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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next 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 an enriching 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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[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
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[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
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max value
Bet667   0
5 minutes ago
Let $a,b$ be a real numbers such that $a^2+ab+b^2\ge a^3+b^3.$Then find maximum value of $a+b$
0 replies
Bet667
5 minutes ago
0 replies
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thank you !
Piwbo   1
N 7 minutes ago by blug
Given positive integers $a,b$ such that $a$ is even , $b$ is odd and $ab(a+b)^{2023}$ is divisible by $a^{2024}+b^{2024}$ .Prove that there exists a prime number $p$ such that $p^{2025}$ is divisible by $a^{2024}+b^{2024}$
1 reply
Piwbo
9 minutes ago
blug
7 minutes ago
J
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Geometry
gggzul   0
43 minutes ago
In trapezoid $ABCD$ segments $AB$ and $CD$ are parallel. Angle bisectors of $\angle A$ and $\angle C$ meet at $P$. Angle bisectors of $\angle B$ and $\angle D$ meet at $Q$. Prove that $ABPQ$ is cyclic
0 replies
gggzul
43 minutes ago
0 replies
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hard problem
Cobedangiu   5
N an hour ago by KhuongTrang
$a,b,c>0$ and $a+b+c=7$. CM:
$\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}+abc \ge ab+bc+ca-2$
5 replies
Cobedangiu
Yesterday at 4:24 PM
KhuongTrang
an hour ago
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Number Theory
AnhQuang_67   3
N Apr 24, 2025 by GreekIdiot
Source: HSGSO 2024
Let $p$ be an odd prime number and a sequence $\{a_n\}_{n=1}^{+\infty}$ satisfy $$a_1=1, a_2=2$$and $$a_{n+2}=2\cdot a_{n+1}+3\cdot a_n, \forall n \geqslant 1$$Prove that always exists positive integer $k$ satisfying for all positive integers $n$, then $a_n \ne k \mod{p}$.

P/s: $\ne$ is "not congruence"
3 replies
AnhQuang_67
Apr 24, 2025
GreekIdiot
Apr 24, 2025
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Number Theory
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Reveal topic tags
number theory
prime numbers
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G H BBookmark kLocked kLocked NReply
Source: HSGSO 2024
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AnhQuang_67
57 posts
AnhQuang_67
#1 h  Apr 24, 2025, 1:44 PM
Y by
Let $p$ be an odd prime number and a sequence $\{a_n\}_{n=1}^{+\infty}$ satisfy $$a_1=1, a_2=2$$and $$a_{n+2}=2\cdot a_{n+1}+3\cdot a_n, \forall n \geqslant 1$$Prove that always exists positive integer $k$ satisfying for all positive integers $n$, then $a_n \ne k \mod{p}$.

P/s: $\ne$ is "not congruence"
This post has been edited 1 time. Last edited by AnhQuang_67, Apr 24, 2025, 2:54 PM
Reason: sorry, my bad
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GreekIdiot
218 posts
GreekIdiot
#2 h  Apr 24, 2025, 2:26 PM
Y by
$p$ is an even prime? Then $p=2$ thus $a_n=0,1 \: mod \: 2$
and since $a_1,a_2$ are not of the same parity there doesnt exist such $k$
I assume you mean 'odd' prime
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AnhQuang_67
57 posts
AnhQuang_67
#3 h  Apr 24, 2025, 2:55 PM
Y by
GreekIdiot wrote:
$p$ is an even prime? Then $p=2$ thus $a_n=0,1 \: mod \: 2$
and since $a_1,a_2$ are not of the same parity there doesnt exist such $k$
I assume you mean 'odd' prime
Oops, sorry about that. It is "odd"
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GreekIdiot
218 posts
GreekIdiot
#4 h  Apr 24, 2025, 3:14 PM
Y by
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