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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)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
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[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

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0 replies
jwelsh
Jul 1, 2025
0 replies
J
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Quadratic equations
Streit31415   6
N 11 minutes ago by TedBot
Let (p) and (q) be integers. Knowing that x² + px + q is positive for all integer (x), prove that the equation x² + px + q = 0 has no real solution.
6 replies
+1 w
Streit31415
5 hours ago
TedBot
11 minutes ago
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My first proof problem
OWOW   10
N 2 hours ago by TedBot
In the equation $\sum_{i=0}^n i$ prove (or disprove) that there exists infinitely many positive integers n in which $\sum_{i=0}^n i$ sums to k! where k is a positive integer. (examples, 1+2+3=3! and 1+2+3+4+5+6+7+8+9+10+11+12+13+14+15=5!

Also this is my first proof problem so don't get mad at me if it's really bad. (Technically I'm a middle schooler so I should post this in the middle school threads but I'm thinking proofs are more high school level so I'm posting it here.)
10 replies
OWOW
Yesterday at 11:44 PM
TedBot
2 hours ago
J
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A nice identity
Synchrone   1
N 3 hours ago by GreenKeeper
Source: Math&Maroc Competition 2025 Day1 Problem 3
For pairwise-distinct real numbers $a_1, \ldots, a_n$ prove that :
$$ \sum_{i =1}^n a_j^2 \prod_{k \neq j} \frac{a_j + a_k}{a_j - a_k} = (a_1 + \ldots + a_n)^2 $$
1 reply
Synchrone
Today at 1:18 PM
GreenKeeper
3 hours ago
J
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Angle bisector, medians
xeroxia   0
3 hours ago
Let $AM$ be a median and $AN$ be an angle bisector of $\triangle ABC$. Let $BK$ be a median and $BH$ be an altitude of $\triangle ABN$. If $AN=6$ and $MK=4$, what is $MH$?
0 replies
xeroxia
3 hours ago
0 replies
J
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Trigonometry equation practice
ehz2701   26
N 4 hours ago by vanstraelen
There is a lack of trigonometric bash practice, and I want to see techniques to do these problems. So here are 10 kinds of problems that are usually out in the wild. How do you tackle these problems? (I had ChatGPT write a code for this.). Please give me some general techniques to solve these kinds of problems, especially set 2b. I’ll add more later.

Leaderboard and Solved Problems

problem set 1a (1-10)

problem set 2a (1-20)

problem set 2b (1-20)
answers 2b

General techniques so far:

Trick 1: one thing to keep in mind is

[center] $\frac{1}{2} = \cos 36 - \sin 18$. [/center]

Many of these seem to be reducible to this. The half can be written as $\cos 60 = \sin 30$, and $\cos 36 = \sin 54$, $\sin 18 = \cos 72$. This is proven in solution 1a-1. We will refer to this as Trick 1.
26 replies
ehz2701
Jul 12, 2025
vanstraelen
4 hours ago
J
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Countable sets
We2592   1
N 5 hours ago by KAME06
Q) prove that union of two countable sets is countable.

proof:

let \[ A = \{a_1, a_2, a_3, \dots\} \]and \[ B = \{b_1, b_2, b_3, \dots\} \]be countable infinite sets

