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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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0 replies
jlacosta
Apr 2, 2025
0 replies
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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
J
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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
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Something nice
KhuongTrang   25
N an hour ago by KhuongTrang
Source: own
Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$$
25 replies
KhuongTrang
Nov 1, 2023
KhuongTrang
an hour ago
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Beautiful problem
luutrongphuc   12
N an hour ago by luutrongphuc
(Phan Quang Tri) Let triangle $ABC$ be circumscribed about circle $(I)$, and let $H$ be the orthocenter of $\triangle ABC$. The circle $(I)$ touches line $BC$ at $D$. The tangent to the circle $(BHC)$ at $H$ meets $BC$ at $S$. Let $J$ be the midpoint of $HI$, and let the line $DJ$ meet $(I)$ again at $X$. The tangent to $(I)$ parallel to $BC$ meets the line $AX$ at $T$. Prove that $ST$ is tangent to $(I)$.
12 replies
1 viewing
luutrongphuc
Apr 4, 2025
luutrongphuc
an hour ago
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2011-gon
3333   25
N 2 hours ago by Marcus_Zhang
Source: All-Russian 2011
A convex 2011-gon is drawn on the board. Peter keeps drawing its diagonals in such a way, that each newly drawn diagonal intersected no more than one of the already drawn diagonals. What is the greatest number of diagonals that Peter can draw?
25 replies
3333
May 17, 2011
Marcus_Zhang
2 hours ago
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Navid FE on R+
Assassino9931   0
2 hours ago
Source: Bulgaria Balkan MO TST 2025
Determine all functions $f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ such that
\[ f(x)f\left(x + 4f(y)\right) = xf\left(x + 3y\right) + f(x)f(y) \]for any positive real numbers $x,y$.
0 replies
Assassino9931
2 hours ago
0 replies
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Combinatorics on progressions
Assassino9931   0
2 hours ago
Source: Bulgaria Balkan MO TST 2025
Let \( p > 1 \) and \( q > 1 \) be coprime integers. Call a set $a_1 < a_2 < \cdots < a_{p+q}$ balanced if the numbers \( a_1, a_2, \ldots, a_p \) form an arithmetic progression with difference \( q \), and the numbers \( a_p, a_{p+1}, \ldots, a_{p+q} \) form an arithmetic progression with difference \( p \).

