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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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Interesting inequalities
sqing   2
N a few seconds 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
$$ \frac{13}{ 10 }> \frac{1}{ a^2+1 }+ \frac{1}{ b^2+1 } \geq \frac{1}{ 2 }$$$$ \frac{6}{ 5 }>\frac{1}{ a^4+1 }+ \frac{1}{ b^4+1 } \geq \frac{1}{ 5 }$$$$ \frac{1}{ a^6+1 }+ \frac{1}{ b^6+1 } \geq \frac{1}{ 14 }$$
2 replies
1 viewing
sqing
3 hours ago
sqing
a few seconds ago
J
H
A cyclic inequality
KhuongTrang   11
N an hour ago by KhuongTrang
Source: own-CRUX
IMAGE
Link
11 replies
KhuongTrang
Apr 2, 2025
KhuongTrang
an hour ago
J
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Divisibility NT FE
CHESSR1DER   1
N an hour ago by CHESSR1DER
Source: Own
Find all functions $f$ $N \iff N$ such for any $a,b$:
$(a+b)|a^{f(b)} + b^{f(a)}$.
1 reply
CHESSR1DER
Yesterday at 7:07 PM
CHESSR1DER
an hour ago
J
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hard problem
Cobedangiu   3
N an hour ago by lbh_qys
Let $x,y>0$ and $\dfrac{1}{x}+\dfrac{1}{y}+1=\dfrac{10}{x+y+1}$. Find max $A$ (and prove):
$A=\dfrac{x^2}{y}+\dfrac{y^2}{x}+\dfrac{1}{xy}$
3 replies
+1 w
Cobedangiu
Today at 5:19 AM
lbh_qys
an hour ago
J
No more topics!
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1/a+1/b<1/f+ 1/g , angle trisectors related
parmenides51   0
May 9, 2022
Source: 2017 - 3rd Tournament of Mathematical Battles "League of Winners" Seniors 1.8 Лига Победителей
In a triangle, denote by $f$ and $g$ the lengths of segments dividing the angle between sides with lengths $a$ and $b$ into $3$ equal parts. Prove that ${1\over a}+{1\over b}<{1\over f}+{1\over g}$.

D. Shiryaev, based on Kozepiskolai Matematikai Lapok, 2017, No 3, B.4865
0 replies
parmenides51
May 9, 2022
0 replies
J
1/a+1/b<1/f+ 1/g , angle trisectors related
G H J
Reveal topic tags
geometric inequality
Geometric Inequalities
geometry
trisector
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G H BBookmark kLocked kLocked NReply
Source: 2017 - 3rd Tournament of Mathematical Battles "League of Winners" Seniors 1.8 Лига Победителей
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parmenides51
30630 posts
parmenides51
#1 h  May 9, 2022, 5:18 PM
Y by
In a triangle, denote by $f$ and $g$ the lengths of segments dividing the angle between sides with lengths $a$ and $b$ into $3$ equal parts. Prove that ${1\over a}+{1\over b}<{1\over f}+{1\over g}$.

D. Shiryaev, based on Kozepiskolai Matematikai Lapok, 2017, No 3, B.4865
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