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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
number theory diophantic with factorials and primes
skellyrah   4
N 15 minutes ago by skellyrah
Source: by me
find all triplets of non negative integers (a,b,p) where p is prime such that $$ a! + b! + 7ab = p^2 $$
4 replies
skellyrah
Feb 16, 2025
skellyrah
15 minutes ago
Inequality em981
oldbeginner   17
N 31 minutes ago by sqing
Source: Own
Let $a, b, c>0, a+b+c=3$. Prove that
\[\sqrt{a+\frac{9}{b+2c}}+\sqrt{b+\frac{9}{c+2a}}+\sqrt{c+\frac{9}{a+2b}}+\frac{2(ab+bc+ca)}{9}\ge\frac{20}{3}\]
17 replies
2 viewing
oldbeginner
Sep 22, 2016
sqing
31 minutes ago
primes,exponentials,factorials
skellyrah   6
N 38 minutes ago by skellyrah
find all primes p,q such that $$ \frac{p^q+q^p-p-q}{p!-q!} $$is a prime number
6 replies
skellyrah
Apr 30, 2025
skellyrah
38 minutes ago
Serbian selection contest for the IMO 2025 - P5
OgnjenTesic   2
N an hour ago by GreenTea2593
Source: Serbian selection contest for the IMO 2025
Determine the smallest positive real number $\alpha$ such that there exists a sequence of positive real numbers $(a_n)$, $n \in \mathbb{N}$, with the property that for every $n \in \mathbb{N}$ it holds that:
\[
        a_1 + \cdots + a_{n+1} < \alpha \cdot a_n.
    \]Proposed by Pavle Martinović
2 replies
OgnjenTesic
May 22, 2025
GreenTea2593
an hour ago
centroid wanted, point that minimizes sum of squares of distances from sides
parmenides51   1
N an hour ago by SuperBarsh
Source: Oliforum Contest V 2017 p9 https://artofproblemsolving.com/community/c2487525_oliforum_contes
Given a triangle $ABC$, let $ P$ be the point which minimizes the sum of squares of distances from the sides of the triangle. Let $D, E, F$ the projections of $ P$ on the sides of the triangle $ABC$. Show that $P$ is the barycenter of $DEF$.

(Jack D’Aurizio)
1 reply
parmenides51
Sep 29, 2021
SuperBarsh
an hour ago
Strictly monotone polynomial with an extra condition
Popescu   11
N an hour ago by Iveela
Source: IMSC 2024 Day 2 Problem 2
Let $\mathbb{R}_{>0}$ be the set of all positive real numbers. Find all strictly monotone (increasing or decreasing) functions $f:\mathbb{R}_{>0} \to \mathbb{R}$ such that there exists a two-variable polynomial $P(x, y)$ with real coefficients satisfying
$$
f(xy)=P(f(x), f(y))
$$for all $x, y\in\mathbb{R}_{>0}$.

Proposed by Navid Safaei, Iran
11 replies
Popescu
Jun 29, 2024
Iveela
an hour ago
Hard geometry
Lukariman   1
N an hour ago by ricarlos
Given triangle ABC, any line d intersects AB at D, intersects AC at E, intersects BC at F. Let O1,O2,O3 be the centers of the circles circumscribing triangles ADE, BDF, CFE. Prove that the orthocenter of triangle O1O2O3 lies on line d.
1 reply
Lukariman
May 12, 2025
ricarlos
an hour ago
Russian Diophantine Equation
LeYohan   1
N an hour ago by Natrium
Source: Moscow, 1963
Find all integer solutions to

