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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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0 replies
jlacosta
Apr 2, 2025
0 replies
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Inequalities
sqing   20
N 24 minutes ago by sqing
Let $ a,b,c> 0 $ and $ ab+bc+ca\leq 3abc . $ Prove that
$$ a+ b^2+c\leq a^2+ b^3+c^2 $$$$ a+ b^{11}+c\leq a^2+ b^{12}+c^2 $$
20 replies
+1 w
sqing
Apr 22, 2025
sqing
24 minutes ago
J
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Nested Permutations
P_Groudon   1
N an hour ago by P_Groudon
Let $S = \{1, 2, 3, 4, 5\}$ and let $\sigma_1 : S \to S$ and $\sigma_2 : S \to S$ be permutations of $S$. Suppose there exists a permutation $\tau : S \to S$ such that $\sigma_1(\tau(s)) = \tau(\sigma_2(s))$ for all $s$ in $S$.

If $N$ is the number of possible pairs of permutations $(\sigma_1, \sigma_2)$, find the remainder when $N$ is divided by 1000.
1 reply
P_Groudon
an hour ago
P_Groudon
an hour ago
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Inequalities from SXTX
sqing   13
N 3 hours ago by sqing
T702. Let $ a,b,c>0 $ and $ a+2b+3c=\sqrt{13}. $ Prove that $$ \sqrt{a^2+1} +2\sqrt{b^2+1} +3\sqrt{c^2+1} \geq 7$$S
T703. Let $ a,b $ be real numbers such that $ a+b\neq 0. $. Find the minimum of $ a^2+b^2+(\frac{1-ab}{a+b} )^2.$
T704. Let $ a,b,c>0 $ and $ a+b+c=3. $ Prove that $$ \frac{a^2+7}{(c+a)(a+b)} + \frac{b^2+7}{(a+b)(b+c)} +\frac{c^2+7}{(b+c)(c+a)} \geq 6$$S
13 replies
sqing
Feb 18, 2025
sqing
3 hours ago
J
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Combinatorics problem
lgx57   1
N 5 hours ago by alexheinis
There are $100$ positive integer arrays $\{a_{1,n}\},\{a_{2,n}\},\cdots,\{a_{100,n}\}$, and they are all infinity arrays.
Prove that: There exists $ m<k$, that for every integer $1 \le i \le 100, a_{i,m}<a_{i,k}$
1 reply
lgx57
Apr 14, 2025
alexheinis
5 hours ago
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stat moment
fruitmonster97   5
N Apr 16, 2025 by aidan0626
Define xiooix deviation of a set as the average positive difference from each element and the average of said set. Prove that the xiooix deviation of a set is less than or equal to standard deviation.
5 replies
fruitmonster97
Apr 15, 2025
aidan0626
Apr 16, 2025
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stat moment
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fruitmonster97
2479 posts
fruitmonster97
#1 h  Apr 15, 2025, 9:33 PM • 1 Y
Y by WiseTigerJ1
Define xiooix deviation of a set as the average positive difference from each element and the average of said set. Prove that the xiooix deviation of a set is less than or equal to standard deviation.
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aidan0626
1860 posts
aidan0626
#2 h  Apr 15, 2025, 10:29 PM
Y by
cauchy schwarz nukes this
i'll write it up later (maybe)
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Kempu33334
609 posts
Kempu33334
#3 h  Apr 15, 2025, 11:07 PM
Y by
fruitmonster97 wrote:
Define xiooix deviation of a set as the average positive difference from each element and the average of said set. Prove that the xiooix deviation of a set is less than or equal to standard deviation.

Let the set $S = \{a_1, a_2, \dots, a_n\}$, and the average be $A$. We are trying to prove that \[\sum_{i=1}^n\left| a_i-A\right| \le \sqrt{\sum_{i=1}^n (a_i-A)^2}.\]This is equivalent to \[\sum_{i=1}^n b_i \le \sqrt{\sum_{i=1}^n b_i^2}\]where $b_i = \left| a_i-A\right|$. This is true by Cauchy-Schwarz.
This post has been edited 4 times. Last edited by Kempu33334, Apr 16, 2025, 12:51 AM
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maromex
163 posts
maromex
#4 h  Apr 15, 2025, 11:26 PM
Y by
The statement should be \[\frac{1}{n}\sum_{i=1}^n\left| a_i-A\right| \le \sqrt{\frac{1}{n}\sum_{i=1}^n (a_i-A)^2}.\]This is also exactly QM-AM.

Though I'm not sure if this argument works because the problem statement never said the set had to be finite?
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BS2012
1027 posts
BS2012
#5 h  Apr 15, 2025, 11:26 PM
Y by
isnt it called the mean absolute deviation or something
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aidan0626
1860 posts
aidan0626
#6 h  Apr 16, 2025, 12:22 AM
Y by
@above yea
funny thing is that we went over the mean deviation and the standard deviation in math class a few days ago
and i proved the exact problem in op cuz i was bored
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