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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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0 replies
jlacosta
May 1, 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
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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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My unsolved problem
ZeltaQN2008   0
2 minutes ago
Source: Belarus 2017
Find all funcition $f:(0,\infty)\rightarrow (0,\infty)$ such that for all any $x,y\in (0,\infty)$ :
$f(x+f(xy))=xf(1+f(y))$
0 replies
ZeltaQN2008
2 minutes ago
0 replies
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Minimizing Triangle Area
v_Power   1
N 6 minutes ago by lele0305
Hi,

I am trying to solve this question:
A square piece of toast ABCD of side length 1 and centre O is cut in half to form two equal pieces ABC and CDA. If the triangle ABC has to be cut into two parts of equal area, one would usually cut along the line of symmetry BO. However, there are other ways of doing this. Find, with justification, the length and location of the shortest straight cut which divides the triangle ABC into two parts of equal area.
I have worked out that the length is sqrt(sqrt(2)-1) using calculus, but I was wondering if there was a way to solve it without using calculus?
1 reply
v_Power
an hour ago
lele0305
6 minutes ago
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Hard limits
Snoop76   6
N 15 minutes ago by Snoop76
$a_n$ and $b_n$ satisfies the following recursion formulas: $a_{0}=1, $ $b_{0}=1$, $ a_{n+1}=a_{n}+b_{n}$$ $ and $ $$ b_{n+1}=(2n+3)b_{n}+a_{n}$. Find $ \lim_{n \to \infty} \frac{a_n}{(2n-1)!!}$ $ $ and $ $ $\lim_{n \to \infty} \frac{b_n}{(2n+1)!!}.$
6 replies
Snoop76
Mar 25, 2025
Snoop76
15 minutes ago
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Real numbers
Jackson0423   0
19 minutes ago

Among any \( n \) distinct real numbers, there exist two numbers \( a \) and \( b \) such that
\[ a^2 + b^2 < \sqrt{3}(1 + a^2b^2). \]Find the smallest possible value of \( n \).
0 replies
Jackson0423
19 minutes ago
0 replies
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Sequences problem
BBNoDollar   1
N 25 minutes ago by BBNoDollar
Source: Mathematical Gazette Contest
Determine the general term of the sequence of non-zero natural numbers (a_n)n≥1, with the property that gcd(a_m, a_n, a_p) = gcd(m^2 ,n^2 ,p^2), for any distinct non-zero natural numbers m, n, p.

⁡Note that gcd(a,b,c) denotes the greatest common divisor of the natural numbers a,b,c .
1 reply
1 viewing
BBNoDollar
Yesterday at 5:53 PM
BBNoDollar
25 minutes ago
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Prove XBY equal to angle C
nataliaonline75   1
N 44 minutes ago by cj13609517288
Let $M$ be the midpoint of $BC$ on triangle $ABC$. Point $X$ lies on segment $AC$ such that $AX=BX$ and $Y$ on line $AM$ such that $XY//AB$. Prove that $\angle XBY = \angle ACB$.
1 reply
nataliaonline75
an hour ago
cj13609517288
44 minutes ago
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Inequality related to geometry
ducthien   0
an hour ago
Let \( ABCD \) be a convex quadrilateral. The diagonals \( AC \) and \( BD \) intersect at \( P \), with \( \angle APD = 60^\circ \). Let \( E, F, G, \) and \( H \) be the midpoints of sides \( AB, BC, CD \), and \( DA \), respectively. Find the greatest positive real number \( k \) such that
\[ EG + 3FH \geq k \cdot d + (1 - k) \cdot s, \]where \( s \) is the semiperimeter of \( ABCD \) and \( d \) is the sum of the lengths of its diagonals (i.e., \( d = AC + BD \)). Determine when equality holds.

Im trying to slove this question
0 replies
ducthien
an hour ago
0 replies
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Hard diophant equation
MuradSafarli   6
N an hour ago by iniffur
Find all positive integers $x, y, z, t$ such that the equation