$\frac{xy}{z} + \frac{xz}{y} + \frac{yz}{x} = 3$.
1 reply
LeYohan
Yesterday at 2:59 PM
Natrium
an hour ago
Simple Geometry
AbdulWaheed   6
N an hour ago by Gggvds1
Source: EGMO
Try to avoid Directed angles
Let ABC be an acute triangle inscribed in circle $\Omega$. Let $X$ be the midpoint of the arc $\overarc{BC}$ not containing $A$ and define $Y, Z$ similarly. Show that the orthocenter of $XYZ$ is the incenter $I$ of $ABC$.
6 replies
AbdulWaheed
May 23, 2025
Gggvds1
an hour ago
Bosnia and Herzegovina JBMO TST 2013 Problem 4
gobathegreat   4
N an hour ago by FishkoBiH
Source: Bosnia and Herzegovina Junior Balkan Mathematical Olympiad TST 2013
It is given polygon with $2013$ sides $A_{1}A_{2}...A_{2013}$. His vertices are marked with numbers such that sum of numbers marked by any $9$ consecutive vertices is constant and its value is $300$. If we know that $A_{13}$ is marked with $13$ and $A_{20}$ is marked with $20$, determine with which number is marked $A_{2013}$
4 replies
gobathegreat
Sep 16, 2018
FishkoBiH
an hour ago
A geometry problem
Lttgeometry   2
N an hour ago by Acrylic3491
Triangle $ABC$ has two isogonal conjugate points $P$ and $Q$. The circle $(BPC)$ intersects circle $(AP)$ at $R \neq P$, and the circle $(BQC)$ intersects circle $(AQ)$ at $S\neq Q$. Prove that $R$ and $S$ are isogonal conjugates in triangle $ABC$.
Note: Circle $(AP)$ is the circle with diameter $AP$, Circle $(AQ)$ is the circle with diameter $AQ$.
2 replies
Lttgeometry
Today at 4:03 AM
Acrylic3491
an hour ago
anglechasing , circumcenter wanted
parmenides51   1
N an hour ago by Captainscrubz
Source: Sharygin 2011 Final 9.2
In triangle $ABC, \angle B = 2\angle C$. Points $P$ and $Q$ on the medial perpendicular to $CB$ are such that $\angle CAP = \angle PAQ = \angle QAB = \frac{\angle A}{3}$ . Prove that $Q$ is the circumcenter of triangle $CPB$.
1 reply
parmenides51
Dec 16, 2018
Captainscrubz
an hour ago
Nice FE over R+
doanquangdang   4
N 2 hours ago by jasperE3
Source: collect
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ such that
\[x+f(yf(x)+1)=xf(x+y)+yf(yf(x))\]for all $x,y>0.$
4 replies
doanquangdang
Jul 19, 2022
jasperE3
2 hours ago
right triangle, midpoints, two circles, find angle
star-1ord   0
2 hours ago
Source: Estonia Final Round 2025 8-3
In the right triangle $ABC$, $M$ is the midpoint of the hypotenuse $AB$. Point $D$ is chosen on the leg $BC$ so that the line segment $DM$ meets $(ACD)$ again at $K$ ($K\neq D$). Let $L$ be the reflection of $K$ in $M$. The circles $(ACD)$ and $(BCL)$ meet again at $N$ ($N\neq C$). Find the measure of $\angle KNL$.
0 replies
star-1ord
2 hours ago
0 replies
Iran second round 2025-q1
mohsen   8
N May 18, 2025 by Autistic_Turk
Find all positive integers n>2 such that sum of n and any of its prime divisors is a perfect square.
8 replies
mohsen
Apr 19, 2025
Autistic_Turk
May 18, 2025
Iran second round 2025-q1
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G H BBookmark kLocked kLocked NReply
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mohsen
16 posts
#1 • 1 Y
Y by sami1618
Find all positive integers n>2 such that sum of n and any of its prime divisors is a perfect square.
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sami1618
913 posts
#2 • 1 Y
Y by Sedro
Answer: No such number exists.

Solution. We start by proving the following two claims:

Claim 1. We have that $n$ is squarefree.
Proof. If there exists a prime $p$ for which $p^2|n$, then $n+p$ must be a perfect square. But $v_p(n+p)=1$, a contradiction.

