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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.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 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
If ab+1 is divisible by A then so is a+b
ravengsd   0
7 minutes ago
Source: Romania EGMO TST 2025 Day 2, Problem 4
Find the greatest positive integer $A$ such that, for all positive integers $a$ and $b$, if $A$ divides $ab+1$, then $A$ divides $a+b$.
0 replies
ravengsd
7 minutes ago
0 replies
4-var inequality
RainbowNeos   1
N 11 minutes ago by sqing
Given $a,b,c,d>0$, show that
\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]
1 reply
+1 w
RainbowNeos
5 hours ago
sqing
11 minutes ago
Do not try to bash on beautiful geometry
ItzsleepyXD   4
N 16 minutes ago by Tkn
Source: Own , Mock Thailand Mathematic Olympiad P9
Let $ABC$be triangle with point $D,E$ and $F$ on $BC,AB,CA$
such that $BE=CF$ and $E,F$ are on the same side of $BC$
Let $M$ be midpoint of segment $BC$ and $N$ be midpoint of segment $EF$
Let $G$ be intersection of $BF$ with $CE$ and $\dfrac{BD}{DC}=\dfrac{AC}{AB}$
Prove that $MN\parallel DG$
4 replies
ItzsleepyXD
Yesterday at 9:30 AM
Tkn
16 minutes ago
Surjective number theoretic functional equation
snap7822   1
N 22 minutes ago by internationalnick123456
Source: 2025 Taiwan TST Round 3 Independent Study 2-N
Let $f:\mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying the following conditions:
[list=i]
[*] For all $m, n \in \mathbb{N}$, if $m > n$ and $f(m) > f(n)$, then $f(m-n) = f(n)$;
[*] $f$ is surjective.
[/list]
Find the maximum possible value of $f(2025)$.

Proposed by snap7822
1 reply
snap7822
2 hours ago
internationalnick123456
22 minutes ago
Balkan 2005-3
zhaoli   36
N 22 minutes ago by sqing
Let $a,b,c$ be positive real numbers. Prove the inequality
\[\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\geq a+b+c+\frac{4(a-b)^2}{a+b+c}.\]
When does equality occur?
36 replies
1 viewing
zhaoli
May 6, 2005
sqing
22 minutes ago
P>2D
gwen01   4
N an hour ago by Rohit-2006
Source: Baltic Way 1992 #18
Show that in a non-obtuse triangle the perimeter of the triangle is always greater than two times the diameter of the circumcircle.
4 replies
gwen01
Feb 18, 2009
Rohit-2006
an hour ago
BMO 2024 SL A1
MuradSafarli   9
N an hour ago by zhaoli
A1.

Let \( u, v, w \) be positive reals. Prove that there is a cyclic permutation \( (x, y, z) \) of \( (u, v, w) \) such that the inequality:

\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]
holds for all positive real numbers \( a, b \) and \( c \).
9 replies
MuradSafarli
Apr 27, 2025
zhaoli
an hour ago
Canmoo the construction master
MathLuis   9
N an hour ago by cj13609517288
Source: USEMO 2023 Problem 3
Canmoo is trying to do constructions, but doesn't have a ruler or compass. Instead, Canmoo has a device that, given four distinct points $A$, $B$, $C$, $P$ in the plane, will mark the isogonal conjugate of $P$ with respect to triangle $ABC$, if it exists. Show that if two points are marked on the plane, then Canmoo can construct their midpoint using this device, a pencil for marking additional points, and no other tools.

(Recall that the isogonal conjugate of $P$ with respect to triangle $ABC$ is the point $Q$ such that lines $AP$ and $AQ$ are reflections around the bisector of $\angle BAC$, lines $BP$ and $BQ$ are reflections around the bisector of $\angle CBA$, lines $CP$ and $CQ$ are reflections around the bisector of $\angle ACB$. Additional points marked by the pencil can be assumed to be in general position, meaning they don't lie on any line through two existing points or any circle through three existing points.)


