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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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a combinatorial geometry problem
xyz123456   5
N 2 minutes ago by Fishheadtailbody
$A_i\left(x_i{,}y_i\right){,}0\le x_i{,}y_i\le 1{,}1\le i\le 6.prove\ that:\exists \ 1\le i<j\le 6\ {,}\left|A_iA_j\right|\le \frac{\sqrt{13}}{6}$
5 replies
+1 w
xyz123456
Mar 3, 2025
Fishheadtailbody
2 minutes ago
J
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problem....
Cobedangiu   2
N 6 minutes ago by sqing
$a,b,c>0$ and $a+b+c=1$
Prove that: $Q=ab+bc+ca-2abc\le\dfrac{7}{27}$
2 replies
Cobedangiu
3 hours ago
sqing
6 minutes ago
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Find all pairs (x; y)
sqing   5
N 21 minutes ago by EVKV
Source: 7th European Mathematical Cup , Junior Category, Q2
Find all pairs $ (x; y) $ of positive integers such that
$$xy | x^2 + 2y -1.$$
5 replies
sqing
Dec 25, 2018
EVKV
21 minutes ago
J
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Why is the old one deleted?
EeEeRUT   10
N 36 minutes ago by kamatadu
Source: EGMO 2025 P1
For a positive integer $N$, let $c_1 < c_2 < \cdots < c_m$ be all positive integers smaller than $N$ that are coprime to $N$. Find all $N \geqslant 3$ such that $$\gcd( N, c_i + c_{i+1}) \neq 1$$for all $1 \leqslant i \leqslant m-1$

Here $\gcd(a, b)$ is the largest positive integer that divides both $a$ and $b$. Integers $a$ and $b$ are coprime if $\gcd(a, b) = 1$.

Proposed by Paulius Aleknavičius, Lithuania
10 replies
EeEeRUT
Apr 16, 2025
kamatadu
36 minutes ago
J
No more topics!
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4 points collinear wanted, incircle, circle of diameter AB, angle bisectors
parmenides51   0
Feb 16, 2021
Source: 2020 Croatian Junior MO - HJMO 2.3 https://artofproblemsolving.com/community/c3015761_croatia_juniors_mo_hjmo___geometry
Let $ABC$ be a triangle with an obtuse angle at the vertex $C$ and $k$ be a circle of diameter $AB$. Angle bisector of $\angle CAB$ intersects the circle $k$ at the point $D$ ($D\ne A$), and the angle bisector of $\angle ABC$ intersects that circle at the point $E$ ($E\ne B$). A circle inscribed in a triangle $ABC$ touches the sides $BC$ and $AC$ at points $F$ and $G$, respectively. Prove that the points $D,E,F$ and $G$ lie in the same line.
0 replies
parmenides51
Feb 16, 2021
0 replies
J
4 points collinear wanted, incircle, circle of diameter AB, angle bisectors
G H J
Reveal topic tags
geometry
collinear
diameter
angle bisector
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Source: 2020 Croatian Junior MO - HJMO 2.3 https://artofproblemsolving.com/community/c3015761_croatia_juniors_mo_hjmo___geometry
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parmenides51
30629 posts
parmenides51
#1 h  Feb 16, 2021, 12:33 PM • 1 Y
Y by Mango247
Let $ABC$ be a triangle with an obtuse angle at the vertex $C$ and $k$ be a circle of diameter $AB$. Angle bisector of $\angle CAB$ intersects the circle $k$ at the point $D$ ($D\ne A$), and the angle bisector of $\angle ABC$ intersects that circle at the point $E$ ($E\ne B$). A circle inscribed in a triangle $ABC$ touches the sides $BC$ and $AC$ at points $F$ and $G$, respectively. Prove that the points $D,E,F$ and $G$ lie in the same line.
This post has been edited 2 times. Last edited by parmenides51, Mar 24, 2024, 11:30 PM
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