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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
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]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]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
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0 replies
jwelsh
Jul 1, 2025
0 replies
Show that every prime number n has property P
orl   9
N 5 minutes ago by Adi1005247
Source: IMO Shortlist 1993, India 5
A natural number $n$ is said to have the property $P,$ if, for all $a, n^2$ divides $a^n - 1$ whenever $n$ divides $a^n - 1.$

a.) Show that every prime number $n$ has property $P.$

b.) Show that there are infinitely many composite numbers $n$ that possess property $P.$
9 replies
+1 w
orl
Mar 15, 2006
Adi1005247
5 minutes ago
Integer sequences with a two-step recurrence
Assassino9931   1
N 30 minutes ago by Assassino9931
Source: Bulgaria Autumn Tournament 2009 Grade 12
Fix an integer $m$. Determine the number of sequences $(a_n)_{n\geq 1}$ of integers such that $a_na_{n+2} = n^2 + m$ for all $n\geq 1$.
1 reply
Assassino9931
Today at 1:02 AM
Assassino9931
30 minutes ago
old product!
teomihai   2
N 33 minutes ago by teomihai
It is posible to prove ,without induction :
$(\frac{1}{2}\frac{3}{4}...\frac{2n-1}{2n})^2\leq{\frac{1}{3n+1}} $ for any positiv integer number $n$.?
2 replies
1 viewing
teomihai
Yesterday at 3:51 PM
teomihai
33 minutes ago
2025 IMO Results
ilikemath247365   5
N 33 minutes ago by cheltstudent
Source: https://www.imo-official.org/year_info.aspx?year=2025
Congrats to China for getting 1st place! Congrats to USA for getting 2nd and congrats to South Korea for getting 3rd!
5 replies
1 viewing
ilikemath247365
Yesterday at 4:48 PM
cheltstudent
33 minutes ago
Axiomatic real numbers x^0
Safal   2
N Yesterday at 6:50 PM by Safal
Source: Discussion
Here is an interesting question for you all:

Assume $\mathbb{R}$ is an ordered field with all field axioms holding.

$\textbf{Question:}$ Suppose $x>0$ is a real number and $0\in\mathbb{R}$(Set of real numbers). Prove that if $x^0\in\mathbb{R}$. Then it (that is $x^0$) must be $1$.

Also show same thing is true if $x<0$ is a real number.

Proof

$\textbf{Question:}$ If $z\neq 0$ be any complex number and $0\in\mathbb{C}$. Show that if $z^0\in\mathbb{C}$ , then $z^0=1$.

$\textbf{Remark:}$ Order property is not true for all complex numbers.

Hint
2 replies
Safal
Yesterday at 3:20 PM
Safal
Yesterday at 6:50 PM
D1054 : A measure porblem
Dattier   1
N Yesterday at 5:59 PM by greenturtle3141
Source: les dattes à Dattier
$M=\bigcup\limits_{n\in\mathbb N^*} \{0,1\}^n$, for $m \in M$, $\overline m=\{mx : x\in \{0,1\}^{\mathbb N} \}$

Let $A \subset M$ with $\forall (a,b) \in A,a\neq b$, $\overline a \cap \overline b=\emptyset$.

Is it true that $\sum\limits_{a\in A} 2^{-|a|}\leq 1$ ?

PS : for $m \in M$, $|m|$ is the length of $m$, hence $|0011|=4$
1 reply
Dattier
Yesterday at 4:36 PM
greenturtle3141
Yesterday at 5:59 PM
Analytic Number Theory
EthanWYX2009   0
Yesterday at 2:11 PM
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
0 replies
EthanWYX2009
Yesterday at 2:11 PM
0 replies
OMOUS-2025 (Team Competition) P10
enter16180   4
N Yesterday at 8:33 AM by enter16180
Source: Open Mathematical Olympiad for University Students (OMOUS-2025)
Let $f: \mathbb{N} \rightarrow \mathbb{N}$ and $g: \mathbb{N} \rightarrow\{A, G\}$ functions are given with following properties:
(a) $f$ is strict increasing and for each $n \in \mathbb{N}$ there holds $f(n)=\frac{f(n-1)+f(n+1)}{2}$ or $f(n)=\sqrt{f(n-1) \cdot f(n+1)}$.
(b) $g(n)=A$ if $f(n)=\frac{f(n-1)+f(n+1)}{2}$ holds and $g(n)=G$ if $f(n)=\sqrt{f(n-1) \cdot f(n+1)}$ holds.

Prove that there exist $n_{0} \in \mathbb{N}$ and $d \in \mathbb{N}$ such that for all $n \geq n_{0}$ we have $g(n+d)=g(n)$
4 replies
enter16180
Apr 18, 2025
enter16180
Yesterday at 8:33 AM
Putnam 2012 A3
Kent Merryfield   9
N Yesterday at 6:29 AM by AngryKnot
Let $f:[-1,1]\to\mathbb{R}$ be a continuous function such that

(i) $f(x)=\frac{2-x^2}{2}f\left(\frac{x^2}{2-x^2}\right)$ for every $x$ in $[-1,1],$

(ii) $ f(0)=1,$ and

(iii) $\lim_{x\to 1^-}\frac{f(x)}{\sqrt{1-x}}$ exists and is finite.

