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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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IMO ShortList 1998, number theory problem 5
orl   63
N 25 minutes ago by ATM_
Source: IMO ShortList 1998, number theory problem 5
Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$.
63 replies
orl
Oct 22, 2004
ATM_
25 minutes ago
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Inequalities
sqing   8
N 31 minutes ago by sqing
Let $ a,b,c,d\geq 0 ,a-b+d=21 $ and $ a+3b+4c=101 $. Prove that
$$ - \frac{1681}{3}\leq ab - cd \leq 820$$$$ - \frac{16564}{9}\leq ac -bd \leq 420$$$$ - \frac{10201}{48}\leq ad- bc \leq\frac{1681}{3}$$
8 replies
sqing
Apr 11, 2025
sqing
31 minutes ago
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IMO Shortlist 2013, Number Theory #1
lyukhson   150
N 33 minutes ago by MuradSafarli
Source: IMO Shortlist 2013, Number Theory #1
Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that
\[ m^2 + f(n) \mid mf(m) +n \]
for all positive integers $m$ and $n$.
150 replies
lyukhson
Jul 10, 2014
MuradSafarli
33 minutes ago
J
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Interesting inequality
A_E_R   0
33 minutes ago
Let a,b,c,d are positive real numbers, if the following inequality holds ab^2+ac^2>=5bcd. Find the minimum value of explanation: (a^2+b^2+c^2+d^2)(1/a^2+1/b^2+1/c^2+1/d^2)
0 replies
A_E_R
33 minutes ago
0 replies
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abcd is not a perfect square if a,b,c,d are in arithmetic progression
adityaguharoy   1
N 37 minutes ago by Mathzeus1024
If $a,b,c,d$ are in arithmetic progression and $a \ne b , b \ne c , c \ne d,$ then show that the product $a\cdot b \cdot c \cdot d$ is not a perfect square
1 reply
adityaguharoy
Jun 24, 2022
Mathzeus1024
37 minutes ago
J
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Dividing Pairs
Jackson0423   1
N an hour ago by ND_
Source: Own
Let \( a \) and \( b \) be positive integers.
Suppose that \( a \) is a divisor of \( b^2 + 1 \) and \( b \) is a divisor of \( a^2 + 1 \).
Find all such pairs \( (a, b) \).
1 reply
Jackson0423
2 hours ago
ND_
an hour ago
J
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Number Theory Chain!
JetFire008   49
N an hour ago by r7di048hd3wwd3o3w58q
I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!

Question 1
49 replies
JetFire008
Apr 7, 2025
r7di048hd3wwd3o3w58q
an hour ago
J
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Inspired by my own results
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b\geq 0 $ and $ a^2-ab+b^2=1$ . Prove that
$$ (a+b-ab)\left( \frac{a}{b+1} + \frac{b}{a+1}\right)\leq \frac{13}{12} $$$$ (a+b-ab)\left( \frac{a}{b+1} + \frac{b}{a+1}+ab\right)\leq 2$$$$ (a+b+ab)\left( \frac{a}{b+1} + \frac{b}{a+1}\right)\leq 3$$$$ (a+b)\left( \frac{a}{b+1} + \frac{b}{a+1}+ab\right)\leq 4$$
1 reply
sqing
an hour ago
sqing
an hour ago
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Fun & Simple puzzle
Kscv   3
N an hour ago by vanstraelen
$\angle DCA=45^{\circ},$ $\angle BDC=15^{\circ},$ $\overline{AC}=\overline{CB}$

$\angle ADC=?$
3 replies
Kscv
Today at 1:46 AM
vanstraelen
an hour ago
J
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An upper bound for Iran TST 1996
Nguyenhuyen_AG   0
an hour ago
Let $a, \ b, \ c$ be the side lengths of a triangle. Prove that
\[\frac{ab+bc+ca}{(a+b)^2} + \frac{ab+bc+ca}{(b+c)^2} + \frac{ab+bc+ca}{(c+a)^2} \leqslant \frac{85}{36}.\]
0 replies
Nguyenhuyen_AG
an hour ago
0 replies
J
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Sharygin 2025 CR P2
Gengar_in_Galar   5
N an hour ago by NicoN9
Source: Sharygin 2025
Four points on the plane are not concyclic, and any three of them are not collinear. Prove that there exists a point $Z$ such that the reflection of each of these four points about $Z$ lies on the circle passing through three remaining points.
Proposed by:A Kuznetsov
5 replies
Gengar_in_Galar
Mar 10, 2025
NicoN9
an hour ago
J
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Kinda lookimg Like AM-GM
Atillaa   1
N 2 hours ago by Natrium
Show that for all positive real numbers \( a, b, c \), the following inequality always holds:
\[ \frac{ab}{b+1} + \frac{bc}{c+1} + \frac{ca}{a+1} \geq \frac{3abc}{1 + abc} \]
1 reply
Atillaa
2 hours ago
Natrium
2 hours ago
J
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Find all tuples of real numbers
nhathhuyyp5c   0
2 hours ago
Let \( n \geq 2 \) be an integer. Find all tuples of real numbers \( (a_1, a_2, \ldots, a_n) \) such that
\[ 2a_1 - 3a_2,\ 2a_2 - 3a_3,\ \ldots,\ 2a_n - 3a_1 \]is a permutation of \( a_1, a_2, \ldots, a_n \).
0 replies
nhathhuyyp5c
2 hours ago
0 replies
J
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NC State Math Contest Wake Tech Regional Problems and Solutions
mathnerd_101   9
N Today at 4:02 AM by BackToSchool
Problem 1: Determine the area enclosed by the graphs of $$y=|x-2|+|x-4|-2, y=-|x-3|+4.$$ Hint
Solution to P1