Claim 2. We have that $n$ has at most two distinct prime divisors.
Proof. Assume otherwise that $p$, $q$, and $r$ are three distinct prime divisors of $n$. Without loss of generality, assume that $p<q<r$. It is easy to see that both $p$ and $q$ must be strictly smaller than $\sqrt{n}$. On the other hand, $$\sqrt{n+q}-\sqrt{n+p}\geq 1\Rightarrow q-p\geq \sqrt{n+q}+\sqrt{n+p}\geq 2\sqrt{n},$$which is a contradiction.

By the Claims, we must have that either $n=p$ for some prime $p>2$ or $n=pq$ for some distinct primes $p$ and $q$. In the first case, $2p$ must be a perfect square, which is absurd. In the second case, both $p(q+1)$ and $q(p+1)$ must be perfect squares. This implies that $p|q+1$ and that $q|p+1$. Assume that $p<q$. Then it must be that $p=q-1$ which implies that $p=2$ and $q=3$, which fails to work.
This post has been edited 1 time. Last edited by sami1618, Apr 19, 2025, 5:33 PM
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missionjoshi.65
2 posts
#3 • 2 Y
Y by Calc_1, sami1618
Posting though very similar as above!
Claim 1: $n$ can't be a power of prime.

Suppose, $n = p^k$ for some prime $p$, then we have $v_p(p^k+p) =1+v_p(p^{k-1}+1),$ which can't be a square when $k \geq 2$. And, $k = 1$ gives $n = 2$ which is a contradiction.

Claim 2: $n$ cannot have three or more prime factors.

Suppose $p,q,r | n$ and $p<q<r$, then

$$n +p = a^2 \quad \text{and} \quad n+q = b^2$$
This implies:
$$ q-p = (b-a)(b+a)$$
Similarly,
$$ q-p \geq b+a = \sqrt{n+p} + \sqrt{n+q} > \sqrt{n}+\sqrt{n} = 2 \sqrt{n}$$
which is a contradiction since both $p, q < \sqrt{n}.$

