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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

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Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
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0 replies
jlacosta
Mar 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
Find the number of integral solutions
Mathlover08092002   1
N 6 minutes ago by quasar_lord
Source: MTRP 2019 Class 11-Multiple Choice Question: Problem 2 :-
What is the number of integral solutions of the equation $a^{b^2}=b^{2a}$, where a > 0 and $|b|>|a|$
[list=1]
[*] 3
[*] 4
[*] 6
[*] 8
[/list]
1 reply
Mathlover08092002
Apr 9, 2020
quasar_lord
6 minutes ago
What can you say about f(x)
Mathlover08092002   3
N 10 minutes ago by quasar_lord
Source: MTRP 2019 Class 11-Multiple Choice Question: Problem 1 :-
Let $f : (0, \infty) \to \mathbb{R}$ is differentiable such that $\lim \limits_{x \to \infty} f(x)=2019$ Then which of the following is correct?
[list=1]
[*] $\lim \limits_{x \to \infty} f'(x)$ always exists but not necessarily zero.
[*] $\lim \limits_{x \to \infty} f'(x)$ always exists and is equal to zero.
[*] $\lim \limits_{x \to \infty} f'(x)$ may not exist.
[*] $\lim \limits_{x \to \infty} f'(x)$ exists if $f$ is twice differentiable.
[/list]
3 replies
Mathlover08092002
Apr 9, 2020
quasar_lord
10 minutes ago
FE on Stems
mathscrazy   4
N 18 minutes ago by SatisfiedMagma
Source: STEMS 2025 Category B4, C3
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y\in \mathbb{R}$, \[xf(y+x)+(y+x)f(y)=f(x^2+y^2)+2f(xy)\]Proposed by Aritra Mondal
4 replies
+1 w
mathscrazy
Dec 29, 2024
SatisfiedMagma
18 minutes ago
xyz(x+y+z)=4
RnstTrjyn   5
N 18 minutes ago by sqing
Let $x, y, z$ be positive real numbers such that $xyz(x + y + z) = 4$. Prove that
$(x+y)^2+3(z+y)^2+(x+z)^2 \geq 8\sqrt7$
5 replies
1 viewing
RnstTrjyn
Feb 3, 2019
sqing
18 minutes ago
One inequality 1
prof.   2
N 19 minutes ago by invisibleman
If $x>0$ prove inequality $$\sqrt{x}\cdot (x+1)+x\cdot (x-4)+1\ge0.$$
2 replies
prof.
3 hours ago
invisibleman
19 minutes ago
Inequality
JK1603JK   1
N 20 minutes ago by CHESSR1DER
Source: unknown
Let a,b,c>=0 and ab+bc+ca>0 then prove \sqrt{a+b}+\sqrt{c+b}+\sqrt{a+c}\ge 2\sqrt[4]{ab+bc+ca}+\sqrt{\frac{a(b-c)^2+b(c-a)^2+c(a-b)^2}{ab+bc+ca}}
1 reply
1 viewing
JK1603JK
Today at 2:58 AM
CHESSR1DER
20 minutes ago
geometry coordinates
CHESSR1DER   0
29 minutes ago
Source: simplified version of Belarus TST
Points $A, B, C$ with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points $D, E, F$ with integer coordinates such that $AB = DE$, $AC = DF$, $BC = EF$
0 replies
CHESSR1DER
29 minutes ago
0 replies
Cutting a big square into smaller squares
nAalniaOMliO   5
N 31 minutes ago by RagvaloD
Source: Belarusian National Olympiad 2020
A $20 \times 20$ checkered board is cut into several squares with integer side length. The size of a square is it's side length.
What is the maximum amount of different sizes this squares can have?
5 replies
nAalniaOMliO
Jan 29, 2025
RagvaloD
31 minutes ago
Ah yes, very interesting
Quidditch   23
N 39 minutes ago by quantam13
Source: EGMO 2024 P4
For a sequence $a_1<a_2<\cdots<a_n$ of integers, a pair $(a_i,a_j)$ with $1\leq i<j\leq n$ is called interesting if there exists a pair $(a_k,a_l)$ of integers with $1\leq k<l\leq n$ such that $$\frac{a_l-a_k}{a_j-a_i}=2.$$For each $n\geq 3$, find the largest possible number of interesting pairs in a sequence of length $n$.
23 replies
Quidditch
Apr 14, 2024
quantam13
39 minutes ago
Nice FE as the First Day Finale
swynca   3
N 43 minutes ago by PerfectPlayer
Source: 2025 Turkey TST P3
Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for all $x,y \in \mathbb{R}-\{0\}$,
$$ f(x) \neq 0 \text{ and } \frac{f(x)}{f(y)} + \frac{f(y)}{f(x)} - f \left( \frac{x}{y}-\frac{y}{x} \right) =2 $$
3 replies
swynca
Mar 18, 2025
PerfectPlayer
43 minutes ago
keep one card and discard the other
Scilyse   1
N an hour ago by g0USinsane777
Source: CGMO 2024 P2
There are $8$ cards on which the numbers $1$, $2$, $\dots$, $8$ are written respectively. Alice and Bob play the following game: in each turn, Alice gives two cards to Bob, who must keep one card and discard the other. The game proceeds for four turns in total; in the first two turns, Bob cannot keep both of the cards with the larger numbers, and in the last two turns, Bob also cannot keep both of the cards with the larger numbers. Let $S$ be the sum of the numbers written on the cards that Bob keeps. Find the greatest positive integer $N$ for which Bob can guarantee that $S$ is at least $N$.
1 reply
Scilyse
Jan 28, 2025
g0USinsane777
an hour ago
Simultaneous eqs. with matrix
RenheMiResembleRice   7
N an hour ago by RenheMiResembleRice
Source: Ningyi Hou
Solve the attached with steps
7 replies
RenheMiResembleRice
4 hours ago
RenheMiResembleRice
an hour ago
Find the minimum
sqing   6
N an hour ago by sqing
Source: 2019 China Mathematical Olympiad Hope League Summer Camp
Let $x,y,z $ be positive real number such that $xyz(x+y+z)=4.$ Find the minimum value of $(x+y)^2+2(y+z)^2+3(z+x)^2.$
6 replies
1 viewing
sqing
Aug 10, 2019
sqing
an hour ago
Interesting inequality
sqing   2
N 2 hours ago by sqing
Source: Own
Let $ a,b\geq 0  . $ Prove that
$$ a^4+b^4 +kab\geq\left(\sqrt{k(k+2)}-k\right)ab(a+b+k)$$Where $ k>0 . $
$$ a^4+b^4 +ab\geq (\sqrt 3-1)ab(a+b+1)$$$$ a^4+b^4 +2ab\geq 2(\sqrt 2-1)ab(a+b+2)$$
2 replies
sqing
5 hours ago
sqing
2 hours ago
Triangulating three-coloured polygon
Ankoganit   4
N Apr 14, 2021 by L567
Source: IMOTC PT 2 2018 P3, India
A convex polygon has the property that its vertices are coloured by three colors, each colour occurring at least once and any two adjacent vertices having different colours. Prove that the polygon can be divided into triangles by diagonals, no two of which intersect in the interior of the polygon, in such a way that all the resulting triangles have vertices of all three colours.
4 replies
Ankoganit
Jul 18, 2018
L567
Apr 14, 2021
Triangulating three-coloured polygon
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G H BBookmark kLocked kLocked NReply
Source: IMOTC PT 2 2018 P3, India
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Ankoganit
3070 posts
#1 • 2 Y
Y by Adventure10, Mango247
A convex polygon has the property that its vertices are coloured by three colors, each colour occurring at least once and any two adjacent vertices having different colours. Prove that the polygon can be divided into triangles by diagonals, no two of which intersect in the interior of the polygon, in such a way that all the resulting triangles have vertices of all three colours.
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Supercali
1260 posts
#2 • 2 Y
Y by Adventure10, Mango247
We will use induction. Just notice that there exist 3 consecutive vertices with different colours.
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biomathematics
2564 posts
#3 • 3 Y
Y by MathBoy23, Adventure10, Mango247
Supercali wrote:
We will use induction. Just notice that there exist 3 consecutive vertices with different colours.

