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k a August Highlights and 2025 AoPS Online Class Information
jwelsh   0
Friday at 2:14 PM
CONGRATULATIONS to all the competitors at this year’s International Mathematical Olympiad (IMO)! The US Team took second place with 5 gold medals and 1 silver - we are proud to say that each member of the 2025 IMO team has participated in an AoPS WOOT (Worldwide Online Olympiad Training) class!

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0 replies
jwelsh
Friday at 2:14 PM
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
Inequality on distinct positive integers
JustPostNorthKoreaTST   0
5 minutes ago
Source: 2016 North Korea TST P5
Find the maximum possible value of $\lambda$, such that for any positive integer $n$ and distinct positive integers $k_1,k_2,\ldots,k_n$, we have
$$ \left(\sum_{i=1}^n \frac{1}{k_i}\right)\left(\sum_{i=1}^n \sqrt{k_i^6+k_i^3}\right)-\left(\sum_{i=1}^n k_i\right)^2 \ge \lambda n^2(n^2-1). $$
0 replies
JustPostNorthKoreaTST
5 minutes ago
0 replies
Perimeter bisectors
JustPostNorthKoreaTST   0
9 minutes ago
Source: 2016 North Korea TST P4
Given a triangle $ABC$, if the line connecting a point $X$ on a side of $\triangle ABC$ and its corresponding vertex bisects the perimeter, we write $(X \to \triangle ABC)$.

In a convex quadrilateral $ABCD$, let $X,Y,Z,M$ be points on sides $AB,AD,DC,CB$, respectively, such that $BM=CM$, $AX=AY$, $YD=DZ$, $ZC=BX$, and $(X \to \triangle ABM)$, $(Y \to \triangle AMD)$, $(Z \to \triangle CDM)$.

