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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
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external bisector in 2 angle
crocodilepradita   5
N 21 minutes ago by ihategeo_1969
Given a $\triangle ABC$ with incenter $I$. Line $BI$ and $CI$ intersects $CA$ and $AB$ at $E$ and $F$, respectively. Let $M$ and $N$ be the midpoints of $BI$ and $CI$. Line $FM$ meets the external bisector of angle $B$ at $K$ and line $EN$ meets the external bisector of angle $C$ at $L$. Prove that $K, B, L, C$ are concylic.
5 replies
crocodilepradita
Aug 22, 2024
ihategeo_1969
21 minutes ago
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Interesting inequalities
sqing   6
N an hour ago by MathPerson12321
Source: Own
Let $ a,b\geq 0 $ and $ a+b=2. $ Prove that
$$ ab( a^2+ b^2)^2 \leq \frac{128}{27}$$$$ ab( a^2-ab+ b^2)^2 \leq \frac{256}{81}$$$$ ab\sqrt{ab}( a^2+ b^2)^2 \leq \frac{1536}{343}\sqrt{\frac{6}{7}}$$$$ ab\sqrt{ab}( a^2-ab+ b^2)^2 \leq \frac{2048}{343\sqrt{7}}$$
6 replies
1 viewing
sqing
2 hours ago
MathPerson12321
an hour ago
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Difficult combinatorics problem about distinct sums under shifts
CBMaster   0
an hour ago
Source: Korea
Problem. Let $a_1, ..., a_n$ be the nonnegative integers in $\{0, 1, ..., m\}$ where $m=\left\lceil \frac{n^{2/3}}{4} \right\rceil $. Define $A=\{a_i+a_j+(j-i)|1\leq i<j\leq n\}$. Prove that $|A|\geq m$.

Bonus problem (Open). Can we prove a tighter result than the one above? That is, is there a function $f(n)$ such that $f(n)=O(n^\alpha)$ where $\alpha>\frac{2}{3}$, and the statement is still true when $m=f(n)$?
Or, is there a function $f(n)$ such that $f(n)\geq C \cdot n^{2/3}$ where $C>\frac{1}{4}$, and the statement is still true when $m=f(n)$?.
0 replies
CBMaster
an hour ago
0 replies
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4 variables with quadrilateral sides
mihaig   5
N an hour ago by mihaig
Source: VL
Let $a,b,c,d\geq0$ satisfying
$$\frac1{a+1}+\frac1{b+1}+\frac1{c+1}+\frac1{d+1}=2.$$Prove
$$4\left(abc+abd+acd+bcd\right)\geq3\left(a+b+c+d\right)+4.$$
5 replies
mihaig
Apr 25, 2025
mihaig
an hour ago
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computational in a 90-60-30 triangle ABC, <AOB=120^o, OC=1, OB=4, OA=?
parmenides51   0
Jul 7, 2020
Source: 2007 Ukraine MO grade IX P4
Inside the triangle $ABC$ with angles $\angle C = 90^o$ and $\angle A = 60^o$ there is a point $O$such that $ \angle AOB = 12^o$, $OC = 1$, $OB = 4$. Find the length of the segment $AO $.

(Alexey Chubenko)
0 replies
parmenides51
Jul 7, 2020
0 replies
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computational in a 90-60-30 triangle ABC, <AOB=120^o, OC=1, OB=4, OA=?
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Reveal topic tags
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Source: 2007 Ukraine MO grade IX P4
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parmenides51
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parmenides51
#1 h  Jul 7, 2020, 11:18 AM
Y by
Inside the triangle $ABC$ with angles $\angle C = 90^o$ and $\angle A = 60^o$ there is a point $O$such that $ \angle AOB = 12^o$, $OC = 1$, $OB = 4$. Find the length of the segment $AO $.

(Alexey Chubenko)
This post has been edited 1 time. Last edited by parmenides51, Jul 14, 2020, 9:31 PM
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