ a^2+b^2+c^2=2. $ Prove that$$a+b+c\leq 2+\sqrt{\frac32\bigl(5\sqrt6-12\bigr)abc}$$
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