Prove that $f$ is unique, and express $f(x)$ in closed form.
9 replies
Kent Merryfield
Dec 3, 2012
AngryKnot
Yesterday at 6:29 AM
AMM problem section
Khalifakhalifa   1
N Yesterday at 4:41 AM by Khalifakhalifa
Does anyone have access to the current AMM edition? I’d like to see the problems section. If so, could someone please share it with me via PM?
1 reply
Khalifakhalifa
Tuesday at 11:17 AM
Khalifakhalifa
Yesterday at 4:41 AM
an integral
Svyatoslav   0
Yesterday at 2:27 AM
How do we prove analytically that
$$\int_0^{\pi/2}\frac{\ln(1+\cos x)-x}{\sqrt{\sin x}}\,dx=0\quad?$$The sourse: Quora

Numeric evaluation
0 replies
Svyatoslav
Yesterday at 2:27 AM
0 replies
Putnam 2014 A4
Kent Merryfield   38
N Yesterday at 1:35 AM by eg4334
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E[X]=1,$ $E[X^2]=2,$ and $E[X^3]=5.$ (Here $E[Y]$ denotes the expectation of the random variable $Y.$) Determine the smallest possible value of the probability of the event $X=0.$
38 replies
Kent Merryfield
Dec 7, 2014
eg4334
Yesterday at 1:35 AM
the limit
Svyatoslav   2
N Tuesday at 10:32 PM by ysharifi
How do you find
$$\lim_{r\to+0}\left(\sum_{k=1}^\infty\frac1{k\,\sqrt{1+\pi^2r^2k^2}}+\ln r\right)\,\,?$$
2 replies
Svyatoslav
Tuesday at 3:58 PM
ysharifi
Tuesday at 10:32 PM
Putnam 2001 B5
ahaanomegas   8
N Tuesday at 9:39 PM by Assassino9931
Let $a$ and $b$ be real numbers in the interval $\left(0,\tfrac{1}{2}\right)$, and let $g$ be a continuous real-valued function such that $g(g(x))=ag(x)+bx$ for all real $x$. Prove that $g(x)=cx$ for some constant $c$.
8 replies
1 viewing
ahaanomegas
Feb 27, 2012
Assassino9931
Tuesday at 9:39 PM
Nice problem of concurrency
deraxenrovalo   2
N May 26, 2025 by deraxenrovalo
Let $(I)$ be an inscribed circle of $\triangle$$ABC$ and touching $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively. Let $EE'$ and $FF'$ be diameters of $(I)$. Let $X$ and $Y$ be the pole of $DE'$ and $DF'$ with respect to $(I)$, respectively. $BE$ cuts $(I)$ again at $K$. $CF$ cuts $(I)$ again at $L$. The tangent at $K$ of $(I)$ cuts $AX$ at $M$. The tangent at $L$ of $(I)$ cuts $AY$ at $N$. Let $U$ and $V$ be midpoint of $IM$ and $IN$, respectively.

Show that : $UV$, $E'F'$ and perpendicular bisector of $ID$ are concurrent.
2 replies
deraxenrovalo
May 24, 2025
deraxenrovalo
May 26, 2025
Nice problem of concurrency
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deraxenrovalo
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Let $(I)$ be an inscribed circle of $\triangle$$ABC$ and touching $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively. Let $EE'$ and $FF'$ be diameters of $(I)$. Let $X$ and $Y$ be the pole of $DE'$ and $DF'$ with respect to $(I)$, respectively. $BE$ cuts $(I)$ again at $K$. $CF$ cuts $(I)$ again at $L$. The tangent at $K$ of $(I)$ cuts $AX$ at $M$. The tangent at $L$ of $(I)$ cuts $AY$ at $N$. Let $U$ and $V$ be midpoint of $IM$ and $IN$, respectively.

Show that : $UV$, $E'F'$ and perpendicular bisector of $ID$ are concurrent.
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Funcshun840
61 posts
#2 • 1 Y
Y by deraxenrovalo
cool!

Let's define some points: $M=DE^{\prime} \cap EF$, $J= DF^{\prime} \cap EF$, $G$, $H$ as the reflections of $E$, $F$ over $FI$ and $EI$ respectively. Additionally, define $O$ as the center of $(IE^{\prime} F^{\prime}$ and $N = MK \cap JL$.

Claim: $M$, $K$, $H$ and $J$, $L$, $G$ are collinear.
Proof: Note that $(K,E;F,D)=-1=(E^{\prime}, E; F; H)$, so $EF$, $KH$ and $DE^{\prime}$ are concurrent, and similarly for $J$, $L$, $G$.

Claim:$E^{\prime}H \cap F^{\prime}G = O$
Proof:simple omitted angle chase. (typing angles on aops so tedious grr)

Now the cool part: Consider the three segments $MJ$, $E^{\prime}F^{\prime}$, $GH$, call them $\kappa _1$, $\kappa _2$, $\kappa _3$. We have that $D$ is the exsimilicenter of $\kappa _2$ and $\kappa _3$, $O$ as the insimilicenter of $\kappa _ 3$ and $\kappa _1$ and $N$ as the insimilicenter of $\kappa _1$ and $\kappa _3$. Hence by Monge d'Alembert, we find that $D$, $O$, $N$ are collinear.

Now we recognise $AX$, $AY$ as the polars of $M$, $J$. This implies that $M=KK\cap HH$ and $N=LL\cap GG$. Hence the pole of $Z=BC \cap MN$ is the line $DN$. We wish to show that the midpoint of $IZ$ lies $(E^{\prime} F^{\prime}$. Inverting this about the incircle, we need to show that the reflection of $I$ over the inverted $Z$ lies on $(IE^{\prime} F^{\prime}$, however this follows from the fact that $O$ lies on the polar of $Z$ (as the inverted $Z$ is just the foot of $I$ on the polar of $Z$).
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deraxenrovalo
9 posts
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Wow! The solution is really amazing! :-D
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