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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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Quadric function
soryn   0
3 minutes ago
If f(x)=ax^2+bx+c, a,b,c integers, |a|>=3, and M îs the set of integers x for which f(x) is a prime number and f has exactly one integers solution,probe that M has at mist three elemente.
0 replies
soryn
3 minutes ago
0 replies
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2 var inquality
sqing   0
10 minutes ago
Source: Own
Let $ a,b\geq 0 $ and $ a+ b+ab=3 . $ Prove that
$$ (a^2+b^2+a+b-4)(2 - ab)\ge\sqrt 7(a-1)(b-1)(a-b)$$$$ 3 (a^2+b^2+a+b-4)(5 - 2ab)\ge 20(a-1)(b-1)(a-b)$$$$ 6(a^2+b^2-2)(5 - 2ab)\ge 35(a-1)(b-1)(a-b)$$$$ 2(a^2+b^2-2)(3 - ab)\ge 7(a-1)(b-1)(a-b)$$
0 replies
1 viewing
sqing
10 minutes ago
0 replies
J
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2 var inquality
sqing   0
25 minutes ago
Source: Own
Let $ a,b\geq 0 $ and $ a+ b+ab+a^2+b^2=5. $ Prove that
$$ (9+\sqrt{21} ) (a+b-2)(5- 2ab) \ge 10(a-1)(b-1)(a-b)$$$$ (9+\sqrt{21} ) (a+b-2)(3- ab) \ge6(a-1)(b-1)(a-b)$$
0 replies
sqing
25 minutes ago
0 replies
J
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2 var inquality
sqing   1
N 38 minutes ago by sqing
Source: Own
Let $ a,b\geq 0 $ and $ a+ b+2ab=4 . $ Prove that
$$ 3(a+b-2)(2 - ab) \ge (a-1)(b-1)(a-b)$$$$ 9 (a+b-2)(3 - 2ab) \ge 2\sqrt 5(a-1)(b-1)(a-b)$$$$9(a+b-2)(6 - 5ab) \ge2\sqrt {14} (a-1)(b-1)(a-b)$$
1 reply
sqing
an hour ago
sqing
38 minutes ago
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Inequalities
hn111009   1
N an hour ago by hn111009
Source: Maybe anywhere?
Let $a,b,c>0;r,s\in\mathbb{R}$ satisfied $a+b+c=1.$ Find minimum and maximum of $$P=a^rb^s+b^rc^s+c^ra^s.$$
1 reply
hn111009
Apr 13, 2025
hn111009
an hour ago
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Function equation
luci1337   1
N 2 hours ago by jasperE3
find all function $f:R \rightarrow R$ such that:
$2f(x)f(x+y)-f(x^2)=\frac{x}{2}(f(2x)+f(f(y)))$ with all $x,y$ is real number
1 reply
luci1337
Yesterday at 3:01 PM
jasperE3
2 hours ago
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Integer-Valued FE comes again
lminsl   204
N 2 hours ago by Sedro
Source: IMO 2019 Problem 1
Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $$f(2a)+2f(b)=f(f(a+b)).$$Proposed by Liam Baker, South Africa
204 replies
lminsl
Jul 16, 2019
Sedro
2 hours ago
J
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Quadratic system
juckter   31
N 2 hours ago by blueprimes
Source: Mexico National Olympiad 2011 Problem 3
Let $n$ be a positive integer. Find all real solutions $(a_1, a_2, \dots, a_n)$ to the system:

\[a_1^2 + a_1 - 1 = a_2\]\[ a_2^2 + a_2 - 1 = a_3\]\[\hspace*{3.3em} \vdots \]\[a_{n}^2 + a_n - 1 = a_1\]
31 replies
juckter
Jun 22, 2014
blueprimes
2 hours ago
J
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The old one is gone.
EeEeRUT   5
N 2 hours ago by Thelink_20
Source: EGMO 2025 P2
An infinite increasing sequence $a_1 < a_2 < a_3 < \cdots$ 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 $a_n = b_n$.
5 replies
1 viewing
EeEeRUT
Apr 16, 2025
Thelink_20
2 hours ago
J
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Another factorisation problem
kjhgyuio   2
N 3 hours ago by kjhgyuio
........
2 replies
kjhgyuio
3 hours ago
kjhgyuio
3 hours ago
J
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Beautiful geometry
m4thbl3nd3r   1
N 3 hours ago by m4thbl3nd3r
Let $\omega$ be the circumcircle of triangle $ABC$, $M$ is the midpoint of $BC$ and $E$ be the second intersection of $AM$ and $\omega$. Tangent line of $\omega$ at $E$ intersects $BC$ at $P$, let $PKL$ be a transversal of $\omega$ and $X,Y$ be intersections of $AK,AL$ with $BC$. Let $PF$ be a tangent line of $\omega$. Prove that $LYFP$ is cyclic
1 reply
m4thbl3nd3r
Yesterday at 4:41 PM
m4thbl3nd3r
3 hours ago
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integers inequality
azzam2912   0
4 hours ago

