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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.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. 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][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]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
J
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problem interesting
Cobedangiu   1
N a few seconds ago by Cobedangiu
Let $a=3k^2+3k+1 (a,k \in N)$
$i)$ Prove that: $a^2$ is the sum of $3$ square numbers
$ii)$ Let $b \vdots a$ and $b$ is the sum of $3$ square numbers. Prove that: $b^n$ is the sum of $3$ square numbers
1 reply
Cobedangiu
an hour ago
Cobedangiu
a few seconds ago
J
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find f
ali666   3
N 10 minutes ago by Blackbeam999
find all valued functions $f$ such that for all real $x,y$:
$f(x-y)=f(x)f(y)$
3 replies
ali666
Aug 19, 2006
Blackbeam999
10 minutes ago
J
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Find f
Redriver   4
N 14 minutes ago by Blackbeam999
Find all $: R \to R : \ \ f(x^2+f(y))=y+f^2(x)$
4 replies
Redriver
Jun 25, 2006
Blackbeam999
14 minutes ago
J
H
2^x+3^x = yx^2
truongphatt2668   7
N 15 minutes ago by Jackson0423
Prove that the following equation has infinite integer solutions:
$$2^x+3^x = yx^2$$
7 replies
truongphatt2668
Apr 22, 2025
Jackson0423
15 minutes ago
J
No more topics!
H
J
H
thank you
Piwbo   3
N Apr 13, 2025 by Safal
Let $p_n$ be the n-th prime number in increasing order for $n\geq 1$. Prove that there exists a sequence of distinct prime numbers $q_n$ satisfying $q_1+q_2+...+q_n=p_n$ for all $n\geq 1 $
3 replies
Piwbo
Apr 10, 2025
Safal
Apr 13, 2025
J
thank you
G H J
Reveal topic tags
NumberTheory
number theory
L
G H BBookmark kLocked kLocked NReply
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Piwbo
3 posts
Piwbo
#1 h  Apr 10, 2025, 11:22 AM
Y by
Let $p_n$ be the n-th prime number in increasing order for $n\geq 1$. Prove that there exists a sequence of distinct prime numbers $q_n$ satisfying $q_1+q_2+...+q_n=p_n$ for all $n\geq 1 $
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CHESSR1DER
57 posts
CHESSR1DER
#2 h  Apr 11, 2025, 5:31 PM
Y by
$p_2 = 3$. However $3$ can't be represented as sum of $2$ primes.
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Piwbo
3 posts
Piwbo
#3 h  Apr 13, 2025, 3:20 PM
Y by
sory it is $\vdots p_n$ and i have a solution using Dirichlet's theorem
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Safal
168 posts
Safal
#5 h  Apr 13, 2025, 5:20 PM
Y by
For example, take $p_{n+1}-p_{n}=q_{n+1}$.
Sorry of being rude in my previous comment, today my temper is little high. Also if your Question is $q_1+q_2+...+q_{n}|p_n\cdots(*)$ and $q_1+q_2+...+q_n+q_{n+1}|p_{n}\cdots(**)$. Since according to question $q_i$ are distinct primes then $(*)$ and $(**)$ cannot both hold at the same time beacuse of odd even parity and $q_1=2$ is forced.
This post has been edited 5 times. Last edited by Safal, Apr 13, 2025, 5:29 PM
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