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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jun 2, 2025
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!

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0 replies
jlacosta
Jun 2, 2025
0 replies
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Beware the degeneracies!
Rijul saini   5
N 23 minutes ago by atdaotlohbh
Source: India IMOTC 2025 Day 1 Problem 1
Let $a,b,c$ be real numbers satisfying $$\max \{a(b^2+c^2),b(c^2+a^2),c(a^2+b^2) \} \leqslant 2abc+1$$Prove that $$a(b^2+c^2)+b(c^2+a^2)+c(a^2+b^2) \leqslant 6abc+2$$and determine all cases of equality.

Proposed by Shantanu Nene
5 replies
Rijul saini
Yesterday at 6:30 PM
atdaotlohbh
23 minutes ago
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polonomials
Ducksohappi   3
N 35 minutes ago by Ducksohappi
Let $P(x)$ be the real polonomial such that all roots are real and distinct. Prove that there is a rational number $r\ne 0 $ that all roots of $Q(x)=$ $P(x+r)-P(x)$ are real numbers
3 replies
Ducksohappi
Apr 10, 2025
Ducksohappi
35 minutes ago
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Easy Diff NT
xToiletG   0
an hour ago
Prove that for every $n \geq 2$ there exists positive integers $x, y, z$ such that
$$\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$
0 replies
xToiletG
an hour ago
0 replies
J
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Might be slightly generalizable
Rijul saini   5
N an hour ago by guptaamitu1
Source: India IMOTC Day 3 Problem 1
Let $ABC$ be an acute angled triangle with orthocenter $H$ and $AB<AC$. Let $T(\ne B,C, H)$ be any other point on the arc $\stackrel{\LARGE\frown}{BHC}$ of the circumcircle of $BHC$ and let line $BT$ intersect line $AC$ at $E(\ne A)$ and let line $CT$ intersect line $AB$ at $F(\ne A)$. Let the circumcircles of $AEF$ and $ABC$ intersect again at $X$ ($\ne A$). Let the lines $XE,XF,XT$ intersect the circumcircle of $(ABC)$ again at $P,Q,R$ ($\ne X$). Prove that the lines $AR,BC,PQ$ concur.
5 replies
Rijul saini
Yesterday at 6:39 PM
guptaamitu1
an hour ago
J
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functional analysis
ILOVEMYFAMILY   1
N Yesterday at 2:27 PM by alexheinis
Let $E$, $F$ be normed spaces with $E$ a Banach space. Suppose $\{A_n: E \to F\}$ is a family of continuous linear maps. Prove that the set
\[ X = \left\{ x \in E \mid \sup_{n\geq 1}|| A_n(x)||< +\infty \right\} \]is either of first category in $E$ or is equal to the whole space $E$.
1 reply
ILOVEMYFAMILY
Jun 2, 2025
alexheinis
Yesterday at 2:27 PM
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Geometry
krrispes   1
N Yesterday at 11:03 AM by Mathzeus1024
Plz solve by using optimization method
1 reply
krrispes
Jan 19, 2025
Mathzeus1024
Yesterday at 11:03 AM
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Limit problem
Martin.s   2
N Tuesday at 8:36 PM by KAME06
Find \(\lim_{n \to \infty} n \sin (2n! e \pi)\)
2 replies
Martin.s
Jun 1, 2025
KAME06
Tuesday at 8:36 PM
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IMC 2021 P8: Maximum number of vectors such that for any 3, 2 are orthogonal
Sumgato   17
N Tuesday at 2:28 PM by lminsl
Source: IMC 2021 P8
Let $n$ be a positive integer. At most how many distinct unit vectors can be selected in $\mathbb{R}^n$ such that from any three of them, at least two are orthogonal?
17 replies
Sumgato
Aug 5, 2021
lminsl
Tuesday at 2:28 PM
J
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D1043 : A general result about diffeomorphisme
Dattier   0
Tuesday at 1:55 PM
Source: les dattes à Dattier
Let $f \in C^2([0,1],[0,1])$ diffeomorphisme bijectif.