3. Let $a, b, c, d, x, y$ be positive integers that satisfy the inequality
\[ \frac{a}{b} < \frac{x}{y} < \frac{c}{d} \]with $bc - ad = 5$. If $b + d = 2025$, determine the minimum value of $y$.


0 replies
azzam2912
4 hours ago
0 replies
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MONT pg 31 example 1.10.40
Jaxman8   0
5 hours ago
Can somebody explain why it works.
0 replies
Jaxman8
5 hours ago
0 replies
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NEPAL TST 2025 DAY 2
Tony_stark0094   9
N 5 hours ago by hectorleo123
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.

$\textbf{Proposed by Kritesh Dhakal, Nepal.}$
9 replies
Tony_stark0094
Apr 12, 2025
hectorleo123
5 hours ago
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J
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Find all triangles
Rushil   6
N Nov 13, 2024 by Mr.Sharkman
Source: IMO 1968 A1
Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.
6 replies
Rushil
Nov 4, 2005
Mr.Sharkman
Nov 13, 2024
J
Find all triangles
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Reveal topic tags
trigonometry
algebra
Triangle
triangle inequality
IMO
IMO 1968
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G H BBookmark kLocked kLocked NReply
Source: IMO 1968 A1
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Rushil
1592 posts
Rushil
#1 h  Nov 4, 2005, 6:58 AM • 2 Y
Y by Adventure10, Mango247
Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.
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campos
411 posts
campos
#2 h  Nov 6, 2005, 3:22 AM • 2 Y
Y by Adventure10, Mango247
Let $ABC$ be a triangle with such conditions, such that $A=2B$. It's very well-known (or it can be deduced from the cosine law and the double-angle formula, or i think it's somewhere in the forum) that $a^2-b^2=bc$
Then, we just have to check the cases $(a,b,c)=(a,a-1,a-2), (a,b,c)=(a,a-2,a-1)$, and $(a,b,c)=(a,a-1,a+1)$.

$(a,b,c)=(a,a-1,a-2)$ gives $a^2-5a+3=0$, but it has no integer roots.
$(a,b,c)=(a,a-1,a+1)$ gives $a^2-2a=0 \Rightarrow a=2 \Rightarrow a+b=3=c \Rightarrow\Leftarrow$
finally, $(a,b,c)=(a,a-2,a-1)$ gives $a^2-7a+6=0$ and since $a=1 \Rightarrow c=0$, then we must have that $a=6$.
then the triangle that satisfies such conditions is $(a,b,c)=(6,4,5)$ with $A=2B$
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sayantanchakraborty
505 posts
sayantanchakraborty
#3 h  Dec 22, 2013, 4:15 AM • 2 Y
Y by Adventure10, Mango247
If <A=2<B, some simple manipulating with the sine rule gives \[ /a^2=b(b+c) /\]. Assuming the lengths of the sides to be \[/ (a-1,a,a+1)/\] and considering all possible cases we get the solutions in (a,b,c) as
\[/(1,2,3) and (4,5,6)/\] and obviously all its permutations.
Z K Y
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sayantanchakraborty
505 posts
sayantanchakraborty
#4 h  Dec 22, 2013, 4:18 AM • 2 Y
Y by Adventure10, Mango247
Campos missed the solution \[(1,2,3)\].
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sunken rock
4383 posts
sunken rock
#5 h  Dec 22, 2013, 5:12 AM • 2 Y
Y by Heisenberg09, Adventure10
sayantanchakraborty wrote:
Campos missed the solution \[(1,2,3)\].

That´s NOT a triangle, but a degenerated one!

Best regards,
sunken rock
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suli
1498 posts
suli
#6 h  May 17, 2015, 4:40 AM • 2 Y
Y by Adventure10, Mango247
Well technically a degenerate triangle is a triangle. The problem should be worded more clearly, or both solutions should be accepted as correct.
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Mr.Sharkman
496 posts
Mr.Sharkman
#8 h  Nov 13, 2024, 8:13 PM
Y by
Bruh AMC 2024 #22
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