Maxim Li
9 replies
MathLuis
Oct 21, 2023
cj13609517288
an hour ago
Inspired by JK1603JK
sqing   2
N an hour ago by sqing
Source: Own
Let $ a,b,c $ be reals such that $  abc\neq 0$ and $ a+b+c=0.  $ Prove that
$$\left|\frac{a-b}{c}\right|+k\left|\frac{b-c}{a} \right|+k^2\left|\frac{c-a}{b} \right|\ge 3(k+1)$$Where $ k\geq 1.$
$$\left|\frac{a-b}{c}\right|+2\left|\frac{b-c}{a} \right|+4\left|\frac{c-a}{b} \right|\ge 9$$
2 replies
1 viewing
sqing
4 hours ago
sqing
an hour ago
Concurrence involving medians.
GeoMetrix   6
N an hour ago by ihategeo_1969
Source: STEMS 2020 CAT A P3
An acute scalene triangle $\triangle{ABC}$ with altitudes $\overline{AD}, \overline{BE},$ and $\overline{CF}$ is inscribed in circle $\Gamma$. Medians from $B$ and $C$ meet $\Gamma$ again at $K$ and $L$ respectively. Prove that the circumcircles of $\triangle{BFK}, \triangle{CEL}$ and $\triangle{DEF}$ concur.
6 replies
GeoMetrix
Jan 24, 2021
ihategeo_1969
an hour ago
all functions satisfying f(x+yf(x))+y = xy + f(x+y)
falantrng   32
N an hour ago by Yagiz_Gundogan
Source: Balkan MO 2025 P3
Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$,
\[f(x+yf(x))+y = xy + f(x+y).\]
Proposed by Giannis Galamatis, Greece
32 replies
+1 w
falantrng
Apr 27, 2025
Yagiz_Gundogan
an hour ago
Cardinality of sets containing both multiples and non-multiples of 3
tom-nowy   0
an hour ago
Source: Own
Let $n$ be a positive integer, and let $N$ be the set $\{ 1,2, \ldots, n\}$.
Let the sets $X_n$ and $Y_n$ be difined as:
\begin{align*}
X_n = &\left\{ (x_1,x_2,x_3) \in N^3  \mid  x_1+x_2+x_3 \text{ is not divisible by } 3. \right\}, \\
Y_n = &\left\{ (y_1,y_2,y_3) \in  N^3  \mid  \text{Among }  y_1-y_2,\, y_2-y_3,\, y_3-y_1, 
\text{ at least one is} \right. \\ 
& \left. \text{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:divisible by }  3  \text{ and at least one is not divisible by } 3. \right\}. 
\end{align*}Which is larger, $\left| X_n \right|$ or $\left| Y_n \right|$ ?
0 replies
tom-nowy
an hour ago
0 replies
Very easy NT
GreekIdiot   8
N an hour ago by GreekIdiot
Prove that there exists no natural number $n>1$ such that $n \mid 2^n-1$.
8 replies
GreekIdiot
Yesterday at 2:49 PM
GreekIdiot
an hour ago
Really hard geometry problem
wimpykid   2
N 2 hours ago by NO_SQUARES
Source: USAMO
USAMO 2025/3

Alice the architect and Bob the builder play a game. First, Alice chooses two points $P$ and $Q$ in the plane and a subset $\mathcal{S}$ of the plane, which are announced to Bob. Next, Bob marks infinitely many points in the plane, designating each a city. He may not place two cities within distance at most one unit of each other, and no three cities he places may be collinear. Finally, roads are constructed between the cities as follows: for each pair $A,\,B$ of cities, they are connected with a road along the line segment $AB$ if and only if the following condition holds: For every city $C$ distinct from $A$ and $B$, there exists $R\in\mathcal{S}$ such that $\triangle PQR$ is directly similar to either $\triangle ABC$ or $\triangle BAC$. Alice wins the game if (i) the resulting roads allow for travel between any pair of cities via a finite sequence of roads and (ii) no two roads cross. Otherwise, Bob wins. Determine, with proof, which player has a winning strategy.
2 replies
wimpykid
2 hours ago
NO_SQUARES
2 hours ago
Sequence...
Jackson0423   0
Apr 21, 2025
Let the sequence \( \{a_n\} \) be defined as follows:
\( a_0 = 1 \), and for all positive integers \( n \),
\[
a_n = a_{\left\lfloor \frac{n}{3} \right\rfloor} + a_{\left\lfloor \frac{n}{2} \right\rfloor}.
\]Find the sum of all values \( k \leq 100 \) for which there exists a unique positive integer \( n \) such that \( a_n = k \).
0 replies
Jackson0423
Apr 21, 2025
0 replies
Sequence...
G H J
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Jackson0423
56 posts
#1
Y by
Let the sequence \( \{a_n\} \) be defined as follows:
\( a_0 = 1 \), and for all positive integers \( n \),
\[
a_n = a_{\left\lfloor \frac{n}{3} \right\rfloor} + a_{\left\lfloor \frac{n}{2} \right\rfloor}.
\]Find the sum of all values \( k \leq 100 \) for which there exists a unique positive integer \( n \) such that \( a_n = k \).
This post has been edited 1 time. Last edited by Jackson0423, Apr 21, 2025, 3:21 PM
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