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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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EGMO magic square
Lukaluce   8
N 4 minutes ago by Yiyj1
Source: EGMO 2025 P6
In each cell of a $2025 \times 2025$ board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to $1$, and the sum of the numbers in each column is equal to $1$. Define $r_i$ to be the largest value in row $i$, and let $R = r_1 + r_2 + ... + r_{2025}$. Similarly, define $c_i$ to be the largest value in column $i$, and let $C = c_1 + c_2 + ... + c_{2025}$.
What is the largest possible value of $\frac{R}{C}$?

Proposed by Paulius Aleknavičius, Lithuania
8 replies
+3 w
Lukaluce
Today at 11:03 AM
Yiyj1
4 minutes ago
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NEPAL TST 2025 DAY 2
Tony_stark0094   7
N 25 minutes ago by lolsamo
Consider an acute triangle $\Delta ABC$. Let $D$ and $E$ be the feet of the altitudes from $A$ to $BC$ and from $B$ to $AC$ respectively.

Define $D_1$ and $D_2$ as the reflections of $D$ across lines $AB$ and $AC$, respectively. Let $\Gamma$ be the circumcircle of $\Delta AD_1D_2$. Denote by $P$ the second intersection of line $D_1B$ with $\Gamma$, and by $Q$ the intersection of ray $EB$ with $\Gamma$.

If $O$ is the circumcenter of $\Delta ABC$, prove that $O$, $D$, and $Q$ are collinear if and only if quadrilateral $BCQP$ can be inscribed within a circle.

$\textbf{Proposed by Kritesh Dhakal, Nepal.}$
7 replies
Tony_stark0094
Apr 12, 2025
lolsamo
25 minutes ago
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Italian WinterCamps test07 Problem5
mattilgale   57
N 44 minutes ago by Marcus_Zhang
Source: ISL 2006, A1, AIMO 2007, TST 1, P1
A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula
\[ a_{i + 1} = \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0; \]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large.

Proposed by Harmel Nestra, Estionia
57 replies
mattilgale
Jan 29, 2007
Marcus_Zhang
44 minutes ago
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number theory
mohsen   0
an hour ago
show that there exist natural numbers a,b such that none of the numbers a+1, a+2,...a+100 is divisible by none of b+1, b+2,..., b+100 but product of them is divisible by product of b+1,...,b+100.
0 replies
mohsen
an hour ago
0 replies
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Inequality while on a trip
giangtruong13   4
N an hour ago by GeoMorocco
Source: Trip
I find this inequality while i was on a trip, it was pretty fun and i have some new experience:
Let $a,b,c \geq -2$ such that: $a^2+b^2+c^2 \leq 8$. Find the maximum: $$A= \sum_{cyc} \frac{1}{16+a^3}$$
4 replies
giangtruong13
Apr 12, 2025
GeoMorocco
an hour ago
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A nice collinearity problem
April   25
N an hour ago by bin_sherlo
Source: IMO Shortlist 2007, G8, AIMO 2008, TST 7, P2
Point $ P$ lies on side $ AB$ of a convex quadrilateral $ ABCD$. Let $ \omega$ be the incircle of triangle $ CPD$, and let $ I$ be its incenter. Suppose that $ \omega$ is tangent to the incircles of triangles $ APD$ and $ BPC$ at points $ K$ and $ L$, respectively. Let lines $ AC$ and $ BD$ meet at $ E$, and let lines $ AK$ and $ BL$ meet at $ F$. Prove that points $ E$, $ I$, and $ F$ are collinear.

Author: Waldemar Pompe, Poland
25 replies
April
Jul 13, 2008
bin_sherlo
an hour ago
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Divisibility NT FE
CHESSR1DER   0
an hour ago
Source: Own
Find all functions $f$ $N \iff N$ such for any $a,b$:
$(a+b)^n|a^{f(b)} + b^{f(a)}$ where a natural number n is given.
0 replies
CHESSR1DER
an hour ago
0 replies
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Equation in naturals
Ahiles   50
N an hour ago by ray66
Source: BMO 2009 Problem 1
Solve the equation
\[ 3^x - 5^y = z^2.\]
in positive integers.

Greece
50 replies
Ahiles
Apr 30, 2009
ray66
an hour ago
J
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Turbo's en route to visit each cell of the board
Lukaluce   12
N an hour ago by cj13609517288
Source: EGMO 2025 P5
Let $n > 1$ be an integer. In a configuration of an $n \times n$ board, each of the $n^2$ cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate $90^{\circ}$ counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of $n$, the maximum number of good cells over all possible starting configurations.