In terms of $p$ and $q$, determine the maximum size of a collection of balanced sets such that every two of them have a non-empty intersection.
0 replies
1 viewing
Assassino9931
2 hours ago
0 replies
J
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Linear recurrence fits with factorial finitely often
Assassino9931   0
2 hours ago
Source: Bulgaria Balkan MO TST 2025
Let $k\geq 3$ be an integer. The sequence $(a_n)_{n\geq 1}$ is defined via $a_1 = 1$, $a_2 = k$ and
\[ a_{n+2} = ka_{n+1} + a_n \]for any positive integer $n$. Prove that there are finitely many pairs $(m, \ell)$ of positive integers such that $a_m = \ell!$.
0 replies
1 viewing
Assassino9931
2 hours ago
0 replies
J
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Projective training on circumscribds
Assassino9931   0
2 hours ago
Source: Bulgaria Balkan MO TST 2025
Let $ABCD$ be a circumscribed quadrilateral with incircle $k$ and no two opposite angles equal. Let $P$ be an arbitrary point on the diagonal $BD$, which is inside $k$. The segments $AP$ and $CP$ intersect $k$ at $K$ and $L$. The tangents to $k$ at $K$ and $L$ intersect at $S$. Prove that $S$ lies on the line $BD$.
0 replies
1 viewing
Assassino9931
2 hours ago
0 replies
J
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Multiplicative polynomial exactly 2025 times
Assassino9931   0
2 hours ago
Source: Bulgaria Balkan MO TST 2025
Does there exist a polynomial $P$ on one variable with real coefficients such that the equation $P(xy) = P(x)P(y)$ has exactly $2025$ ordered pairs $(x,y)$ as solutions?
0 replies
1 viewing
Assassino9931
2 hours ago
0 replies
J
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Holy inequality
giangtruong13   2
N 3 hours ago by arqady
Source: Club
Let $a,b,c>0$. Prove that:$$\frac{8}{\sqrt{a^2+b^2+c^2+1}} - \frac{9}{(a+b)\sqrt{(a+2c)(b+2c)}} \leq \frac{5}{2}$$
2 replies
giangtruong13
Yesterday at 4:09 PM
arqady
3 hours ago
J
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Inequality with Unhomogenized Condition
Mathdreams   1
N 3 hours ago by arqady
Source: 2025 Nepal Mock TST Day 3 Problem 3
Let $x, y, z$ be positive reals such that $xy + yz + xz + xyz = 4$. Prove that $$3(2 - xyz) \ge \frac{2}{xy+1} + \frac{2}{yz+1} + \frac{2}{xz + 1}.$$(Shining Sun, USA)
1 reply
1 viewing
Mathdreams
4 hours ago
arqady
3 hours ago
J
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Orthocenter config once again
Assassino9931   5
N 3 hours ago by Assassino9931
Source: Bulgaria National Olympiad 2025, Day 2, Problem 4
Let \( ABC \) be an acute triangle with \( AB < AC \), midpoint $M$ of side $BC$, altitude \( AD \) (\( D \in BC \)), and orthocenter \( H \). A circle passes through points \( B \) and \( D \), is tangent to line \( AB \), and intersects the circumcircle of triangle \( ABC \) at a second point \( Q \). The circumcircle of triangle \( QDH \) intersects line \( BC \) at a second point \( P \). Prove that the lines \( MH \) and \( AP \) are perpendicular.
5 replies
1 viewing
Assassino9931
Tuesday at 1:53 PM
Assassino9931
3 hours ago
J
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Scanner on squarefree integers
Assassino9931   2
N 4 hours ago by Assassino9931
Source: Bulgaria National Olympiad 2025, Day 2, Problem 5
Let $n$ be a positive integer. Prove that there exists a positive integer $a$ such that exactly $\left \lfloor \frac{n}{4} \right \rfloor$ of the integers $a + 1, a + 2, \ldots, a + n$ are squarefree.
2 replies
1 viewing
Assassino9931
Tuesday at 1:54 PM
Assassino9931
4 hours ago
J
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Poly with sequence give infinitely many prime divisors
Assassino9931   5
N 4 hours ago by Assassino9931
Source: Bulgaria National Olympiad 2025, Day 1, Problem 3
Let $P(x)$ be a non-constant monic polynomial with integer coefficients and let $a_1, a_2, \ldots$ be an infinite sequence. Prove that there are infinitely many primes, each of which divides at least one term of the sequence $b_n = P(n)^{a_n} + 1$.
5 replies
Assassino9931
Tuesday at 1:51 PM
Assassino9931
4 hours ago
J
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Connecting chaos in a grid
Assassino9931   2
N 4 hours ago by Assassino9931
Source: Bulgaria National Olympiad 2025, Day 1, Problem 2
Exactly \( n \) cells of an \( n \times n \) square grid are colored black, and the remaining cells are white. The cost of such a coloring is the minimum number of white cells that need to be recolored black so that from any black cell \( c_0 \), one can reach any other black cell \( c_k \) through a sequence \( c_0, c_1, \ldots, c_k \) of black cells where each consecutive pair \( c_i, c_{i+1} \) are adjacent (sharing a common side) for every \( i = 0, 1, \ldots, k-1 \). Let \( f(n) \) denote the maximum possible cost over all initial colorings with exactly \( n \) black cells. Determine a constant $\alpha$ such that
\[ \frac{1}{3}n^{\alpha} \leq f(n) \leq 3n^{\alpha} \]for any $n\geq 100$.
2 replies
Assassino9931
Tuesday at 1:50 PM
Assassino9931
4 hours ago
J
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Problems for v_p(n)
xytunghoanh   15
N Apr 5, 2025 by AshAuktober
Hello everyone. I need some easy problems for $v_p(n)$ (use in a problem for junior) to practice. Can anyone share to me?
Thanks :>
15 replies
xytunghoanh
Apr 5, 2025
AshAuktober
Apr 5, 2025
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Problems for v_p(n)
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Reveal topic tags
number theory
Junior
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G H BBookmark kLocked kLocked NReply
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xytunghoanh
23 posts
xytunghoanh
#1 h  Apr 5, 2025, 1:28 PM
Y by
Hello everyone. I need some easy problems for $v_p(n)$ (use in a problem for junior) to practice. Can anyone share to me?
Thanks :>
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whwlqkd
89 posts
whwlqkd
#2 h  Apr 5, 2025, 2:03 PM
Y by
What about this one?
Find all positive integer pair $(k,n)$ that satisfies $k!=(2^n-1)(2^n-2)(2^n-4)(2^n-8)…(2^n-2^{n-1})$
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Sadigly
133 posts
Sadigly
#3 h  Apr 5, 2025, 2:50 PM
Y by
whwlqkd wrote:
What about this one?
Find all positive integer pair $(k,n)$ that satisfies $k!=(2^n-1)(2^n-2)(2^n-4)(2^n-8)…(2^n-2^{n-1})$

My man, he is junior,and asks for easy problems,why IMO question?
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whwlqkd
89 posts
whwlqkd
#4 h  Apr 5, 2025, 2:55 PM
Y by
What is the aops scale of easy question?
(6=IMO P1, 7.5=P2, 4=Junior Balkan P1, 5=Junior Balkan P2)
This post has been edited 1 time. Last edited by whwlqkd, Apr 5, 2025, 2:56 PM
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mathnerd_101
1477 posts
mathnerd_101
#5 h  Apr 5, 2025, 2:56 PM
Y by
I'm pretty new to $v_p(n)$ stuff, so here's a bold shot. Someone tell me where I went wrong.

