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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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units as many as nilpotent elements implies there are a power of 2 of elements
CatalinBordea   1
N 11 minutes ago by FFA21
In a certain ring there are as many units as there are nilpotent elements. Prove that the order of the ring is a power of $ 2. $

Dinu Şerbănescu
1 reply
CatalinBordea
Feb 2, 2024
FFA21
11 minutes ago
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primes in a sequence
kapilpavase   2
N 3 hours ago by ihategeo_1969
Source: STEMS 2021 Math Cat C Q3
Let $p \in \mathbb{N} \setminus \{0, 1\}$ be a fixed positive integer. Prove that for every $K > 0$, there exist infinitely many $n$ and $N$ such that there are atleast $\dfrac{KN}{\log(N)}$ primes among the following $N$ numbers given by
\[n + 1, n + 2^p, n + 3^p, \cdots, n + N^p.\]
Proposed by Bimit Mandal
2 replies
kapilpavase
Jan 25, 2021
ihategeo_1969
3 hours ago
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Angle oriented geometry
Problems_eater   0
3 hours ago
Let $A, B, C,D$ be four distinct points in the plane.
Which of the following statements, expressed using oriented angles, are always true?

1.If lines $AB$ and $CD$ are distinct and parallel, then
the oriented angle $ABC$ is equal to the oriented angle DCB.

2.If $B$ lies on the segment $AC$, then
the oriented angle $DBA$ plus the oriented angle $DBC $equals $180°$.

3.If the oriented angle$ ABC$ plus the oriented angle $BCD$ equals 0°, then
lines $AB $and $CD$ are parallel.

4.If the oriented angle $ABC$ plus the oriented angle $BCD$ equals $180°$, then
lines $AB$ and $CD$are parallel.
0 replies
Problems_eater
3 hours ago
0 replies
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Group theory
FFA21   6
N 3 hours ago by RobertRogo
Source: MSU 2001 olimpiad P6
$|G|\leq 2001$. $|G|$ not divisible by 2. Prove $\exists S\subset G$ ($S$ is a subset, not necessarily a subgroup) that $|S|\leq 20$ and $\forall g\in G$ $\exists n$ that $\exists s_1....s_n\in S$ that$s_i=s_j \implies i=j$ and $s_1*s_2*.....*s_n=g$
6 replies
FFA21
Today at 12:42 AM
RobertRogo
3 hours ago
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how many quadrilaterals ?
Ecrin_eren   6
N Today at 5:31 PM by mathprodigy2011
"All the diagonals of an 11-gon are drawn. How many quadrilaterals can be formed using these diagonals as sides? (The vertices of the quadrilaterals are selected from the vertices of the 11-gon.)"
6 replies
Ecrin_eren
Apr 13, 2025
mathprodigy2011
Today at 5:31 PM
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Limit finding
CuriousBabu   8
N Today at 5:14 PM by 3ch03s
Let \( f(x) = \tan x \), and define

\[ g_r(x) = \frac{1}{\tan^{r+1} x} \cdot \frac{d^r}{dx^r} (\tan x) \]
Find
\[ \lim_{x \to \frac{\pi}{2}} g_r(x) \]
8 replies
CuriousBabu
Yesterday at 4:12 PM
3ch03s
Today at 5:14 PM
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Plane geometry problem with inequalities
ReticulatedPython   3
N Today at 2:48 PM by vanstraelen
Let $A$ and $B$ be points on a plane such that $AB=1.$ Let $P$ be a point on that plane such that $$\frac{AP^2+BP^2}{(AP)(BP)}=3.$$Prove that $$AP \in \left[\frac{5-\sqrt{5}}{10}, \frac{-1+\sqrt{5}}{2}\right] \cup \left[\frac{5+\sqrt{5}}{10}, \frac{1+\sqrt{5}}{2}\right].$$
Source: Own
3 replies
ReticulatedPython
Apr 10, 2025
vanstraelen
Today at 2:48 PM
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Inequalities
sqing   1
N Today at 1:55 PM by sqing
Let $ a,b $ be reals such that $ a^2-ab+b^2 =3$ . Prove that
$$ \frac{13}{ 10 }> \frac{1}{ a^2+1 }+ \frac{1}{ b^2+1 } \geq \frac{1}{ 2 }$$$$ \frac{6}{ 5 }>\frac{1}{ a^4+1 }+ \frac{1}{ b^4+1 } \geq \frac{1}{ 5 }$$$$ \frac{1}{ a^6+1 }+ \frac{1}{ b^6+1 } \geq \frac{1}{ 14 }$$
1 reply
sqing
Today at 8:59 AM
sqing
Today at 1:55 PM
J
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idk12345678 Math Contest
idk12345678   21
N Today at 1:25 PM by idk12345678
Welcome to the 1st idk12345678 Math Contest.
You have 4 hours. You do not have to prove your answers.
Post \signup username to sign up. Post your answers in a hide tag and I will tell you your score.*