Problem 2: Calculate the sum of the real solutions to the equation $x^\frac{3}{2} -9x-16x^\frac{1}{2} +144=0.$
Hint
Solution to P2



Problem 3: List the two transformations needed to convert the graph $\frac{x-1}{x+2}$ to $\frac{3x-6}{x-1}.$
Hint
Solution to P3

Problem 4: Let $a,b$ be positive integers such that $a^2-b^2=20,$ and $a^3-b^3=120.$ Determine the value of $a+\frac{b^2}{a+b}.$
Hint
Solution for P4

Problem 5: Eve and Oscar are playing a game where they roll a fair, six-sided die. If an even number occurs on two consecutive rolls, then Eve wins. If an odd number is immediately followed by an even number, Oscar wins. The die is rolled until one person wins. What is the probability that Oscar wins?
Hint
Solution to P5

Problem 6: In triangle $ABC,$ $M$ is on point $\overline{AB}$ such that $AM = x+32$ and $MB=x+12$ and $N$ is a point on $\overline{AC}$ such that $MN=2x+1$ and $BC=x+22.$ Given that $\overline{MN} || \overline{BC},$ calculate $MN.$
Hint
Solution to P6

Problem 7: Determine the sum of the zeroes of the quadratic of polynomial $Q(x),$ given that $$Q(0)=72, Q(1) = 75, Q(3) = 63.$$
Hint

Solution to Problem 7

Problem 8:
Hint
Solution to P8

Problem 9:
Find the sum of all real solutions to $$(x-4)^{log_8(4x-16)} = 2.$$ Hint
Solution to P9

Problem 10:
Define the function
\[f(x) = \begin{cases} x - 9, & \text{if } x > 100 \\ f(f(x + 10)), & \text{if } x \leq 100 \end{cases}\]
Calculate \( f(25) \).

Hint

Solution to P10

Problem 11:
Let $a,b,x$ be real numbers such that $$log_{a-b} (a+b) = 3^{a+b}, log_{a+b} (a-b) = 125 \cdot 15^{b-a}, a^2-b^2=3^x. $$Find $x.$
Hint

Solution to P11

Problem 12: Points $A,B,C$ are on circle $Q$ such that $AC=2,$ $\angle AQC = 180^{\circ},$ and $\angle QAB = 30^{\circ}.$ Determine the path length from $A$ to $C$ formed by segment $AB$ and arc $BC.$

Hint
Solution to P12

Problem 13: Determine the number of integers $x$ such that the expression $$\frac{\sqrt{522-x}}{\sqrt{x-80}} $$is also an integer.
Hint

Solution to Problem 13

Problem 14: Determine the smallest positive integer $n$ such that $n!$ is a multiple of $2^15.$

Hint
Solution to Problem 14

Problem 15: Suppose $x$ and $y$ are real numbers such that $x^3+y^3=7,$ and $xy(x+y)=-2.$ Calculate $x-y.$
Funnily enough, I guessed this question right in contest.

Hint
Solution to Problem 15

Problem 16: A sequence of points $p_i = (x_i, y_i)$ will follow the rules such that
\[ p_1 = (0,0), \quad p_{i+1} = (x_i + 1, y_i) \text{ or } (x_i, y_i + 1), \quad p_{10} = (4,5). \]How many sequences $\{p_i\}_{i=1}^{10}$ are possible such that $p_1$ is the only point with equal coordinates?

Hint
Solution to P16

Problem 18: (Also stolen from akliu's blog post)
Calculate

$$\sum_{k=0}^{11} (\sqrt{2} \sin(\frac{\pi}{4}(1+2k)))^k$$
Hint
Solution to Problem 18

Problem 19: Determine the constant term in the expansion of $(x^3+\frac{1}{x^2})^{10}.$

Hint
Solution to P19

Problem 20:

In a magical pond there are two species of talking fish: trout, whose statements are always true, and \emph{flounder}, whose statements are always false. Six fish -- Alpha, Beta, Gamma, Delta, Epsilon, and Zeta -- live together in the pond. They make the following statements:
Alpha says, "Delta is the same kind of fish as I am.''
Beta says, "Epsilon and Zeta are different from each other.''
Gamma says, "Alpha is a flounder or Beta is a trout.''
Delta says, "The negation of Gamma's statement is true.''
Epsilon says, "I am a trout.''
Zeta says, "Beta is a flounder.''