Bogus Solution
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AshAuktober
972 posts
AshAuktober
#6 h  Apr 5, 2025, 2:58 PM
Y by
Well the first equation you used is incorrect. Since $\nu_p$ extracts an exponent, you can expect it to behave somewhat like a logarithm.
In particular, \[\nu_p(mn)=\nu_p(m)+\nu_p(n).\]
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Maximilian113
529 posts
Maximilian113
#7 h  Apr 5, 2025, 2:59 PM
Y by
yea the IMO is made easier by knowing you have to use v_p, but i guess its still not the easiest

Try:
https://artofproblemsolving.com/community/c6h3041815
https://artofproblemsolving.com/community/c6h2317634s1_ncg_returns
https://artofproblemsolving.com/community/c6h220915p1225024

Suppose that $m$ and $k$ are non-negative inegers and $p=2^{2^m}+1$ is a prime number. Prove that $2^{m+1}p^k$ is the smallest positive integer $n$ satisfying the congruence equation $2^n \equiv 1 \pmod {p^{k+1}}.$
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mathnerd_101
1477 posts
mathnerd_101
#8 h  Apr 5, 2025, 2:59 PM
Y by
Ohhh... oops! Looks like Stack Exchange is wrong... lol. I'll think about it some more, then.
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AshAuktober
972 posts
AshAuktober
#9 h  Apr 5, 2025, 3:01 PM • 2 Y
Y by scannose, whwlqkd
Here's an easy problem you can try.
Let $n$ be a positive integer. Evaluate \[\nu_3(2^{3^n}+1).\]
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whwlqkd
89 posts
whwlqkd
#10 h  Apr 5, 2025, 3:11 PM
Y by
lol…it is so easy
Click to reveal hidden text
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mathnerd_101
1477 posts
mathnerd_101
#11 h  Apr 5, 2025, 3:13 PM • 1 Y
Y by whwlqkd
Solution

Edit: Thanks so much. This was actually a really nice introduction to p-adic for me.

Also @above, please note there's people not as smart as you. What may seem easy to you may not be as easy to others. Please do be considerate and kind. Thanks C:
This post has been edited 2 times. Last edited by mathnerd_101, Apr 5, 2025, 3:14 PM
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AshAuktober
972 posts
AshAuktober
#12 h  Apr 5, 2025, 3:30 PM
Y by
mathnerd_101 wrote:
Solution

Edit: Thanks so much. This was actually a really nice introduction to p-adic for me.

Also @above, please note there's people not as smart as you. What may seem easy to you may not be as easy to others. Please do be considerate and kind. Thanks C:

Hmm try this without LTE. :D
Also to be tried:
If $a,b,c, \frac ab +\frac bc+\frac ca \in \mathbb{N}$, prove that $abc$ is a perfect cube.
This post has been edited 1 time. Last edited by AshAuktober, Apr 5, 2025, 3:32 PM
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Sadigly
133 posts
Sadigly
#13 h  Apr 5, 2025, 3:41 PM
Y by
Prove that for all $n>1$, the expression $\mathbb{H}_n=\sum_{i=1}^{n}\frac{1}{i}$ is not a integer
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xytunghoanh
23 posts
xytunghoanh
#14 h  Apr 5, 2025, 3:41 PM
Y by
Oh, I'm new to $v_p(n)$. I need some articles, problems to practice and learn.
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mathnerd_101
1477 posts
mathnerd_101
#15 h  Apr 5, 2025, 3:48 PM
Y by
@above I used this one here. It had helpful practice problems which I enjoyed.
This post has been edited 1 time. Last edited by mathnerd_101, Apr 5, 2025, 3:48 PM
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AshAuktober
972 posts
AshAuktober
#16 h  Apr 5, 2025, 3:59 PM
Y by
Hmm, the one you sent doesn't have a proof, does it? https://cjquines.com/files/liftingtheexponent.pdf might be better.
(Note that proving LTE is a good exercise so try it solo.)
This post has been edited 1 time. Last edited by AshAuktober, Apr 5, 2025, 4:00 PM
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