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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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3 var inquality
sqing   0
2 minutes ago
Source: Own
Let $ a,b,c> 0 $ and $ 4(a+b) +3c-ab \geq10$ . Prove that$$a^2+b^2+c^2+abc\geq 4$$
0 replies
1 viewing
sqing
2 minutes ago
0 replies
J
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The old one is gone.
EeEeRUT   3
N 9 minutes ago by ItzsleepyXD
Source: EGMO 2025 P2
An infinite increasing sequence $a_1 < a_2 < a_3 < \cdots$ of positive integers is called central if for every positive integer $n$ , the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$.

Show that there exists an infinite sequence $b_1, b_2, b_3, \dots$ of positive integers such that for every central sequence $a_1, a_2, a_3, \dots, $ there are infinitely many positive integers $n$ with $a_n = b_n$.
3 replies
EeEeRUT
an hour ago
ItzsleepyXD
9 minutes ago
J
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A Projection Theorem
buratinogigle   1
N 16 minutes ago by aidan0626
Source: VN Math Olympiad For High School Students P1 - 2025
In triangle $ABC$, prove that
\[ a = b\cos C + c\cos B. \]
1 reply
+1 w
buratinogigle
an hour ago
aidan0626
16 minutes ago
J
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Interesting inequalities
sqing   4
N 16 minutes ago by sqing
Source: Own
Let $ a,b $ be reals such that $ a^2+ab+b^2 =1$ . Prove that
$$ \frac{8}{ 5 }> \frac{1}{ a^2+1 }+ \frac{1}{ b^2+1 } \geq 1$$$$ \frac{9}{ 5 }\geq\frac{1}{ a^4+1 }+ \frac{1}{ b^4+1 } \geq 1$$$$ \frac{27}{ 14 }\geq \frac{1}{ a^6+1 }+ \frac{1}{ b^6+1 } \geq 1$$Let $ a,b $ be reals such that $ a^2+ab+b^2 =3$ . Prove that
$$ \frac{13}{ 10 }> \frac{1}{ a^2+1 }+ \frac{1}{ b^2+1 } \geq \frac{1}{ 2 }$$$$ \frac{6}{ 5 }>\frac{1}{ a^4+1 }+ \frac{1}{ b^4+1 } \geq \frac{1}{ 5 }$$$$ \frac{1}{ a^6+1 }+ \frac{1}{ b^6+1 } \geq \frac{1}{ 14 }$$
4 replies
sqing
Yesterday at 8:32 AM
sqing
16 minutes ago
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NT Problem
tiendat004   0
Apr 4, 2025
Let $a\in\mathbb{N}_+$ with $a$ is coprime to $21$. It is known that for every $s\in\mathbb{N}_+$, there always exist $r,t\in\mathbb{N}_+$ satisfying $$a+7s^3=r^3+7t^3.$$Prove that $a$ is a cube.
0 replies
tiendat004
Apr 4, 2025
0 replies
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NT Problem
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tiendat004
#1 h  Apr 4, 2025, 5:09 AM
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Let $a\in\mathbb{N}_+$ with $a$ is coprime to $21$. It is known that for every $s\in\mathbb{N}_+$, there always exist $r,t\in\mathbb{N}_+$ satisfying $$a+7s^3=r^3+7t^3.$$Prove that $a$ is a cube.
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