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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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Circumcircle excircle chaos
CyclicISLscelesTrapezoid   25
N 2 hours ago by bin_sherlo
Source: ISL 2021 G8
Let $ABC$ be a triangle with circumcircle $\omega$ and let $\Omega_A$ be the $A$-excircle. Let $X$ and $Y$ be the intersection points of $\omega$ and $\Omega_A$. Let $P$ and $Q$ be the projections of $A$ onto the tangent lines to $\Omega_A$ at $X$ and $Y$ respectively. The tangent line at $P$ to the circumcircle of the triangle $APX$ intersects the tangent line at $Q$ to the circumcircle of the triangle $AQY$ at a point $R$. Prove that $\overline{AR} \perp \overline{BC}$.
25 replies
CyclicISLscelesTrapezoid
Jul 12, 2022
bin_sherlo
2 hours ago
J
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hard problem
Cobedangiu   7
N 2 hours ago by arqady
Let $x,y,z>0$ and $xy+yz+zx=3$ : Prove that :
$\sum \ \frac{x}{y+z}\ge\sum \frac{1}{\sqrt{x+3}}$
7 replies
Cobedangiu
Apr 2, 2025
arqady
2 hours ago
J
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Absolute value
Silverfalcon   8
N 2 hours ago by zhoujef000
This problem seemed to be too obvious.. And I think I"m wrong.. :D

Problem:

Consider the sequence $x_0, x_1, x_2,...x_{2000}$ of integers satisfying

\[x_0 = 0, |x_n| = |x_{n-1} + 1|\]

for $n = 1,2,...2000$.

Find the minimum value of the expression $|x_1 + x_2 + ... x_{2000}|$.

My idea

Pretty sure I'm wrong but where did I go wrong?
8 replies
+1 w
Silverfalcon
Jun 27, 2005
zhoujef000
2 hours ago
J
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Combo problem
soryn   2
N 2 hours ago by Anulick
The school A has m1 boys and m2 girls, and ,the school B has n1 boys and n2 girls. Each school is represented by one team formed by p students,boys and girls. If f(k) is the number of cases for which,the twice schools has,togheter k girls, fund f(k) and the valute of k, for which f(k) is maximum.
2 replies
soryn
Today at 6:33 AM
Anulick
2 hours ago
J
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Tetrahedrons and spheres
ReticulatedPython   3
N 2 hours ago by vanstraelen
Let $OABC$ be a tetrahedron such that $\angle{AOB}=\angle{AOC}=\angle{BOC}=90^\circ.$ A sphere of radius $r$ is circumscribed about tetrahedron $OABC.$ Given that $OA=a$, $OB=b$, and $OC=c$, prove that $$r^2+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge \frac{9\sqrt[3]{4}}{4}$$with equality at $a=b=c=\sqrt[3]{2}.$
3 replies
ReticulatedPython
Yesterday at 6:39 PM
vanstraelen
2 hours ago
J
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Calculate the distance of chess king!!
egxa   4
N 2 hours ago by Primeniyazidayi
Source: All Russian 2025 9.4
A chess king was placed on a square of an \(8 \times 8\) board and made $64$ moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called $\emph{pleasant}$ if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)
4 replies
egxa
Apr 18, 2025
Primeniyazidayi
2 hours ago
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As some nations like to say "Heavy theorems mostly do not help"
Assassino9931   9
N 3 hours ago by EVKV
Source: European Mathematical Cup 2022, Senior Division, Problem 2
We say that a positive integer $n$ is lovely if there exist a positive integer $k$ and (not necessarily distinct) positive integers $d_1$, $d_2$, $\ldots$, $d_k$ such that $n = d_1d_2\cdots d_k$ and $d_i^2 \mid n + d_i$ for $i=1,2,\ldots,k$.

a) Are there infinitely many lovely numbers?

b) Is there a lovely number, greater than $1$, which is a perfect square of an integer?
9 replies
Assassino9931
Dec 20, 2022
EVKV
3 hours ago
J
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congruence
moldovan   5
N 3 hours ago by EVKV
Source: Canada 2004
Let $p$ be an odd prime. Prove that:
\[\displaystyle\sum_{k=1}^{p-1}k^{2p-1} \equiv \frac{p(p+1)}{2} \pmod{p^2}\]
5 replies
moldovan
Jun 26, 2009
EVKV
3 hours ago
J
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Checking a summand property for integers sufficiently large.
DinDean   1
N 3 hours ago by Double07
For any fixed integer $m\geqslant 2$, prove that there exists a positive integer $f(m)$, such that for any integer $n\geqslant f(m)$, $n$ can be expressed by a sum of positive integers $a_i$'s as
\[n=a_1+a_2+\dots+a_m,\]where $a_1\mid a_2$, $a_2\mid a_3$, $\dots$, $a_{m-1}\mid a_m$.
1 reply
DinDean
4 hours ago
Double07
3 hours ago
J
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Σ to ∞
phiReKaLk6781   3
N 4 hours ago by Maxklark
Evaluate: $ \sum\limits_{k=1}^\infty \frac{1}{k\sqrt{k+2}+(k+2)\sqrt{k}}$
3 replies
phiReKaLk6781
Mar 20, 2010
Maxklark
4 hours ago
J
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real+ FE
pomodor_ap   4
N 4 hours ago by jasperE3
Source: Own, PDC001-P7
Let $f : \mathbb{R}^+ \to \mathbb{R}^+$ be a function such that
$$f(x)f(x^2 + y f(y)) = f(x)f(y^2) + x^3$$for all $x, y \in \mathbb{R}^+$. Determine all such functions $f$.
4 replies
pomodor_ap
Yesterday at 11:24 AM
jasperE3
4 hours ago
J
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FE solution too simple?
Yiyj1   8
N 4 hours ago by lksb
Source: 101 Algebra Problems from the AMSP
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that the equality $$f(f(x)+y) = f(x^2-y)+4f(x)y$$holds for all pairs of real numbers $(x,y)$.