let define a mapping a bijection mapping from $f:\mathbb{N} \to A$ like
\[ f(n) = \begin{cases} a_{\frac{n+1}{2}} & \text{if } n \text{ is odd} \\ a_{\frac{n}{2} + 1} & \text{if } n \text{ is even} \end{cases} \]let define a mapping a bijection mapping from $g:\mathbb{N} \to B$ like
\[ f(n) = \begin{cases} b_{\frac{n+1}{2}} & \text{if } n \text{ is odd} \\ b_{\frac{n}{2} + 1} & \text{if } n \text{ is even} \end{cases} \]
now how to get a intuition to pick a choice for the bijection map for $\phi:\mathbb{N}\to A\cup B$
please give a idea of doing this problem help
1 reply
We2592
Today at 2:58 PM
KAME06
5 hours ago
J
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Existence of a special polynomial
Synchrone   0
6 hours ago
Source: Math&Maroc Competition 2025 Day1 Problem 4
Let $n \geq 3 :$
- We denote by : $$E = \{ \sum_{1 \leq i \leq j \leq n} a_{i,j} X_i X_j, \forall 1 \leq i \leq j \leq n, a_{i,j} \in \mathbb{R} \} $$The set of real homogeneous polynomials of degree $2$ in $n$ indeterminates.
- For $P = P(X_1, \ldots, X_n) \in E$ and $M = (m_{i,j})_{1 \leq i \leq j \leq n} \in M_n(\mathbb{R})$, we done by $P \circ M \in E$ their composition, defined by :
$$ P \circ M(X_1, \ldots, X_n) = P(m_{1,1}X_1 + \ldots + m_{1,n}X_n, \ldots, m_{n,1}X_1 + \ldots + m_{n,n}X_n). $$- For $P \in E$, we denote by :
$$ V(P) = \{x \in \mathbb{R}^n, P(x) = 0\} $$Show that there exists a polynomial \(P_0 \in E\), which we will exhibit, such that for all \(P \in E\), the following equivalence holds :
$$ V(P) \text{ is NOT an } \mathbb{R}-\text{vector space} \Longleftrightarrow \exists M \in M_n(\mathbb{R}), P \circ M = P_0 $$
0 replies
Synchrone
6 hours ago
0 replies
J
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Floor of a square root sequence
Synchrone   1
N Today at 3:29 PM by KAME06
Source: Math&Maroc Competition 2025 Day 1 Problem 1
The sequence $(a_n)$ is defined by :
$$ a_1 =1, a_2 = 1, a_n = \lfloor \sqrt{2a_{n-1} + a_{n-2} + \ldots + a_1} \rfloor \text{ if } n > 2 $$Find $a_{2025}$
1 reply
Synchrone
Today at 1:02 PM
KAME06
Today at 3:29 PM
J
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Probabilistic Method with Main Inequality Difficulty
EthanWYX2009   0
Today at 3:24 PM
Source: 2024 January 谜之竞赛-6
Given integer \( k \geq 2 \). Determine the largest real number \( \lambda \), such that there exists a constant \( C \) depending only on \( k \), with the following property: For any finite simple graph \( G \), its vertex set can be partitioned into \( k \) parts, satisfying that at least
\[ \left(1 - \frac{1}{k}\right)e(G) + \lambda \sqrt{e(G)} + C \]edges of \( G \) have their two vertices in different parts, where $e(G)$ denotes the number of edges in $G$.
0 replies
EthanWYX2009
Today at 3:24 PM
0 replies
J
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Moroccan Sets
Synchrone   0
Today at 1:12 PM
Source: Math&Maroc Competition 2025 Day1 Problem 2
For any non-empty subsets $X$ and $Y$ of $\mathbb{N}_{\geq 1}$, we define $X \cdot Y := \{xy, x \in X, y \in Y\}$.
For instance, if : $$X = \{1,2,4\}, Y = \{3,4,6\}$$Then : $$ X \cdot Y = \{3,4,6,8,12,16,24\} $$We say that a subset $S$ of $\mathbb{N}_{\geq 1}$ is Moroccan if there exists two subsets $A$ and $B$ of $\mathbb{N}_{\geq 1}$ each containing at least two elements, such that $A \cdot B$ and $S$ are equal.
Prove that the set of perfect powers which are greater or equal to $2025$ is Moroccan
(A perfect power is an integer $n^k$, where $n > 1$ and $k > 1$ are integers)
0 replies
Synchrone
Today at 1:12 PM
0 replies
J
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Question about Fubini integration in Euclidean spaces
euclides05   0
Today at 10:07 AM
Fubini's theorem in Lebesgue integration uses the product measure. My question is, how can this be applied in Euclidean spaces, where if $d= d_1+d_2$, the lebesgue measure of $R^d$ is the completion of the product measures
of $R^{d_1}$, $R^{d_2}$
and not the actual product measure?
I would be very pleased if someone provided a thorough answer cause i'm studying convolution stuff where it's used heavily and i haven't been able to find a stict explanation for it.
0 replies
euclides05
Today at 10:07 AM
0 replies
J
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On the number of isomorphism classes of subgroups and normal subgroups
Sivin   3
N Today at 4:02 AM by Sivin
Let $(G,\circ)$ be a group. Let $\mathcal{I}(G)$ denote the number of subgroups of $G$ up to isomorphism (i.e. the number of isomorphism classes of subgroups of $G$) and let $\mathcal{N}(G)$ denote the number of normal subgroups of $G$ (not necessarily up to isomorphism). Find all groups $G$ with $\mathcal{I}(G)=\mathcal{N}(G).$

