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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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Very tight inequalities
KhuongTrang   2
N 18 minutes ago by SunnyEvan
Source: own
Problem. Given non-negative real numbers $a,b,c$ satisfying $ab+bc+ca=1.$ Prove that $$\color{black}{\frac{1}{35a+12b+2}+\frac{1}{35b+12c+2}+\frac{1}{35c+12a+2}\ge \frac{4}{39}.}$$$$\color{black}{\frac{1}{4a+9b+6}+\frac{1}{4b+9c+6}+\frac{1}{4c+9a+6}\le \frac{2}{9}.}$$When does equality hold?
2 replies
KhuongTrang
May 17, 2024
SunnyEvan
18 minutes ago
J
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Sum of First, Second, and Third Powers
Brut3Forc3   47
N 31 minutes ago by cubres
Source: 1973 USAMO Problem 4
Determine all roots, real or complex, of the system of simultaneous equations
\begin{align*} x+y+z &= 3, \\ x^2+y^2+z^2 &= 3, \\ x^3+y^3+z^3 &= 3.\end{align*}
47 replies
Brut3Forc3
Mar 7, 2010
cubres
31 minutes ago
J
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Modified Sum of floors
prMoLeGend42   2
N 41 minutes ago by cubres
Find the closed form of : $\sum _{k=0}^{n-1} \left\lfloor \frac{ak+b}{n}\right \rfloor$ where $\gcd(a,n)=1$
2 replies
prMoLeGend42
Yesterday at 9:09 AM
cubres
41 minutes ago
J
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6 tangents to 1 circle
moony_   3
N 43 minutes ago by whwlqkd
Source: own
Let $P$ be a point inside the triangle $ABC$. $AP$ intersects $BC$ at $A_0$. Points $B_0$ and $C_0$ are defined similarly. Line $B_0C_0$ intersects $(ABC)$ at points $A_1$, $A_2$. The tangents at these points to $(ABC)$ intersect BC at points $A_3$, $A_4$. Points $B_3$, $B_4$, $C_3$, $C_4$ are defined similarly. Prove that points $A_3$, $A_4$, $B_3$, $B_4$, $C_3$, $C_4$ lie on one conic
3 replies
moony_
Yesterday at 9:30 AM
whwlqkd
43 minutes ago
J
No more topics!
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for any choice of P the points D_P , E_P and F are collinear
parmenides51   1
N Oct 31, 2022 by demmy
Source: 2022 Austrian Federal Competition For Advanced Students, Part 2 p5
Let $ABC$ be an isosceles triangle with base $AB$. We choose a point $P$ inside the triangle on altitude through $C$. The circle with diameter $CP$ intersects the straight line through $B$ and $P$ again at the point $D_P$ and the Straight through $A$ and $C$ one more time at point $E_P$. Prove that there is a point $F$ such that for any choice of $P$ the points $D_P , E_P$ and $F$ lie on a straight line.

(Walther Janous)
1 reply
parmenides51
Oct 5, 2022
demmy
Oct 31, 2022
J
for any choice of P the points D_P , E_P and F are collinear
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Reveal topic tags
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isosceles
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Source: 2022 Austrian Federal Competition For Advanced Students, Part 2 p5
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parmenides51
30630 posts
parmenides51
#1 h  Oct 5, 2022, 12:16 PM • 1 Y
Y by Mango247
Let $ABC$ be an isosceles triangle with base $AB$. We choose a point $P$ inside the triangle on altitude through $C$. The circle with diameter $CP$ intersects the straight line through $B$ and $P$ again at the point $D_P$ and the Straight through $A$ and $C$ one more time at point $E_P$. Prove that there is a point $F$ such that for any choice of $P$ the points $D_P , E_P$ and $F$ lie on a straight line.

(Walther Janous)
Z K Y
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demmy
133 posts
demmy
#2 h  Oct 31, 2022, 8:59 AM
Y by
Let M be a midpoint of AB. We need to prove that D , E , M collinear which is easy to proof .
This post has been edited 1 time. Last edited by demmy, Oct 31, 2022, 8:59 AM
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