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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
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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
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Lower bound on a product
anantmudgal09   11
N 19 minutes ago by AshAuktober
Source: RMO Maharashtra and Goa 2016, P5
Let $x,y,z$ be non-negative real numbers such that $xyz=1$. Prove that $$(x^3+2y)(y^3+2z)(z^3+2x) \ge 27.$$
11 replies
anantmudgal09
Oct 11, 2016
AshAuktober
19 minutes ago
J
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Product of three positive reals is less than 1/8
kk108   47
N 23 minutes ago by Adywastaken
Source: RMO Hyderabad 2016 , P2 .
Let $a,b,c$ be positive real numbers such that $$\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1.$$Prove that $abc \le \frac{1}{8}$.
47 replies
1 viewing
kk108
Oct 12, 2016
Adywastaken
23 minutes ago
J
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Infinite Sum
P162008   1
N 42 minutes ago by cazanova19921
Find $\Omega = \lim_{n \to \infty} \frac{1}{n^2} \left(\sum_{i + j + k + l = n} ijkl\right) \left(\sum_{i + j + k = n} ijk\right)^{-1}.$
1 reply
P162008
4 hours ago
cazanova19921
42 minutes ago
J
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4 variables with quadrilateral sides
mihaig   0
an hour ago
Source: VL
Let $a,b,c,d\geq0$ satisfying
$$\frac1{a+1}+\frac1{b+1}+\frac1{c+1}+\frac1{d+1}=2.$$Prove
$$4\left(abc+abd+acd+bcd\right)\geq3\left(a+b+c+d\right)+4.$$
0 replies
1 viewing
mihaig
an hour ago
0 replies
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ratio of areas (OMCN),(ABCD), ABCD with perpendicular diagonals
parmenides51   1
N Jul 10, 2020 by WolfusA
Source: Moldova JBMO TST 2003 1.3
The quadrilateral $ABCD$ with perpendicular diagonals is inscribed in the circle with center $O$, the points $M,N$ are the midpoints of $[BC]$ and $[CD]$ respectively. Find the ratio of areas of the figures $OMCN$ and $ABCD$
1 reply
parmenides51
Jul 10, 2020
WolfusA
Jul 10, 2020
J
ratio of areas (OMCN),(ABCD), ABCD with perpendicular diagonals
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Source: Moldova JBMO TST 2003 1.3
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parmenides51
30631 posts
parmenides51
#1 h  Jul 10, 2020, 6:32 AM
Y by
The quadrilateral $ABCD$ with perpendicular diagonals is inscribed in the circle with center $O$, the points $M,N$ are the midpoints of $[BC]$ and $[CD]$ respectively. Find the ratio of areas of the figures $OMCN$ and $ABCD$
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WolfusA
1900 posts
WolfusA
#2 h  Jul 10, 2020, 12:48 PM • 1 Y
Y by Mango247
Let $C’$ be the reflection of $C$ wrt $O$. $AC’\perp AC\perp BD$ so $[ABD]=[C’BD]=4[OMN]$ since $\triangle OMN\sim\triangle C’BD$. Thus $[ABCD]=4[OMCN]$
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