For example, the dihedral group of degree $10$:
$$D_{10}=\langle s,t\mid s^{10},t^{2},stst\rangle$$has $7$ isomorphism classes of subgroups, namely $C_1,C_2,K,C_5,C_{10},D_5,D_{10},$ so $\mathcal{I}(D_{10})=7.$ It also has $7$ normal subgroups (see https://groupprops.subwiki.org/wiki/Dihedral_group:D20#Subgroups). Hence $\mathcal{N}(D_{10})=7.$ It follows that $D_{10}$ is a group with $\mathcal{I}(G)=\mathcal{N}(G).$
3 replies
Sivin
Aug 17, 2024
Sivin
Today at 4:02 AM
J
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Frankl Theorem?
EthanWYX2009   0
Today at 2:55 AM
Source: 2024 February 谜之竞赛-3
Let \( n \) be a positive integer, \( f(n) \) denote the maximum possible size of a family \(\mathcal{F}\) of subsets of \(\{1, 2, \cdots, n\}\) such that for any two subsets \( X, Y \in \mathcal{F} \), \( |X \cap Y| \) is a perfect square.

Show that $2^{\lfloor \sqrt{n} \rfloor} \leq f(n) \leq 100^{\sqrt{n} \ln n}.$

Proposed by Qianhong Cheng, Guangdong Experimental High School
0 replies
EthanWYX2009
Today at 2:55 AM
0 replies
J
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On f(x) and f([x])
ajovanovic   3
N Yesterday at 8:39 PM by ajovanovic
For which $f:\mathbb{R}\to\mathbb{R}$ does $f(x)$ ~$f([x])$ hold?

Note: I have proved this for concave, unbounded, monotone increasing $f$. Is this a necessary condition?
3 replies
ajovanovic
Yesterday at 8:18 PM
ajovanovic
Yesterday at 8:39 PM
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J
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Plz help
Bet667   3
N May 13, 2025 by K1mchi_
f:R-->R for any integer x,y
f(yf(x)+f(xy))=(x+f(x))f(y)
find all function f
(im not good at english)
3 replies
Bet667
Jan 28, 2024
K1mchi_
May 13, 2025
J
Plz help
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Reveal topic tags
function
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Bet667
169 posts
Bet667
#1 h  Jan 28, 2024, 3:55 PM • 1 Y
Y by PikaPika999
f:R-->R for any integer x,y
f(yf(x)+f(xy))=(x+f(x))f(y)
find all function f
(im not good at english)
Z K Y
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Mathzeus1024
1073 posts
Mathzeus1024
#2 h  May 11, 2025, 10:17 AM • 1 Y
Y by PikaPika999
It works for $\textcolor{red}{f(x)=0}$ and $\textcolor{red}{f(x)=x}$.
Z K Y
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jasperE3
11449 posts
jasperE3
#3 h  May 12, 2025, 7:42 AM
Y by
https://artofproblemsolving.com/community/c6h3518642p34163857
in my opinion the phrase "for any integer x,y" is not what the problem actually says
This post has been edited 2 times. Last edited by jasperE3, May 13, 2025, 7:38 PM
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K1mchi_
189 posts
K1mchi_
#4 h  May 13, 2025, 6:50 PM
Y by
Bet667 wrote:
f:R-->R for any integer x,y
f(yf(x)+f(xy))=(x+f(x))f(y)
find all function f
(im not good at english)

just use translate?
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