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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.

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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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Inequalities
sqing   15
N a minute ago by sqing
Let $ a,b>0 $ and $ a+ b^2=\frac{3}{4} $.Prove that
$$ \frac{1}{a^3(a+b)} + \frac{2}{b^3(2b+1)} + \frac{16}{2a+1} \geq 24$$Let $ a,b>0 $ and $a^2+b^2=\frac{1}{2} $.Prove that
$$ \frac{1}{a^3(a+b)} + \frac{2}{b^3(2b+1)} + \frac{16}{2a+1} \geq 24$$
15 replies
sqing
Nov 29, 2024
sqing
a minute ago
J
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Inequalities
sqing   1
N an hour ago by sqing
Let $ a,b,c>0 . $ Prove that
$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq 4\left(\frac{a+b}{b+c}+ \frac{b+c}{a+b}\right)$$$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq \frac{32}{9}\left(\frac{a+b}{b+c}+ \frac{c+a}{a+b}\right)$$$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq \frac{8}{3}\left( \frac{a+b}{b+c}+ \frac{b+c}{c+a}+ \frac{c+a}{a+b}\right)$$$$ \left(1 +\frac{a^2}{b^2}\right)\left(1+\frac{b^2}{c^2}\right)\left(1+\frac{c^2}{a^2}\right )\geq \frac{8}{3}\left( \frac{a^2+bc}{b^2+ca}+\frac{b^2+ca}{c^2+ab}+\frac{c^2+ab}{a^2+bc}\right)$$
1 reply
sqing
Today at 12:20 AM
sqing
an hour ago
J
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Inequalities
sqing   6
N an hour ago by sqing
Let $a,b,c> 0$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1.$ Prove that
$$ (1-abc) (1-a)(1-b)(1-c) \ge 208 $$$$ (1+abc) (1-a)(1-b)(1-c) \le -224 $$$$(1+a^2b^2c^2) (1-a)(1-b)(1-c) \le -5840 $$
6 replies
sqing
Jul 12, 2024
sqing
an hour ago
J
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A problem with a rectangle
Raul_S_Baz   11
N 2 hours ago by Raul_S_Baz
On the sides AB and AD of the rectangle ABCD, points M and N are taken such that MB = ND. Let P be the intersection of BN and CD, and Q be the intersection of DM and CB. How can we prove that PQ || MN?
IMAGE
11 replies
Raul_S_Baz
Apr 26, 2025
Raul_S_Baz
2 hours ago
J
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(14n+25)/(2n+1) 'is a perfect square - Portugal OPM 2017 p1
parmenides51   3
N 2 hours ago by justaguy_69
Determine all integer values of n for which the number $\frac{14n+25}{2n+1}$ 'is a perfect square.
3 replies
parmenides51
May 15, 2024
justaguy_69
2 hours ago
J
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(3x-1)^2/x+(3y-1)^2/y >=1, for x,y>0, x+y=1 Austria Beginners' 2010 p3
parmenides51   22
N 3 hours ago by justaguy_69
Let $x$ and $y$ be positive real numbers with $x + y =1 $. Prove that
$$\frac{(3x-1)^2}{x}+ \frac{(3y-1)^2}{y} \ge1.$$For which $x$ and $y$ equality holds?

(K. Czakler, GRG 21, Vienna)
22 replies
parmenides51
Oct 3, 2021
justaguy_69
3 hours ago
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New but easy
ZETA_in_olympiad   2
N 3 hours ago by jasperE3
Find all functions $f:\mathbb R\to \mathbb R$ such that $$f(f(x)+f(y))=xf(y)+yf(x)$$for all $x,y\in \mathbb R.$
2 replies
ZETA_in_olympiad
Oct 1, 2022
jasperE3
3 hours ago
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2025 CMIMC team p7, rephrased
scannose   4
N 3 hours ago by scannose
In the expansion of $(x^2 + x + 1)^{2024}$, find the number of terms with coefficient divisible by $3$.
4 replies
scannose
Apr 18, 2025
scannose
3 hours ago
J
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DA + AE = KC +CM = AB=BC=CA - All-Russian MO 1997 Regional (R4) 8.3
parmenides51   2
N 4 hours ago by sunken rock
On sides $AB$ and $BC$ of an equilateral triangle $ABC$ are taken points $D$ and $K$, and on the side $AC$ , points $E$ and $M$ so that $DA + AE = KC +CM = AB$. Prove that the angle between lines $DM$ and $KE$ is equal to $60^o$.
2 replies
parmenides51
Sep 23, 2024
sunken rock
4 hours ago
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Frankenstein FE
NamelyOrange   3
N 6 hours ago by jasperE3
[quote = My own problem]Solve the FE $f(x)+f(-x)=2f(x^2)$ over $\mathbb{R}$. Ignore "pathological" solutions.[/quote]

How do I solve this? I made this while messing around, and I have no clue as to what to do...
3 replies
NamelyOrange
Jul 19, 2024
jasperE3
6 hours ago
J
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Find the domain and range of $f(x)=\frac{1}{1-2\cos x}.$
Vulch   1
N Today at 2:22 AM by aidan0626
Find the domain and range of $f(x)=\frac{1}{1-2\cos x}.$
1 reply
Vulch
Today at 2:06 AM
aidan0626
Today at 2:22 AM
J
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Find the domain and range of $f(x)=\frac{4-x}{x-4}.$
Vulch   1
N Today at 2:09 AM by aidan0626
Find the domain and range of $f(x)=\frac{4-x}{x-4}.$
1 reply
Vulch
Today at 2:02 AM
aidan0626
Today at 2:09 AM
J
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Find the domain and range of $f(x)=2-|x-5|.$
Vulch   0
Today at 2:07 AM
Find the domain and range of $f(x)=2-|x-5|.$
0 replies
Vulch
Today at 2:07 AM
0 replies
J
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Amc 10 mock
Mathsboy100   4
N Today at 1:01 AM by pooh123
let \[\lfloor x \rfloor\]denote the greatest integer less than or equal to x . What is the sum of the squares of the real numbers x for which \[ x^2 - 20\lfloor x \rfloor + 19 = 0 \]
4 replies
Mathsboy100
Oct 9, 2024
pooh123
Today at 1:01 AM
J
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J
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glue 2 triangles to get a third (2013-14 Savin Competition 6-8 p14)
parmenides51   1
N Jun 18, 2021 by EVTrainer
Three triangles are given: an acute, a right and an obtuse. It is known that two of them can be attached to each other so that it turns out a triangle equal to the third. Which of the three given triangles is it equal to?
1 reply
parmenides51
Jun 17, 2021
EVTrainer
Jun 18, 2021
J
glue 2 triangles to get a third (2013-14 Savin Competition 6-8 p14)
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Reveal topic tags
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parmenides51
30650 posts
parmenides51
#1 h  Jun 17, 2021, 7:59 PM
Y by
Three triangles are given: an acute, a right and an obtuse. It is known that two of them can be attached to each other so that it turns out a triangle equal to the third. Which of the three given triangles is it equal to?
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EVTrainer
502 posts
EVTrainer
#2 h  Jun 18, 2021, 12:54 AM • 1 Y
Y by Mango247
If you split a right triangle with a line from the vertex with the right angle to any point on the hypotenuse besides the midpoint, you will create an acute triangle and an obtuse one.
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