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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

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[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
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0 replies
jwelsh
Jul 1, 2025
0 replies
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Inequality
Ecrin_eren   4
N 30 minutes ago by Skkkippz
For positive real numbers a, b, c satisfying
ab + ac + bc = 3abc, prove that

bc / (a⁴(b + c)) + ac / (b⁴(a + c)) + ab / (c⁴(a + b)) ≥ 3/2
4 replies
Ecrin_eren
Jul 23, 2025
Skkkippz
30 minutes ago
J
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Find the value of angle C
markosa   11
N 4 hours ago by Razyed
Given a triangle ABC with base BC

angle B = 3x
angle C = x
AP is the bisector of base BC (i.e.) BP = PC
angle APB = 45 degrees

Find x

I know there are multiple methods to solve this problem using cosine law, coord geo
But is there any pure geometrical solution?
11 replies
markosa
Today at 12:45 PM
Razyed
4 hours ago
J
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Polynomials
Roots_Of_Moksha   7
N 5 hours ago by vanstraelen
The polynomial $x^3 - 3(1+\sqrt{2})x^2 + (6\sqrt{2}-55)x -(7+5\sqrt{2})$ has three distinct real roots $\alpha$, $\beta$ and $\gamma$. The polynomial $p(x)=x^3+ax^2+bx+c$ has roots $\sqrt[3]{\alpha}$, $\sqrt[3]{\beta}$, $\sqrt[3]{\gamma}$. Find the integer closest to $a^2 + b^2 + c^2$.
Answer
7 replies
Roots_Of_Moksha
Jul 20, 2025
vanstraelen
5 hours ago
J
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Inequalities
sqing   10
N Today at 2:37 PM by DAVROS
Let $ a,b> 0 $ and $2a+2b+ab=5. $ Prove that
$$ \frac{a^4}{b^4}+\frac{1}{a^4}+42ab-a^4\geq 43$$$$ \frac{a^5}{b^5}+\frac{1}{a^5}+64ab-a^5\geq 65$$$$ \frac{a^6}{b^6}+\frac{1}{a^6}+90ab-a^6\geq 91$$$$ \frac{a^7}{b^7}+\frac{1}{a^7}+121ab-a^7\geq 122$$
10 replies
sqing
May 28, 2025
DAVROS
Today at 2:37 PM
J
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2022 Putnam B6
giginori   11
N Today at 8:03 AM by Ritwin
Find all continuous functions $f:\mathbb{R}^+\rightarrow \mathbb{R}^+$ such that $$f(xf(y))+f(yf(x))=1+f(x+y)$$for all $x, y>0.$
11 replies
giginori
Dec 4, 2022
Ritwin
Today at 8:03 AM
J
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Analytic Number Theory
EthanWYX2009   1
N Today at 6:10 AM by paxtonw
Source: 2024 Jan 谜之竞赛-7
For positive integer \( n \), define \(\lambda(n)\) as the smallest positive integer satisfying the following property: for any integer \( a \) coprime with \( n \), we have \( a^{\lambda(n)} \equiv 1 \pmod{n} \).

