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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)!

Be sure to mark your calendars for the following events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
[*]April 8th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS State Discussion
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
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Problems for v_p(n)
xytunghoanh   13
N 2 minutes ago by xytunghoanh
Hello everyone. I need some easy problems for $v_p(n)$ (use in a problem for junior) to practice. Can anyone share to me?
Thanks :>
13 replies
+2 w
xytunghoanh
2 hours ago
xytunghoanh
2 minutes ago
J
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Modular Matching Pairs
steven_zhang123   3
N 19 minutes ago by flower417477
Source: China TST 2025 P20
Let \( n \) be an odd integer, \( m = \frac{n+1}{2} \). Consider \( 2m \) integers \( a_1, a_2, \ldots, a_m, b_1, b_2, \ldots, b_m \) such that for any \( 1 \leq i < j \leq m \), \( a_i \not\equiv a_j \pmod{n} \) and \( b_i \not\equiv b_j \pmod{n} \). Prove that the number of \( k \in \{0, 1, \ldots, n-1\} \) for which satisfy \( a_i + b_j \equiv k \pmod{n} \) for some \( i \neq j \), $i, j \in \left \{ 1,2,\cdots,m \right \} $ is greater than \( n - \sqrt{n} - \frac{1}{2} \).
3 replies
steven_zhang123
Mar 29, 2025
flower417477
19 minutes ago
J
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Find the function!
Johann Peter Dirichlet   13
N 27 minutes ago by bin_sherlo
Source: Problem 3, Brazilian Olympic Revenge 2005
Find all functions $f: R \rightarrow R$ such that
\[f(x+yf(x))+f(xf(y)-y)=f(x)-f(y)+2xy\]
for all $x,y \in R$
13 replies
Johann Peter Dirichlet
Jun 1, 2005
bin_sherlo
27 minutes ago
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inequality ( 4 var
SunnyEvan   3
N an hour ago by arqady
Let $ a,b,c,d \in R $ , such that $ a+b+c+d=4 . $ Prove that :
$$ a^4+b^4+c^4+d^4+3 \geq \frac{7}{4}(a^3+b^3+c^3+d^3) $$$$ a^4+b^4+c^4+d^4+ \frac{252}{25} \geq \frac{88}{25}(a^3+b^3+c^3+d^3) $$equality cases : ?
3 replies
SunnyEvan
Yesterday at 5:19 AM
arqady
an hour ago
J
No more topics!
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M, G, N are collinear iff MB/MA + NC/NA =1
parmenides51   0
Dec 28, 2021
Source: 2010 Saudi Arabia Pre-TST 1.4
In triangle $ABC$ with centroid $G$, let $M \in (AB)$ and $N \in (AC)$ be points on two of its sides. Prove that points $M, G, N$ are collinear if and only if $\frac{MB}{MA}+\frac{NC}{NA}=1$.
0 replies
parmenides51
Dec 28, 2021
0 replies
J
M, G, N are collinear iff MB/MA + NC/NA =1
G H J
Reveal topic tags
geometry
Centroid
collinear
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G H BBookmark kLocked kLocked NReply
Source: 2010 Saudi Arabia Pre-TST 1.4
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parmenides51
30629 posts
parmenides51
#1 h  Dec 28, 2021, 9:53 AM
Y by
In triangle $ABC$ with centroid $G$, let $M \in (AB)$ and $N \in (AC)$ be points on two of its sides. Prove that points $M, G, N$ are collinear if and only if $\frac{MB}{MA}+\frac{NC}{NA}=1$.
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