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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 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
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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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(3x-1)^2/x+(3y-1)^2/y >=1, for x,y>0, x+y=1 Austria Beginners' 2010 p3
parmenides51   22
N 41 minutes 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
41 minutes ago
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New but easy
ZETA_in_olympiad   2
N an hour 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
an hour ago
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2025 CMIMC team p7, rephrased
scannose   4
N an hour 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
an hour ago
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DA + AE = KC +CM = AB=BC=CA - All-Russian MO 1997 Regional (R4) 8.3
parmenides51   2
N 2 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
2 hours ago
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Frankenstein FE
NamelyOrange   3
N 3 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
3 hours ago
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Find the domain and range of $f(x)=\frac{1}{1-2\cos x}.$
Vulch   1
N 5 hours ago by aidan0626
Find the domain and range of $f(x)=\frac{1}{1-2\cos x}.$
1 reply
Vulch
5 hours ago
aidan0626
5 hours ago
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Find the domain and range of $f(x)=\frac{4-x}{x-4}.$
Vulch   1
N 5 hours ago by aidan0626
Find the domain and range of $f(x)=\frac{4-x}{x-4}.$
1 reply
Vulch
5 hours ago
aidan0626
5 hours ago
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Find the domain and range of $f(x)=2-|x-5|.$
Vulch   0
5 hours ago
Find the domain and range of $f(x)=2-|x-5|.$
0 replies
Vulch
5 hours ago
0 replies
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Inequalities
sqing   5
N Today at 1:13 AM by pooh123
Let $a,b,c> 0$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1.$ Prove that
$$ (1-abc) (1-a)(1-b)(1-c) \ge 208 $$$$ (1+abc) (1-a)(1-b)(1-c) \le -224 $$$$(1+a^2b^2c^2) (1-a)(1-b)(1-c) \le -5840 $$
5 replies
sqing
Jul 12, 2024
pooh123
Today at 1:13 AM
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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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Inequalities
sqing   0
Today at 12:20 AM
Let $ a,b,c>0 . $ Prove that
$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq 4\left(\frac{a+b}{b+c}+ \frac{b+c}{a+b}\right)$$$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq \frac{32}{9}\left(\frac{a+b}{b+c}+ \frac{c+a}{a+b}\right)$$$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq \frac{8}{3}\left( \frac{a+b}{b+c}+ \frac{b+c}{c+a}+ \frac{c+a}{a+b}\right)$$$$ \left(1 +\frac{a^2}{b^2}\right)\left(1+\frac{b^2}{c^2}\right)\left(1+\frac{c^2}{a^2}\right )\geq \frac{8}{3}\left( \frac{a^2+bc}{b^2+ca}+\frac{b^2+ca}{c^2+ab}+\frac{c^2+ab}{a^2+bc}\right)$$
0 replies
sqing
Today at 12:20 AM
0 replies
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Purple Comet High School Math Meet 2024 P1
franklin2013   3
N Today at 12:10 AM by codegirl2013
Joe ate one half of a fifth of a pizza. Gale ate one third of a quarter of that pizza. The difference in the amounts that the two ate was $\frac{1}{n}$ of the pizza, where $n$ is a positive integer. Find $n$.
3 replies
franklin2013
Yesterday at 11:20 PM
codegirl2013
Today at 12:10 AM
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Cool vieta sum
Kempu33334   0
Yesterday at 11:44 PM
Let the roots of \[\mathcal{P}(x) = x^{108}+x^{102}+x^{96}+2x^{54}+3x^{36}+4x^{24}+5x^{18}+6\]be $r_1, r_2, \dots, r_{108}$. Find \[\dfrac{r_1^6+r_2^6+\dots+r_{108}^6}{r_1^6r_2^6+r_1^6r_3^6+\dots+r_{107}^6r_{108}^6}\]without Newton Sums.
0 replies
Kempu33334
Yesterday at 11:44 PM
0 replies
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Inequality, tougher than it looks
tom-nowy   2
N Yesterday at 11:28 PM by pooh123
Prove that for $a,b \in \mathbb{R}$
$$ 2(a^2+1)(b^2+1) \geq 3(a+b). $$Is there an elegant way to prove this?
2 replies
tom-nowy
Apr 29, 2025
pooh123
Yesterday at 11:28 PM
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J
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congruent triangles wanted 2011 Armenia NMO 8.3
parmenides51   0
Aug 3, 2021
Prove that $\vartriangle ABC=\vartriangle A_1B_1C_1$, if it is known that $AB=A_1B_1$, $BC=B_1C_1$ and $\angle A-\angle C= \angle A_1 -\angle C_1 > 0$.
0 replies
parmenides51
Aug 3, 2021
0 replies
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congruent triangles wanted 2011 Armenia NMO 8.3
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parmenides51
30650 posts
parmenides51
#1 h  Aug 3, 2021, 7:31 PM
Y by
Prove that $\vartriangle ABC=\vartriangle A_1B_1C_1$, if it is known that $AB=A_1B_1$, $BC=B_1C_1$ and $\angle A-\angle C= \angle A_1 -\angle C_1 > 0$.
This post has been edited 1 time. Last edited by parmenides51, Aug 3, 2021, 7:44 PM
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