Is it true that $\int_0^1f(t) \text{d}t \leq \dfrac12+ \dfrac{\max(|f''|)}{5\times \min(|f'|^3)}$?
0 replies
Dattier
Tuesday at 1:55 PM
0 replies
J
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IMC 2009 Day 1 P2
joybangla   3
N Tuesday at 11:23 AM by lminsl
Let $A,B,C$ be real square matrices of the same order, and suppose $A$ is invertible. Prove that
\[ (A-B)C=BA^{-1}\implies C(A-B)=A^{-1}B \]
3 replies
joybangla
Jul 15, 2014
lminsl
Tuesday at 11:23 AM
J
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Expand into a Fourier series
Tip_pay   2
N Tuesday at 10:38 AM by Mathzeus1024
Expand the function in a Fourier series on the interval $(-\pi, \pi)$
$$f(x)=\begin{cases} 1, & -1<x\leq 0\\ x, & 0<x<1 \end{cases}$$
2 replies
Tip_pay
Dec 12, 2023
Mathzeus1024
Tuesday at 10:38 AM
J
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D1041 : A generalisation of Tchebychef's Inequality
Dattier   1
N Jun 3, 2025 by Dattier
Source: les dattes à Dattier
Let $f,g \in C^1([0,1])$.

Is it true that : $\min(|f'|)\times \min(|g'|) \leq 12\times \left|\int_0^1f(t)\times g(t) \text{d}t -\int_0^1f(t) \text{d}t\times \int_0^1g(t)\text{d}t\right| \leq \max(|f'|)\times \max(|g'|)$?
1 reply
Dattier
Jun 2, 2025
Dattier
Jun 3, 2025
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Putnam 2013 A5
Kent Merryfield   10
N Jun 2, 2025 by blackbluecar
For $m\ge 3,$ a list of $\binom m3$ real numbers $a_{ijk}$ $(1\le i<j<k\le m)$ is said to be area definite for $\mathbb{R}^n$ if the inequality \[\sum_{1\le i<j<k\le m}a_{ijk}\cdot\text{Area}(\triangle A_iA_jA_k)\ge0\] holds for every choice of $m$ points $A_1,\dots,A_m$ in $\mathbb{R}^n.$ For example, the list of four numbers $a_{123}=a_{124}=a_{134}=1, a_{234}=-1$ is area definite for $\mathbb{R}^2.$ Prove that if a list of $\binom m3$ numbers is area definite for $\mathbb{R}^2,$ then it is area definite for $\mathbb{R}^3.$
10 replies
Kent Merryfield
Dec 9, 2013
blackbluecar
Jun 2, 2025
J
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Reducing the exponents for good
RobertRogo   3
N Jun 2, 2025 by RobertRogo
Source: The national Algebra contest (Romania), 2025, Problem 3/Abstract Algebra (a bit generalized)
Let $A$ be a ring with unity such that for every $x \in A$ there exist $t_x, n_x \in \mathbb{N}^*$ such that $x^{t_x+n_x}=x^{n_x}$. Prove that
a) If $t_x \cdot 1 \in U(A), \forall x \in A$ then $x^{t_x+1}=x, \forall x \in A$
b) If there is an $x \in A$ such that $t_x \cdot 1 \notin U(A)$ then the result from a) may no longer hold.

Authors: Laurențiu Panaitopol, Dorel Miheț, Mihai Opincariu, me, Filip Munteanu
3 replies
RobertRogo
May 20, 2025
RobertRogo
Jun 2, 2025
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handout I wrote on the orthocenter
winnertakeover   4
N Mar 26, 2019 by Kagebaka
Here is something I wrote a while back. It's about lengths related to the orthocenter. I hope you find this useful!
4 replies
winnertakeover
Jan 3, 2019
Kagebaka
Mar 26, 2019
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handout I wrote on the orthocenter
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Reveal topic tags
orthocenter
geometry
article
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G H BBookmark kLocked kLocked NReply
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winnertakeover
1179 posts
winnertakeover
#1 h  Jan 3, 2019, 5:10 AM • 6 Y
Y by Wizard_32, AlastorMoody, Kagebaka, khan.academy, anonman, Adventure10
Here is something I wrote a while back. It's about lengths related to the orthocenter. I hope you find this useful!
Attachments:
Ortho_DIST.pdf (222kb)
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Atajan
95 posts
Atajan
#2 h  Jan 3, 2019, 6:59 AM • 2 Y
Y by Adventure10, Mango247
thanks it is useful
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winnertakeover
1179 posts
winnertakeover
#3 h  Jan 3, 2019, 6:17 PM • 2 Y
Y by Adventure10, Mango247
Here are just the problems, if you wish to solve the example problems yourself.
Attachments:
Ortho_DIST.pdf (160kb)
This post has been edited 1 time. Last edited by winnertakeover, Jan 3, 2019, 6:18 PM
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AlastorMoody
2125 posts
AlastorMoody
#4 h  Jan 6, 2019, 11:41 AM • 2 Y
Y by TheCosineLaw, Adventure10
\bump for others
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Kagebaka
3001 posts
Kagebaka
#5 h  Mar 26, 2019, 3:11 AM • 1 Y
Y by Adventure10
\bump again :o
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