The contest is attached to the post

Clarifications

*I mightve done them wrong feel free to ask about an answer
21 replies
idk12345678
Apr 10, 2025
idk12345678
Today at 1:25 PM
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purple comet math competition question
AVY2024   4
N Today at 1:02 PM by K1mchi_
Given that (1 + tan 1)(1 + tan 2). . .(1 + tan 45) = 2n, find n
4 replies
AVY2024
Today at 11:00 AM
K1mchi_
Today at 1:02 PM
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Inequalities
sqing   25
N Today at 12:06 PM by sqing
Let $ a,b,c,d>0 $ and $(a+c)(b+d)=ac+\frac{3}{2}bd.$ Prove that
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq \frac{20-\sqrt{10}}{3}$$Let $ a,b,c,d>0 $ and $(a+c)(b+d)=ac+\frac{4}{3}bd.$ Prove that
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq \frac{21-\sqrt{6}}{3}$$
25 replies
sqing
Dec 3, 2024
sqing
Today at 12:06 PM
J
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Polynomials
CuriousBabu   3
N Today at 11:40 AM by osszhangbanvan
\[ \frac{(x+y+z)^5 - x^5 - y^5 - z^5}{(x+y)(y+z)(z+x)} = 0 \]
Find the number of real solutions.
3 replies
CuriousBabu
Yesterday at 4:09 PM
osszhangbanvan
Today at 11:40 AM
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Combination
aria123   0
Today at 10:59 AM
Prove that three squares of side length $4$ cannot completely cover a square of side length $5$, if the three smaller squares do not overlap in their interiors (i.e., they may touch at edges or corners, but no part of one lies over another).
0 replies
aria123
Today at 10:59 AM
0 replies
J
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Geo Mock #6
Bluesoul   3
N Today at 3:16 AM by dudade
Consider triangle $ABC$ with $AB=5, BC=8, AC=7$, denote the incenter of the triangle as $I$. Extend $BI$ to meet the circumcircle of $\triangle{AIC}$ at $Q\neq I$, find the length of $QC$.
3 replies
Bluesoul
Apr 1, 2025
dudade
Today at 3:16 AM
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D753 : A strange series
Dattier   4
N Apr 6, 2025 by KevinKV01
Source: les dattes à Dattier
Determinate $\sum \limits_{i=1}^\infty \sqrt[3]{\sin(i)}$.
4 replies
Dattier
Jan 24, 2024
KevinKV01
Apr 6, 2025
J
D753 : A strange series
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Reveal topic tags
real analysis
infinite series
L
G H BBookmark kLocked kLocked NReply
Source: les dattes à Dattier
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Dattier
1476 posts
Dattier
#1 h  Jan 24, 2024, 9:32 PM
Y by
Determinate $\sum \limits_{i=1}^\infty \sqrt[3]{\sin(i)}$.
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m0nk
6 posts
m0nk
#2 h  Apr 5, 2025, 12:10 PM • 1 Y
Y by franklin2013
This clearly diverges
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Alphaamss
234 posts
Alphaamss
#3 h  Apr 6, 2025, 2:49 AM • 1 Y
Y by LawofCosine
Dattier wrote:
Determinate $\sum \limits_{i=1}^\infty \sqrt[3]{\sin(i)}$.
The limit $$\lim_{n\to\infty}\sqrt[3]{\sin(n)}$$doesn't exist(which is equivalent to the exisence of limit $\lim_{n\to\infty}{\sin(n)}$).
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m0nk
6 posts
m0nk
#4 h  Apr 6, 2025, 4:32 AM
Y by
And there are infinitely many $k \in Z$ such that $\sin(k) =1$ summing infinitely many 1's is infinity
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KevinKV01
14 posts
KevinKV01
#5 h  Apr 6, 2025, 4:37 AM
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