The idea is correct but some more detail is required.

Let the three colours be red (R), blue (B), and green (G). Let the convex polygon be $A_1A_2 \cdots A_n$. The colour of a vertex $v$ is denoted by $c(v)$. Suppose some colour, say red, occurs only once as the colour of some vertex, say $c(A_n) = R$. Then the vertices from $A_1$ to $A_{n-1}$ must be coloured alternately with the colours $B$ and $G$. Thus the triangulation $A_nA_1A_2, A_nA_2A_3 , \cdots , A_nA_{n-2}A_{n-1}$ works.

Otherwise for each colour, there exist at least two vertices with that colour. Here, we show that Supercali's claim is true. Indeed, assume that it was not true. Let $c(A_1) = R$ and $c(A_2) = B$, say. Then by our assumption, $c(A_3) = R$, $c(A_4) = B,$ etc, and the whole colouring is determined by only red and blue. But this would lead to a contradiction, as the colour green occurred nowhere. Thus some vertex $v$ has neighbours of different colours. The triangle $T$ that they form has all three colours different. Separate this triangle, and the remaining polygon is $n-1$-sided, and each colour occurs at least once in it ( this is why we need to make cases; we need each colour to occur at least once). Now this is induction.
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MathBoy23
487 posts
#4 • 2 Y
Y by Adventure10, Mango247
I came here so happy, thinking that I had solved the problem and that I would post a solution. Saw bio's solution and told myself that people can think alike. :D
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L567
1184 posts
#5
Y by
Induct on the number of sides, its obvious for $n=3$, so assume it holds for $n$ and we need to prove it for $n+1$

First, see that there must exist three consecutive vertices of different colours. Suppose not, then pick any two adjacent vertices with different colours. Call the colours $A,B$, so we get that the next vertices need to be coloured $A,B,A,B$ and so on, and so all vertices have to be coloured only $A,B$, which is impossible.

So, pick any triple of three consecutive vertices of different colours, call them $v_1,v_2,v_3$ and let them be coloured with colours $A,B,C$. Then, we can draw diagonal $v_1v_3$ and then use the induction hypothesis on all the vertices except $v_2$. This works (almost always) because $v_1, v_3$ are indeed different colours, as required by the problem.

The only problem arises when $v_2$ was the only vertex of colour $B$, because then the remaining polygon does not have any vertex of colour $B$ and so the induction hypothesis doesnt work.

But this case is only possible if the remaining polygon is a $2k$ gon with colours alternating every vertex. Call the remaining vertices $v_1, v_2, ..., v_{2k}$. Then, just draw the diagonals $v_1v_3, v_1v_5, v_1v_7....v_1v_{2k-1}$ and $v_3v_5, v_5v_7,...,v_{2k-3}v_{2k-1}$, which can easily be checked to work.

Therefore, it is always possible to divide the given polygon into triangles such that all triangles have vertices of all three colours.

Edit: Just realised my solution is the same as that of biomathemetics :(
This post has been edited 4 times. Last edited by L567, Apr 12, 2023, 10:28 AM
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