How many of these fish are trout?

Hint
Solution to P20
SHORT ANSWER QUESTIONS:
1. Five people randomly choose a positive integer less than or equal to $10.$ The probability that at least two people choose the same number can be written as $\frac{m}{n}.$ Find $m+n.$

Hint
Solution to S1

2. Define a function $F(n)$ on the positive integers using the rule that for $n=1,$ $F(n)=0.$ For all prime $n$, $F(n) = 1,$ and for all other $n,$ $F(xy)=xF(y) + yF(x).$ Find the smallest possible value of $n$ such that $F(n) = 2n.$

Hint
Solution to S2

3. How many integers $n \le 2025$ can be written as the sum of two distinct, non-negative integer powers of $3?$
Huge shoutout to OTIS for teaching me how to solve problems like this.

Hint

Solution to S3

4. Let $S$ be the set of positive integers of $x$ such that $x^2-5y^2=1$ for some other positive integer $y.$ Find the only three-digit value of $x$ in $S.$
Hint
Solution to S4

5. Let $N$ be a positive integer and let $M$ be the integer that is formed by removing the first three digits from $N.$ Find the value of $N$ with least value such that $N = 2025M.$
Hint

Solution to S5
9 replies
mathnerd_101
Friday at 11:40 AM
BackToSchool
Today at 4:02 AM
J
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J
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multiple summation
gasgous   3
N May 19, 2023 by fungarwai
Prove that:
$\sum_{a+b+c+d=n}{x^ay^bz^cw^d}=\frac{x^{n+3}}{\left(x-y\right)\left(x-z\right)\left(x-w\right)}+\frac{y^{n+3}}{\left(y-x\right)\left(y-z\right)\left(y-w\right)}+\frac{z^{n+3}}{\left(z-x\right)\left(z-y\right)\left(z-w\right)}+\frac{w^{n+3}}{\left(w-x\right)\left(w-y\right)\left(w-z\right)}$ ,where $a,b,c,d,n$ are non negative integers.
3 replies
gasgous
May 15, 2023
fungarwai
May 19, 2023
J
multiple summation
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Reveal topic tags
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gasgous
887 posts
gasgous
#1 h  May 15, 2023, 1:29 AM
Y by
Prove that:
$\sum_{a+b+c+d=n}{x^ay^bz^cw^d}=\frac{x^{n+3}}{\left(x-y\right)\left(x-z\right)\left(x-w\right)}+\frac{y^{n+3}}{\left(y-x\right)\left(y-z\right)\left(y-w\right)}+\frac{z^{n+3}}{\left(z-x\right)\left(z-y\right)\left(z-w\right)}+\frac{w^{n+3}}{\left(w-x\right)\left(w-y\right)\left(w-z\right)}$ ,where $a,b,c,d,n$ are non negative integers.
This post has been edited 1 time. Last edited by gasgous, May 15, 2023, 1:31 AM
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natmath
8219 posts
natmath
#2 h  May 15, 2023, 2:15 AM • 2 Y
Y by gasgous, fungarwai
This is the coefficient of $t^n$ in the expansion of $\frac{1}{(1-xt)(1-yt)(1-zt)(1-wt)}$. Now we do partial fraction decomposition.
$$\frac{1}{(1-xt)(1-yt)(1-zt)(1-wt)}=\sum_{cyc}\frac{A_x}{1-xt}$$$$1=\sum_{cyc}A_x(1-yt)(1-zt)(1-wt)$$Plug in $t=1/x$,
$$1=\frac{A_x}{x^3}(x-y)(x-z)(x-w)$$$$A_x=\frac{x^3}{(x-y)(x-z)(x-w)}$$Symmetric expressions exist for $A_y,\ldots$. The result follows.

@below it's basically the same, just more notation.
This post has been edited 1 time. Last edited by natmath, May 15, 2023, 2:39 AM
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rchokler
2958 posts
rchokler
#3 h  May 15, 2023, 2:35 AM
Y by
More generally:
Prove that $\sum_{a_1+\cdots+a_m=n}\prod_{i=1}^mx_i^{a_i}=\sum_{i=1}^m\frac{x_i^{m+n-1}}{\prod_{j\neq i}(x_i-x_j)}$.
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fungarwai
863 posts
fungarwai
#4 h  May 19, 2023, 12:11 PM
Y by
Best Proof!
I've noted and generialised #2 at Summation with polynomial roots

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