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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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Can you construct the incenter of a triangle ABC?
PennyLane_31   3
N a few seconds ago by cj13609517288
Source: 2023 Girls in Mathematics Tournament- Level B, Problem 4
Given points $P$ and $Q$, Jaqueline has a ruler that allows tracing the line $PQ$. Jaqueline also has a special object that allows the construction of a circle of diameter $PQ$. Also, always when two circles (or a circle and a line, or two lines) intersect, she can mark the points of the intersection with a pencil and trace more lines and circles using these dispositives by the points marked. Initially, she has an acute scalene triangle $ABC$. Show that Jaqueline can construct the incenter of $ABC$.
3 replies
PennyLane_31
Oct 29, 2023
cj13609517288
a few seconds ago
J
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Do not try to bash on beautiful geometry
ItzsleepyXD   3
N 6 minutes ago by FarrukhBurzu
Source: Own , Mock Thailand Mathematic Olympiad P9
Let $ABC$be triangle with point $D,E$ and $F$ on $BC,AB,CA$
such that $BE=CF$ and $E,F$ are on the same side of $BC$
Let $M$ be midpoint of segment $BC$ and $N$ be midpoint of segment $EF$
Let $G$ be intersection of $BF$ with $CE$ and $\dfrac{BD}{DC}=\dfrac{AC}{AB}$
Prove that $MN\parallel DG$
3 replies
ItzsleepyXD
Today at 9:30 AM
FarrukhBurzu
6 minutes ago
J
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Cool functional equation
Rayanelba   2
N 7 minutes ago by ATM_
Source: Own
Find all functions $f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ that verify the following equation for all $x,y\in \mathbb{Z}_{>0}$:
$max(f^{f(y)}(x),f^{f(y)}(y))|min(x,y)$
2 replies
Rayanelba
25 minutes ago
ATM_
7 minutes ago
J
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1 line solution to Inequality
ItzsleepyXD   2
N 13 minutes ago by Vivaandax
Source: Own , Mock Thailand Mathematic Olympiad P8
Let $x_1,x_2,\dots,x_n$ be positive real integer such that $x_1^2+x_2^2+\cdots+x_n^2=2$ Prove that
$$\sum_{i=1}^{n}\frac{1}{x_i^3(x_{i-1}+x_{i+1})}\geqslant \left(\sum_{i=1}^{n}\frac{x_i}{x_{i-1}+x_{i+1}}\right)^3$$such that $x_{n+1}=x_1$ and $x_0=x_n$
2 replies
ItzsleepyXD
Today at 9:27 AM
Vivaandax
13 minutes ago
J
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Generating Functions
greenplanet2050   7
N 3 hours ago by rchokler
So im learning generating functions and i dont really understand why $1+2x+3x^2+4x^3+5x^4+…=\dfrac{1}{(1-x)^2}$

can someone help

thank you :)
7 replies
greenplanet2050
Yesterday at 10:42 PM
rchokler
3 hours ago
J
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Algebraic Manipulation
Darealzolt   1
N 5 hours ago by Soupboy0
Find the number of pairs of real numbers $a, b, c$ that satisfy the equation $a^4 + b^4 + c^4 + 1 = 4abc$.
1 reply
Darealzolt
Today at 1:25 PM
Soupboy0
5 hours ago
J
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BrUMO 2025 Team Round Problem 13
lpieleanu   1
N 5 hours ago by vanstraelen
Let $\omega$ be a circle, and let a line $\ell$ intersect $\omega$ at two points, $P$ and $Q.$ Circles $\omega_1$ and $\omega_2$ are internally tangent to $\omega$ at points $X$ and $Y,$ respectively, and both are tangent to $\ell$ at a common point $D.$ Similarly, circles $\omega_3$ and $\omega_4$ are externally tangent to $\omega$ at $X$ and $Y,$ respectively, and are tangent to $\ell$ at points $E$ and $F,$ respectively.

