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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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inequality
revol_ufiaw   1
N 10 minutes ago by sqing
Prove that that for any real $x \ge 0$ and natural number $n$,
$$x^n (n+1)^{n+1} \le n^n (x+1)^{n+1}.$$
1 reply
revol_ufiaw
23 minutes ago
sqing
10 minutes ago
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Any nice way to do this?
NamelyOrange   3
N 28 minutes ago by pooh123
Source: Taichung P.S.1 math program tryouts

How many ordered pairs $(a,b,c)\in\mathbb{N}^3$ are there such that $c=ab$ and $1\le a\le b\le c\le60$?
3 replies
NamelyOrange
Apr 2, 2025
pooh123
28 minutes ago
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Inequalities
sqing   3
N 29 minutes ago by sqing
Let $ a,b,c> 0 $ and $ \frac{a}{a^2+ab+c}+\frac{b}{b^2+bc+a}+\frac{c}{c^2+ca+b} \geq 1$. Prove that
$$ a+b+c\leq 3 $$
3 replies
sqing
Yesterday at 3:52 AM
sqing
29 minutes ago
J
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Inequalities
sqing   0
an hour ago
Let $a,b$ be real numbers such that $ a^2+b^2+a^3 +b^3=4 . $ Prove that
$$a+b \leq 2$$Let $a,b$ be real numbers such that $a+b + a^2+b^2+a^3 +b^3=6 . $ Prove that
$$a+b \leq 2$$
0 replies
sqing
an hour ago
0 replies
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circle tangent to circle and semicircle (2015 All-Siberian Qualifying 11.4)
parmenides51   1
N May 15, 2021 by natmath
In a semicircle with a radius of $18$ cm, a semicircle of radius $9$ cm is built on one of the halves of the diameter, and a circle is inscribed, touching the larger semicircle from the inside, the smaller semicircle from the outside and the second half of the diameter. Find the radius of this circle.
1 reply
parmenides51
May 15, 2021
natmath
May 15, 2021
J
circle tangent to circle and semicircle (2015 All-Siberian Qualifying 11.4)
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Reveal topic tags
geometry
tangent circles
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parmenides51
30629 posts
parmenides51
#1 h  May 15, 2021, 5:24 PM
Y by
In a semicircle with a radius of $18$ cm, a semicircle of radius $9$ cm is built on one of the halves of the diameter, and a circle is inscribed, touching the larger semicircle from the inside, the smaller semicircle from the outside and the second half of the diameter. Find the radius of this circle.
This post has been edited 2 times. Last edited by parmenides51, May 15, 2021, 5:36 PM
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natmath
8219 posts
natmath
#2 h  May 15, 2021, 5:33 PM • 1 Y
Y by parmenides51
parmenides51 wrote:
In a semicircle with a radius of $18$ cm, a semicircle of radius $9$ cm is built on one of the halves of the diameter, and a circle is inscribed, touching the larger semicircle from the inside, the smaller semicircle from the outside and and the second half of the diameter. Find the radius of this circle.

I think you mean this?
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