Now, for two primes $p<q$ we have $p|q+1$ and $q|p+1$ which implies $p=q-1,$ so only possible values are $p=2$ and $q=3$, which do not work. And, we are done!
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Parsia--
79 posts
#4 • 2 Y
Y by Mahdi_Mashayekhi, amirhsz
Here's a much different and overkill approach.
suppose $n$ has at least two different primes $q,r$ such that $q \equiv r \equiv 3 \mod 4$. Note that $n$ is also square-free.
So we get
$$q(\frac{n}{q}+1) = x^2 \Rightarrow (\frac{r}{q})=1 $$Similarly $ (\frac{q}{r})=1$.
Now note that $$ (\frac{r}{q})(\frac{q}{r})=(-1)^{\frac{r-1}{2}. \frac{q-1}{2}}=-1 \Rightarrow  1=(\frac{r}{q})=-(\frac{q}{r})=-1$$which is a contradiction.
if $n = 2p$, we must have $3p$ is a perfect square which gives $p=3$ which doesn't work.
Now $n$ odd won't work because of mod $4$. so $n$ is even. Consider a prime divisor $p$ of $n$ such $p \equiv 1 \mod 4$ we get $p(\frac{n}{p}+1) =x^2$ but $\frac{n}{p}+1 \equiv 3 \mod 4$.
Thus we conclude that no such $n$ exists.
This post has been edited 2 times. Last edited by Parsia--, May 10, 2025, 3:18 PM
Reason: Shortened the solution
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MathLuis
1555 posts
#5
Y by
Suppose FTSOC there existed such $n$, then if $n$ was not squarefree then there exists a prime $p$ with $p^2 \mid n$ however we have $n+p$ is a perfect square and thus $2 \mid \nu_p(n+p)=1$, a clear contradiction!, therefore $n$ is squarefree.
Now let $n=p_1 \cdot p_2 \cdots p_k$ where $p_1<p_2 \cdots p_k$. Clearly $k \ge 2$ as if $k=1$ then $2p_1$ is a perfect square but $p_1 \ne 2$ in this case so this isn't true, and if $k \ge 3$ then notice we must have $\sqrt{n+p_2}-\sqrt{n+p_1} \ge 1$ which implies that $n+p_2 \ge n+p_1+2\sqrt{n+p_1}+1$ which implies that $p_2^2>(p_2-p_1-1)^2 \ge 4n+4p_1>4p_2^2+4p_1$ which is clearly a contradiction.
And now if $n=p_1p_2$ then $p_2(p_1+1), p_1(p_2+1)$ are both perfect squares then $p_2 \mid p_1+1$ and $p_1 \mid p_2+1$ so $p_1p_2 \mid p_1p_2+p_1+p_2+1$ which means $p_1p_2 \mid p_1+p_2+1$ which means that $2 \le (p_1-1)(p_2-1) \le 2$ where equality is attained when $p_1=2$ and $p_2=3$ however $2 \cdot 4=8$ is not a perfect square thus contradiction either way and we are done :cool:
This post has been edited 1 time. Last edited by MathLuis, Apr 23, 2025, 12:31 AM
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math-helli
13 posts
#6
Y by
Here you can find some solutions of second round
https://t.me/matholampiad123
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Marco22
3 posts
#7
Y by
Answer:such number do not exist.
Suppose on the contrary that a number $n$ satisfying the properties exists, and let $p$ and $q$ be two prime divisors of $n$. Then by the hypothesis there exist $x,y \in \mathbb{N}$ such that $n+p=x^2$ and $n+q=y^2$. clearly we can see that $p|x$ and $p|y$. Then by a size argument it follows:
$p-q=x^2-y^2=(x-y)(x+y)=(ap-bq)(ap+bq)>p+q$ for some integers $a$ and $b$. So the answer clearly follows
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Mathgloggers
90 posts
#8
Y by
Notice that each prime sums with $n$ to a distinct perfect square.
So let $k \in N$ such that $k^2>n$
We have, $(k+3)^2-n<\frac{n}{2} $,because the max prime is $< \frac{n}{2}$
$k^2+9+6k<\frac{3n}{2} $ $\implies$ $n^2<k<\frac{\frac{n}{2}-9}{6} $,now just solve this equation for $n$ you will get $-2<n<2$ ,So you it also does not work.

So it has clearly at most 2 prime factors for $n=p.q>1$ which respectively sums to $n+p=k^2$ and $n+q=(k+1)^2$.
hence one has to be even =$2$ and the other must be $2k+3$ and $n=k^2-2$



Now just put all in that equation you will get :$k^2-4k-8=0$ no integer solution .
(I don't why my solution was so basic:(
This post has been edited 1 time. Last edited by Mathgloggers, May 18, 2025, 6:41 AM
Reason: m
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Autistic_Turk
12 posts
#9
Y by
mohsen wrote:
Find all positive integers n>2 such that sum of n and any of its prime divisors is a perfect square.

Here is my solution that I actually used in contest
Trivialy there exist positive intiger m and non negative integer q so that n=m^2+q and n<(m+1)^2 so q<2m+1 now there are 2 cases
Case 1: n has at least two different prime divisor
Let p_1 and p_2 be prime divisor of n and WLOG p_2>p_1 notice there exist intiger r_1 and r_2 such that n+p_1=(m+r_1)^2 and n+p_2=(m+r_2)^2 and p_2 divide n therfore p_2 divide n+p_2 so p_2 divide (m+r_2)^2 thus p_2 divide m+r_2 but we know p_2=(m+r_2)^2-n=m^2+2(r_2)m+(r_2)^2-m^2-q and we know any natural divisor of a natural number is smaller or equal to that number so we have
2(r_2)m +(r_2)^2-q =< m+r_2
But we have r_2=>2 and q<=2m+1 thus contradiction is trivial
Case 2: n has only one prime factor
Thus there exist prime number p and positive intiger a such that n=p^a therfore p^a+p is square but for any a larger then 1 we know that p divide n and p^2 do not divide n thus n can't be a square so a=1 and for a=1 we would get 2p is square thus 4 divide 2p and p is even thus p=2
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