$$ 2017^x + 6^y + 2^z = 2025^t $$
is satisfied.
6 replies
MuradSafarli
Friday at 6:12 PM
iniffur
an hour ago
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trig relation with two equal circles, each passing through center of other
parmenides51   4
N an hour ago by Captainscrubz
Source: Sharygin 2010 Final 10.2
Each of two equal circles $\omega_1$ and $\omega_2$ passes through the center of the other one. Triangle $ABC$ is inscribed into $\omega_1$, and lines $AC, BC$ touch $\omega_2$ . Prove that $cosA + cosB = 1$.
4 replies
parmenides51
Nov 25, 2018
Captainscrubz
an hour ago
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sum (a^2 + b^2)/2ab + 2(ab + bc + ca)/3 >=5
parmenides51   9
N an hour ago by mudok
Source: 2023 Greece JBMO TST p3/ easy version of Shortlist 2022 A6 https://artofproblemsolving.com/community/c6h3099025p28018726
Let $a, b,$ and $c$ be positive real numbers such that $a^2 + b^2 + c^2 = 3$. Prove that
$$\frac{a^2 + b^2}{2ab} + \frac{b^2 + c^2}{2bc} + \frac{c^2 + a^2}{2ca} + \frac{2(ab + bc + ca)}{3} \ge 5 $$When equality holds?
9 replies
parmenides51
May 17, 2024
mudok
an hour ago
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Cool minimum
giangtruong13   0
an hour ago
Source: my friend
Let $x,y>0$ such that: $x>y>1$ và $(xy+1)^2+(x+y)^2\le2(x+y)(x^2-xy+y^2+1)$. Find min:
$P=\dfrac{\sqrt{x-y}}{y-1}$
0 replies
giangtruong13
an hour ago
0 replies
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angle chasing, tangents at circumcircle of a right triangle
parmenides51   1
N an hour ago by CovertQED
Source: China Northern MO 2015 10.2 CNMO
It is known that $\odot O$ is the circumcircle of $\vartriangle ABC$ wwith diameter $AB$. The tangents of $\odot O$ at points $B$ and $C$ intersect at $P$ . The line perpendicular to $PA$ at point $A$ intersects the extension of $BC$ at point $D$. Extend $DP$ at length $PE = PB$. If $\angle ADP = 40^o$ , find the measure of $\angle E$.
1 reply
parmenides51
Oct 28, 2022
CovertQED
an hour ago
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Inequality
lgx57   3
N an hour ago by lgx57
Source: Own
$a,b,c>0,ab+bc+ca=1$. Prove that

$$\sum \sqrt{8ab+1} \ge 5$$
(I don't know whether the equality holds)
3 replies
lgx57
Yesterday at 3:14 PM
lgx57
an hour ago
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polonomials
Ducksohappi   0
2 hours ago
given $p$ set of numbers:
$A_1=({a_{11}, a_{12}, ..., a_{1q}})$, ..., $A_p=(a_{p1}, ..., a_{pq}) satisfying: \forall k \le q-1, S(i,k)=S(j,k), \forall i<j\le p$
Where $S(i,k) $is k-degree elementary symmetric polonomial of $A_i$
Prove that:
$a_{i1}^k+...+a_{iq}^k=a_{j1}^k+...+a_{jq}^k, \forall 1\le i \le j \le p, 1\le k \le q-1$
0 replies
Ducksohappi
2 hours ago
0 replies
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J
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A problem about sum of all digits of a number
Tung-CHL   1
N Apr 25, 2025 by Tung-CHL
Source: Problems from the book
We say that a number $n$ is good if $n=m+s(m)$ for some positive integer $m$. Here $s(m)$ is the sum of all digits of $m$. Prove that:
$$10^k+n\; \text{is good for infinitely many} \; k \Longleftrightarrow n-1 \; \text{is good}$$
1 reply
Tung-CHL
Jul 16, 2022
Tung-CHL
Apr 25, 2025
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A problem about sum of all digits of a number
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Reveal topic tags
number theory
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Source: Problems from the book
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Tung-CHL
126 posts
Tung-CHL
#1 h  Jul 16, 2022, 8:48 AM
Y by
We say that a number $n$ is good if $n=m+s(m)$ for some positive integer $m$. Here $s(m)$ is the sum of all digits of $m$. Prove that:
$$10^k+n\; \text{is good for infinitely many} \; k \Longleftrightarrow n-1 \; \text{is good}$$
Z K Y
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Tung-CHL
126 posts
Tung-CHL
#2 h  Apr 25, 2025, 2:26 PM
Y by
This is actually not true!

There are infinitely many pairs of positive integers $(t,q)$ such that $9t=10^{q+1}+17$. Consider the number
$$m=99\dots900\dots02$$with $t$ digits $9$'s and $q$ digits $0$'s. We have
$$m+s(m)=m+9t+2=m+10^{q+1}+17+2=100\dots021=10^k+21$$where $k=t+q+1$. This means there are infinitely many $k$ such that $10^k+21$ is good. But $21-1=20$ is not a good number.
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