Proposed by Melek Güngör, Turkey
12 replies
Lukaluce
Today at 11:01 AM
cj13609517288
an hour ago
J
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I found this question really easy, but it is a P4...
Sadigly   3
N 2 hours ago by grupyorum
Take a sequence $(a_n)_{n=1}^\infty$ such that

$a_1=3$

$a_n=a_1a_2a_3...a_{n-1}-1$

a) Prove that there exists infitely many primes that divides at least 1 term of the sequence.
b) Prove that there exists infitely many primes that doesn't divide any term of the sequence.
3 replies
Sadigly
Yesterday at 7:17 PM
grupyorum
2 hours ago
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Parallelograms and concyclicity
Lukaluce   17
N 2 hours ago by cj13609517288
Source: EGMO 2025 P4
Let $ABC$ be an acute triangle with incentre $I$ and $AB \neq AC$. Let lines $BI$ and $CI$ intersect the circumcircle of $ABC$ at $P \neq B$ and $Q \neq C$, respectively. Consider points $R$ and $S$ such that $AQRB$ and $ACSP$ are parallelograms (with $AQ \parallel RB, AB \parallel QR, AC \parallel SP$, and $AP \parallel CS$). Let $T$ be the point of intersection of lines $RB$ and $SC$. Prove that points $R, S, T$, and $I$ are concyclic.
17 replies
Lukaluce
Today at 10:59 AM
cj13609517288
2 hours ago
J
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sequence infinitely similar to central sequence
InterLoop   17
N 2 hours ago by juckter
Source: EGMO 2025/2
An infinite increasing sequence $a_1 < a_2 < a_3 < \dots$ of positive integers is called central if for every positive integer $n$, the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$.

Show that there exists an infinite sequence $b_1$, $b_2$, $b_3$, $\dots$ of positive integers such that for every central sequence $a_1$, $a_2$, $a_3$, $\dots$, there are infinitely many positive integers $n$ with $a_n = b_n$.
17 replies
InterLoop
Yesterday at 12:38 PM
juckter
2 hours ago
J
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Weighted Activity Selection Algorithm
Maximilian113   2
N 2 hours ago by Maximilian113
An interesting problem:

There are $n$ events $E_1, E_2, \cdots, E_n$ that are each continuous and last on a certain time interval. Each event has a weight $w_i.$ However, one can only choose to attend activities that do not overlap with each other. The goal is to maximize the sum of weights of all activities attended. Prove or disprove that the following algorithm allows for an optimal selection:

For each $E_i$ consider $x_i,$ the sum of $w_j$ over all $j$ such that $E_j$ and $E_i$ are not compatible.
1. At each step, delete the event that has the maximal $x_i.$ If there are multiple such events, delete the event with the minimal weight.
2. Update all $x_i$
3. Repeat until all $x_i$ are $0.$
2 replies
Maximilian113
Yesterday at 12:30 AM
Maximilian113
2 hours ago
J
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pairwise coprime sum gcd
InterLoop   31
N 2 hours ago by juckter
Source: EGMO 2025/1
For a positive integer $N$, let $c_1 < c_2 < \dots < c_m$ be all the positive integers smaller than $N$ that are coprime to $N$. Find all $N \ge 3$ such that
$$\gcd(N, c_i + c_{i+1}) \neq 1$$for all $1 \le i \le m - 1$.
31 replies
InterLoop
Yesterday at 12:34 PM
juckter
2 hours ago
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J
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no numbers of the form 80...01 are squares
Marius_Avion_De_Vanatoare   2
N Apr 3, 2025 by EVKV
Source: Moldova JTST 2024 P5
Prove that a number of the form $80\dots01$ (there is at least 1 zero) can't be a perfect square.
2 replies
Marius_Avion_De_Vanatoare
Jun 10, 2024
EVKV
Apr 3, 2025
J
no numbers of the form 80...01 are squares
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Source: Moldova JTST 2024 P5
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Marius_Avion_De_Vanatoare
56 posts
Marius_Avion_De_Vanatoare
#1 h  Jun 10, 2024, 2:37 PM
Y by
Prove that a number of the form $80\dots01$ (there is at least 1 zero) can't be a perfect square.
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navier3072
108 posts
navier3072
#2 h  Jun 10, 2024, 3:00 PM
Y by
Suppose not. Then, for $k\geq 2$, $80\dots01=8 \cdot 10^k +1=X^2$. Then, \[2^{k+3} \cdot 5^k =(X+1)(X-1) \]Hence,
Case 1: $2^{k+2} \cdot 5^m - 2 \cdot 5^n =2 $
Then, $2^{k+1} 5^m - 5^n =1 $. Since, $n>m$ (by size arguments), $5^m \equiv 1 \pmod 5 \implies m=0$, $n=k$. Thus, $2^{k+1} - 5^{k} =1 $, contradiction by, say, Mihailescu.

Case 2: $2 \cdot 5^m - 2^{k+2} \cdot 5^n =2 $
Then, using similar arguments, since $m>n$, $n=0$ and $m=k$, so $5^{k} - 2^{k+1} =1 $, contradiction.
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EVKV
43 posts
EVKV
#3 h  Apr 3, 2025, 2:21 AM
Y by
FTSOC assume it is possible for some k
$2^{k+3}5^{k}$ +$1$ = $X^{2}$
As $gcd(X+1,X-1) = 2$ Only 2 cases possible
Case 1
$2^{k+2}5^{k} -2 = 2$
which is nonsense modulo 5
Case 2
$25^{k} - 2^{k+2}= 2$
again nonsense for $k \geq 2$ (By Catalans or if u want induction)

For $k = 1 $
801 is not a square
This post has been edited 1 time. Last edited by EVKV, Apr 3, 2025, 2:23 AM
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