Given that the radius of $\omega$ is $13,$ the segment $\overline{PQ}$ has a length of $24,$ and $YD=YE,$ find the length of segment $\overline{YF}.$
1 reply
lpieleanu
Apr 27, 2025
vanstraelen
5 hours ago
J
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Inequlities
sqing   33
N Today at 1:50 PM by sqing
Let $ a,b,c\geq 0 $ and $ a^2+ab+bc+ca=3 .$ Prove that$$\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2} \geq \frac{3}{2}$$$$\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2}-bc \geq -\frac{3}{2}$$
33 replies
sqing
Jul 19, 2024
sqing
Today at 1:50 PM
J
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Very tasteful inequality
tom-nowy   1
N Today at 1:39 PM by sqing
Let $a,b,c \in (-1,1)$. Prove that $$(a+b+c)^2+3>(ab+bc+ca)^2+3(abc)^2.$$
1 reply
tom-nowy
Today at 10:47 AM
sqing
Today at 1:39 PM
J
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Inequalities
sqing   8
N Today at 1:31 PM by sqing
Let $x\in(-1,1). $ Prove that
$$ \dfrac{1}{\sqrt{1-x^2}} + \dfrac{1}{2+ x^2} \geq \dfrac{3}{2}$$$$ \dfrac{2}{\sqrt{1-x^2}} + \dfrac{1}{1+x^2} \geq 3$$
8 replies
sqing
Apr 26, 2025
sqing
Today at 1:31 PM
J
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đề hsg toán
akquysimpgenyabikho   1
N Today at 12:16 PM by Lankou
làm ơn giúp tôi giải đề hsg

1 reply
akquysimpgenyabikho
Apr 27, 2025
Lankou
Today at 12:16 PM
J
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Inequalities
sqing   2
N Today at 10:05 AM by sqing
Let $a,b,c> 0$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1.$ Prove that
$$ (1-abc) (1-a)(1-b)(1-c) \ge 208 $$$$ (1+abc) (1-a)(1-b)(1-c) \le -224 $$$$(1+a^2b^2c^2) (1-a)(1-b)(1-c) \le -5840 $$
2 replies
sqing
Jul 12, 2024
sqing
Today at 10:05 AM
J
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9 Physical or online
wimpykid   0
Today at 6:49 AM
Do you think the AoPS print books or the online books are better?

0 replies
wimpykid
Today at 6:49 AM
0 replies
J
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Three variables inequality
Headhunter   6
N Today at 6:08 AM by lbh_qys
$\forall a\in R$ ,$~\forall b\in R$ ,$~\forall c \in R$
Prove that at least one of $(a-b)^{2}$, $(b-c)^{2}$, $(c-a)^{2}$ is not greater than $\frac{a^{2}+b^{2}+c^{2}}{2}$.

I assume that all are greater than it, but can't go more.
6 replies
Headhunter
Apr 20, 2025
lbh_qys
Today at 6:08 AM
J
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J
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concurrent wanted, tangents of circles (AKC), (ABK) where K lies on BC
parmenides51   0
May 28, 2022
Source: 2021 SPbU finals grades 8-9 p4 v1 - Saint Petersburg State University School Olympiad
A point $K$ is marked on side $BC$ of triangle $ABC$. To the circumscribed circle of the triangle $AKC$, the tangent $\ell_1$ is drawn, parallel to the line $AB$ and nearest to it. It touched circle at point $ L$. Line $AL$ intersects the circumscribed circle of triangle $ABK$ at point $M$ ($M \ne A$). The tangent $\ell_2$ is drawn to this circle at the point $M$. Prove that the lines $BK$, $\ell_1$ and $\ell_2$ intersect at one point.
0 replies
parmenides51
May 28, 2022
0 replies
J
concurrent wanted, tangents of circles (AKC), (ABK) where K lies on BC
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Reveal topic tags
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Source: 2021 SPbU finals grades 8-9 p4 v1 - Saint Petersburg State University School Olympiad
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parmenides51
30650 posts
parmenides51
#1 h  May 28, 2022, 7:09 PM
Y by
A point $K$ is marked on side $BC$ of triangle $ABC$. To the circumscribed circle of the triangle $AKC$, the tangent $\ell_1$ is drawn, parallel to the line $AB$ and nearest to it. It touched circle at point $ L$. Line $AL$ intersects the circumscribed circle of triangle $ABK$ at point $M$ ($M \ne A$). The tangent $\ell_2$ is drawn to this circle at the point $M$. Prove that the lines $BK$, $\ell_1$ and $\ell_2$ intersect at one point.
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