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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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[*]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 problem with a rectangle
Raul_S_Baz   12
N 24 minutes ago by george_54
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
12 replies
Raul_S_Baz
Apr 26, 2025
george_54
24 minutes ago
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(14n+25)/(2n+1) 'is a perfect square - Portugal OPM 2017 p1
parmenides51   4
N an hour ago by Namisgood
Determine all integer values of n for which the number $\frac{14n+25}{2n+1}$ 'is a perfect square.
4 replies
parmenides51
May 15, 2024
Namisgood
an hour ago
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Inequalities
sqing   4
N an hour ago by sqing
Let $ a,b,c>0 . $ Prove that
$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq 4\left(\frac{a+b}{b+c}+ \frac{b+c}{a+b}\right)$$$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq \frac{32}{9}\left(\frac{a+b}{b+c}+ \frac{c+a}{a+b}\right)$$$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq \frac{8}{3}\left( \frac{a+b}{b+c}+ \frac{b+c}{c+a}+ \frac{c+a}{a+b}\right)$$$$ \left(1 +\frac{a^2}{b^2}\right)\left(1+\frac{b^2}{c^2}\right)\left(1+\frac{c^2}{a^2}\right )\geq \frac{8}{3}\left( \frac{a^2+bc}{b^2+ca}+\frac{b^2+ca}{c^2+ab}+\frac{c^2+ab}{a^2+bc}\right)$$
4 replies
sqing
Today at 12:20 AM
sqing
an hour ago
J
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Inequalities
sqing   15
N an hour 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
an hour ago
J
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Inequalities
sqing   6
N 3 hours 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
3 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 4 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
4 hours ago
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New but easy
ZETA_in_olympiad   2
N 4 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
4 hours ago
J
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2025 CMIMC team p7, rephrased
scannose   4
N 5 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
5 hours ago
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DA + AE = KC +CM = AB=BC=CA - All-Russian MO 1997 Regional (R4) 8.3
parmenides51   2
N 5 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
5 hours ago
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Frankenstein FE
NamelyOrange   3
N Today at 3:53 AM 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
Today at 3:53 AM
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
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J
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area of a pentagon wanted, 2 right angles (2018 Auckland MO J3)
parmenides51   3
N Dec 31, 2020 by HamstPan38825
Consider the pentagon below. Find its area.
IMAGE
3 replies
parmenides51
Dec 31, 2020
HamstPan38825
Dec 31, 2020
J
area of a pentagon wanted, 2 right angles (2018 Auckland MO J3)
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Reveal topic tags
geometry
pentagon
area
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parmenides51
30650 posts
parmenides51
#1 h  Dec 31, 2020, 7:29 AM
Y by
Consider the pentagon below. Find its area.
https://cdn.artofproblemsolving.com/attachments/7/b/02ad3852b72682513cf62a170ed4aa45c23785.png
This post has been edited 1 time. Last edited by parmenides51, Dec 31, 2020, 7:30 AM
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parmenides51
30650 posts
parmenides51
#2 h  Dec 31, 2020, 7:30 AM
Y by
posted for the image link
Attachments:
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natmath
8219 posts
natmath
#3 h  Dec 31, 2020, 7:58 AM
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This post has been edited 1 time. Last edited by natmath, Dec 31, 2020, 7:59 AM
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HamstPan38825
8857 posts
HamstPan38825
#4 h  Dec 31, 2020, 3:59 PM
Y by
Pretty simple by splitting the thing up.

natmath you should sleep if I presume you are in the correct timezone :P
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