Given an integer \( m \geq \lambda(n) \left( 1 + \ln \frac{n}{\lambda(n)} \right) \), and integers \( a_1, a_2, \cdots, a_m \) all coprime with \( n \), prove that there exists a non-empty subset \( I \) of \(\{1, 2, \cdots, m\}\) such that
\[\prod_{i \in I} a_i \equiv 1 \pmod{n}.\]Proposed by Zhenqian Peng from High School Affiliated to Renmin University of China
1 reply
EthanWYX2009
Jul 23, 2025
paxtonw
Today at 6:10 AM
J
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Putnam 2015 B1
Kent Merryfield   33
N Today at 5:43 AM by smileapple
Let $f$ be a three times differentiable function (defined on $\mathbb{R}$ and real-valued) such that $f$ has at least five distinct real zeros. Prove that $f+6f'+12f''+8f'''$ has at least two distinct real zeros.
33 replies
Kent Merryfield
Dec 6, 2015
smileapple
Today at 5:43 AM
J
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Derivative problem with nonnegative domain
EmilXM   7
N Today at 2:53 AM by MS_asdfgzxcvb
Source: Mock AYT (Turkish entrance exam)
Let $f:\mathbb{R}^+\cup\{0\}\rightarrow\mathbb{R}$ be a differentiable function. If $f(0)=3$, $f'(0)=0$ and $(f(x)-1)f''(x)=x+5$ for all $x\geq0$. Which of the followings are necessarily true:
$i) f'(2)\leq 6$
$ii) f(2)\leq\frac{26}{3}$
$iii)$ f is strictly increasing
7 replies
EmilXM
Jul 19, 2025
MS_asdfgzxcvb
Today at 2:53 AM
J
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Integral inequality with differentiable function
Ciobi_   4
N Yesterday at 4:45 PM by AngryKnot
Source: Romania NMO 2025 12.2
Let $f \colon [0,1] \to \mathbb{R} $ be a differentiable function such that its derivative is an integrable function on $[0,1]$, and $f(1)=0$. Prove that \[ \int_0^1 (xf'(x))^2 dx \geq 12 \cdot \left( \int_0^1 xf(x) dx\right)^2 \]
4 replies
Ciobi_
Apr 2, 2025
AngryKnot
Yesterday at 4:45 PM
J
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On units in a ring with a polynomial property
Ciobi_   5
N Yesterday at 3:18 PM by AngryKnot
Source: Romania NMO 2025 12.1
We say a ring $(A,+,\cdot)$ has property $(P)$ if :
\[ \begin{cases} \text{the set } A \text{ has at least } 4 \text{ elements} \\ \text{the element } 1+1 \text{ is invertible}\\ x+x^4=x^2+x^3 \text{ holds for all } x \in A \end{cases} \]a) Prove that if a ring $(A,+,\cdot)$ has property $(P)$, and $a,b \in A$ are distinct elements, such that $a$ and $a+b$ are units, then $1+ab$ is also a unit, but $b$ is not a unit.
b) Provide an example of a ring with property $(P)$.
5 replies
Ciobi_
Apr 2, 2025
AngryKnot
Yesterday at 3:18 PM
J
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Structure of the group $(\mathbb{Z}/p\mathbb{Z})^{\times}$ and its application t
nayr   1
N Yesterday at 10:05 AM by GreenKeeper
Let $\mathbb{F}_p^{\times} = (\mathbb{Z} / p\mathbb{Z})^{\times}$ be the unit group of $\mathbb{F}_p$. It is well known that this group is cyclic. Let $g$ be a generator of this group and consider the map $\varphi : \mathbb{F}_p^{\times} \rightarrow \mathbb{F}_p^{\times}, x\mapsto x^k$ for a fixed positive integer $k$. I know that the kernel $\ker \varphi$ has oder $d:= (p-1, k)$. By the first isomorphism theorem, $\mathbb{F}_p^{\times} / \ker \varphi \cong \operatorname{im} \varphi$. Since $\mathbb{F}_p^{\times}$ is cyclic, so are its subgroups and hence $\operatorname{im} \varphi$ is cyclic of oder $\frac{p-1}{d}$. Let $H = \operatorname{im} \varphi$. Then $\mathbb{F}_p^{\times}/H$ is cyclic too and hence we have the partition:

$$\mathbb{F}_p = \{0\} \sqcup H \sqcup s^2H \sqcup \cdots \sqcup s^{d-1}H$$
for any $s\notin H$ (for example $g$).

I am trying to use this fact to solve the following question: Show that $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ have non-trivial solution for all primes $p$. Here is my attempt:

For simplicity, we rewrite the original equation for $p>3$, as $x^3+Ay^3+Bz^3\equiv 0 \pmod{p}$ (the case $p=2,3$ is easy).

If $p\equiv 2\pmod{3}$, then everything is a cube (since the cubing map $x\,mapsto x^3$ is an anutomorphism by above) and the equation is solvable.