My solution

I feel like my solution is too simple. Is there something I did wrong or something I missed?
8 replies
Yiyj1
Apr 9, 2025
lksb
4 hours ago
J
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Polynomials in Z[x]
BartSimpsons   16
N 4 hours ago by bin_sherlo
Source: European Mathematical Cup 2017 Problem 4
Find all polynomials $P$ with integer coefficients such that $P (0)\ne 0$ and $$P^n(m)\cdot P^m(n)$$is a square of an integer for all nonnegative integers $n, m$.

Remark: For a nonnegative integer $k$ and an integer $n$, $P^k(n)$ is defined as follows: $P^k(n) = n$ if $k = 0$ and $P^k(n)=P(P(^{k-1}(n))$ if $k >0$.

Proposed by Adrian Beker.
16 replies
BartSimpsons
Dec 27, 2017
bin_sherlo
4 hours ago
J
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Geometric inequality
ReticulatedPython   0
4 hours ago
Let $A$ and $B$ be points on a plane such that $AB=n$, where $n$ is a positive integer. Let $S$ be the set of all points $P$ such that $\frac{AP^2+BP^2}{(AP)(BP)}=c$, where $c$ is a real number. The path that $S$ traces is continuous, and the value of $c$ is minimized. Prove that $c$ is rational for all positive integers $n.$
0 replies
ReticulatedPython
4 hours ago
0 replies
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J
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Prove that
abduqahhor_math   5
N Apr 3, 2025 by krancky22
n^2+3n+5 is not divided to 121
5 replies
abduqahhor_math
Mar 31, 2025
krancky22
Apr 3, 2025
J
Prove that
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abduqahhor_math
5 posts
abduqahhor_math
#1 h  Mar 31, 2025, 5:37 PM
Y by
n^2+3n+5 is not divided to 121
This post has been edited 1 time. Last edited by abduqahhor_math, Mar 31, 2025, 5:38 PM
Reason: ! is extra
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Kempu33334
605 posts
Kempu33334
#2 h  Mar 31, 2025, 6:01 PM
Y by
abduqahhor_math wrote:
n^2+3n+5 is not divided to 121

Is the problem:

Prove that \[n^2+3n+5 \equiv 0\pmod{121}\]has no solutions?
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abduqahhor_math
5 posts
abduqahhor_math
#3 h  Apr 2, 2025, 4:14 PM
Y by
yes it is
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rchokler
2966 posts
rchokler
#4 h  Apr 2, 2025, 5:52 PM
Y by
First solve $n^2+3n+5\equiv 0\mod 11$.

\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c} $n$&0&1&2&3&4&5&6&7&8&9&10\\ \hline $n^2+3n+5$&5&9&4&1&0&1&4&9&5&3&3 \end{tabular}
So we have $n\equiv 4\mod 11$.

Next plug in $n=11k+4$.

$n^2+3n+5=(11k+4)^2+3(11k+4)+5=121k^2+88k+16+33k+12+5=121k^2+121k+33\equiv 33\mod 121$

$\square$
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KevinKV01
14 posts
KevinKV01
#5 h  Apr 3, 2025, 6:19 AM
Y by
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krancky22
1 post
krancky22
#6 h  Apr 3, 2025, 6:21 AM
Y by
Assume that n^2 + 3n + 5 is divisible by 121 when n is an integer, then we have that n^2 + 3n + 5 = 121m for some positive integer m.
n^2 + 3n + 5 - 121m = 0
n = (-3 +- square root of (9 - 4(5 - 121m)) )/2 = (-3 +- square root of (484m - 11))/2,
Now we must prove that 484m - 11 is not a perfect square to then contradict the fact that n is an integer
484m - 11 = 11(44m - 1); now clearly this can not be a perfect square as 11 is prime and 44m - 1 will never be divisible by 11.
Therefore, n^2 + 3n + 5 can not be divisible by 121
This post has been edited 1 time. Last edited by krancky22, Apr 3, 2025, 6:42 AM
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