If $p\equiv 1\pmod{3}$, let $H:=\{x^3|x\in \mathbb{F}_p^{\times}\}$ and $sH, s^2H$ be the cosets where $s \notin H$, then we have the following cases:

Case 1: $A \in H$ or $B\in H$, Without loss of generality, assume $A=4/3$ is a cube, then $4/3=a^3$ or $4=3a^3$ and we may take $(x,y,z)=(a,-1,0)$ as our solution.

Case 2: $A \in sH$ and $B\in sH$, then $A=sa^3$ and $B=sb^3$ and we may take $(x,y,z)=(0,b,-a)$ as our solution.

Case 3: $A \in s^2H$ and $B\in s^2H$, then $A=s^2a^3$ and $B=s^2b^3$ and we may take $(x,y,z)=(0,b,-a)$ as our solution.

Case 4: $A \in sH$ and $B\in s^2H$, then $A=sa^3$ and $B=s^2b^3$. This is the case I am stuck with. If we have $s^3=1$, then we may take $(x,y,z)=(ab,b,a)$ as our solution since $1+s+s^2=0$ for $s^3=1$ and $s$ is not $1$). But it is not always possible to have both $s^3=1$ and $s\notin H$. For example, I can take $s=g^{\frac{p-1}{3}}$, then $s^3=1$, but $g^{\frac{p-1}{3}}\notin H$ iff $9\nmid p-1$.

How should I resolve case 4?
1 reply
nayr
Yesterday at 8:43 AM
GreenKeeper
Yesterday at 10:05 AM
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Group Theory resources
JerryZYang   3
N Yesterday at 4:22 AM by JerryZYang
Can someone give me some resources for group theory. ;)
3 replies
JerryZYang
Jul 23, 2025
JerryZYang
Yesterday at 4:22 AM
J
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Find max(a+√b+∛c) where 0< a, b, c < 1= a+b+c.
elim   7
N Yesterday at 2:25 AM by sqing
Find $\max_{a,\,b,\,c>0\atop a+b+c=1}(a+\sqrt{b}+\sqrt[3]{c})$
7 replies
elim
Feb 7, 2020
sqing
Yesterday at 2:25 AM
J
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Are all solutions normal ?
loup blanc   11
N Wednesday at 9:02 PM by GreenKeeper
This post is linked to this one
https://artofproblemsolving.com/community/c7t290f7h3608120_matrix_equation
Let $Z=\{A\in M_n(\mathbb{C}) ; (AA^*)^2=A^4\}$.
If $A\in Z$ is a normal matrix, then $A$ is unitarily similar to $diag(H_p,S_{n-p})$,
where $H$ is hermitian and $S$ is skew-hermitian.
But are there other solutions? In other words, is $A$ necessarily normal?
I don't know the answer.
11 replies
loup blanc
Jul 17, 2025
GreenKeeper
Wednesday at 9:02 PM
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J
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Factorization Ex.28a Q30
Obvious_Wind_1690   1
N May 27, 2025 by Lankou
Please help with factorization. Given is the question


\begin{align*} a(a+1)x^2+(a+b)xy-b(b-1)y^2\\ \end{align*}
And the given answer is


\begin{align*} [(a+1)x-(b-1)y][ax+by]\\ \end{align*}
But I am unable to reach the answer.
1 reply
Obvious_Wind_1690
May 27, 2025
Lankou
May 27, 2025
J
Factorization Ex.28a Q30
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Reveal topic tags
special factorizations
factorization
factoring polynomials
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Obvious_Wind_1690
5 posts
Obvious_Wind_1690
#1 h  May 27, 2025, 4:17 AM
Y by
Please help with factorization. Given is the question


\begin{align*} a(a+1)x^2+(a+b)xy-b(b-1)y^2\\ \end{align*}
And the given answer is


\begin{align*} [(a+1)x-(b-1)y][ax+by]\\ \end{align*}
But I am unable to reach the answer.
Z K Y
The post below has been deleted. Click to close.
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Lankou
1408 posts
Lankou
#2 h  May 27, 2025, 4:43 PM
Y by
Expand the first expression then add and subtract $abxy$
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