Prove that $\triangle ABM \cong \triangle MDA \cong \triangle DMC$.
0 replies
JustPostNorthKoreaTST
9 minutes ago
0 replies
Classic inequality in retrospect
JustPostNorthKoreaTST   0
16 minutes ago
Source: 2016 North Korea TST P3
Let $a_1,a_2,\ldots,a_n$ be positive real numbers, and denote $a_{n+1}=a_1$. Prove that
$$ \sum_{i=1}^n \frac{a_{i+1}}{a_i} \ge \sum_{i=1}^n \sqrt{\frac{a_{i+1}^2+1}{a_i^2+1}}. $$
0 replies
JustPostNorthKoreaTST
16 minutes ago
0 replies
Infinitely many prime divisors of a recurrence sequence
JustPostNorthKoreaTST   0
19 minutes ago
Source: 2016 North Korea TST P2
Given a sequence $\{a_n\}_{n \ge 1}$ of positive integers, such that $a_1 \in \mathbb{N}_+$, and for $n \ge 1$,
$$ a_{n+1}=\sum_{i=2}^{n+1} \lfloor \sqrt[i]{a_n} \rfloor. $$Show that for any prime $p$, there are infinitely many terms in $\{a_n\}_{n \ge 1}$ that are divisible by $p$.
0 replies
JustPostNorthKoreaTST
19 minutes ago
0 replies
Find sum over permutations
JustPostNorthKoreaTST   0
24 minutes ago
Source: 2016 North Korea TST P1
Given an odd positive integer $n$. Find the value of
$$ \sum_\pi \prod_{i=1}^n (\pi(i)-i), $$where $\{\pi(i)\}_{i=1}^n$ is a permutation of $\{1,2,\ldots,n\}$, and the summation runs over all such permutations.
0 replies
JustPostNorthKoreaTST
24 minutes ago
0 replies
Set family with special conditions
JustPostNorthKoreaTST   0
27 minutes ago
Source: 2015 North Korea Mathematical Olympiad P6
Let $S$ be a set with $n$ ($n \ge 3$) elements. Find all possible integers $n$ such that there exists a family $\mathcal{F}$ of three-element subsets of $S$, satisfying the following conditions:
(1) For any $a,b \in S$, $a \neq b$, exactly one element of $\mathcal{F}$ contains $\{a,b\}$.
(2) For any $a,b,c,x,y,z \in S$, if $\{a,b,z\}, \{b,c,x\}, \{c,a,y\} \in \mathcal{F}$, then $\{x,y,z\} \in \mathcal{F}$.
0 replies
JustPostNorthKoreaTST
27 minutes ago
0 replies
inequality
aktyw19   1
N 36 minutes ago by Mathzeus1024
Let $x,y>0$ and $xy<1$. Prove $\left(\frac{2x}{1+x^{2}}\right)^{2}+\left(\frac{2y}{1+y^{2}}\right)^{2}\le\frac{1}{1-xy}$.
1 reply
aktyw19
Dec 19, 2012
Mathzeus1024
36 minutes ago
Easy geometry with orthocenter
JustPostNorthKoreaTST   0
37 minutes ago
Source: 2015 North Korea Mathematical Olympiad P4
Let $ABC$ be a scalene triangle with circumcircle $\odot O$ and orthocenter $H$, and let $AH,BH,CH$ intersects $\odot O$ at $A_1,B_1,C_1$, respectively. The line passing through $A_1$ and parallel to $BC$ intersects $\odot O$ at $A_2$. Define $B_2,C_2$ similarly. Let $AC_2$ intersects $BC_1$ at $M$, $BA_2$ intersects $CA_1$ at $N$, and $CB_2$ intersects $AB_1$ at $P$. Prove that $\angle MNB=\angle AMP$.
0 replies
JustPostNorthKoreaTST
37 minutes ago
0 replies
Tricolor complete graphs
JustPostNorthKoreaTST   0
42 minutes ago
Source: 2015 North Korea Mathematical Olympiad P3
Consider a complete graph $K_n$ on $n$ vertices, where $n \ge 3$. Each edge is colored with one of three colors, and each color is used on at least one edge. Find the minimum positive integer $k$ such that for any such edge coloring and any color $C$ chosen from the three colors, it is possible to recolor at most $k$ edges to color $C$ so that the subgraph consisting of all edges of color $C$ is connected.
0 replies
JustPostNorthKoreaTST
42 minutes ago
0 replies
How to solve this?
Mr._Calculator   0
an hour ago
In the diagram below, $ABC$ is a triangle, where the angle bisectors of $\angle ABC$ and $\angle ACB$ intersect at point $P$. Let points $D$ and $E$ lie on sides $AB$ and $AC$, respectively, such that $DE$ passes through point $P$ and $\angle AED = \angle ABP$. If the areas of triangles $BDP$, $CEP$ and $BPC$ are equal to $18 \text{ cm}^2$, $36 \text{ cm}^2$ and $57 \text{ cm}^2$, respectively, then what is the area, in $\text{cm}^2$, of $ABC$?
0 replies
Mr._Calculator
an hour ago
0 replies
Number Theory
JetFire008   1
N an hour ago by blug
Source: Elementary Number Theory by David M. Burton
Modify Euclid's proof that there are infinitely many primes by assuming the existence of a largest prime $p$ and using the integer $N=p!+1$ to arrive at a contradiction.
1 reply
JetFire008
2 hours ago
blug
an hour ago
Partial products from p-2 numbers
JustPostNorthKoreaTST   2
N an hour ago by navid
Source: 2015 North Korea Mathematical Olympiad P2
Let $p$ be an odd prime and $a_1,a_2,\ldots,a_{p-2}$ be positive integers (not necessarily different), such that for any $k \in \{1,2,\ldots,p-2\}$, we have $p \nmid a_k$ and $p \nmid (a_k^k-1)$. Show that one can pick some numbers from $a_1,a_2,\ldots,a_{p-2}$ such that their product is $\equiv 2 \pmod p$.
2 replies
JustPostNorthKoreaTST
4 hours ago
navid
an hour ago
Bonza functions
KevinYang2.71   72
N an hour ago by pie854
Source: 2025 IMO P3
Let $\mathbb{N}$ denote the set of positive integers. A function $f\colon\mathbb{N}\to\mathbb{N}$ is said to be bonza if
\[
f(a)~~\text{divides}~~b^a-f(b)^{f(a)}
\]for all positive integers $a$ and $b$.

Determine the smallest real constant $c$ such that $f(n)\leqslant cn$ for all bonza functions $f$ and all positive integers $n$.

Proposed by Lorenzo Sarria, Colombia
72 replies
KevinYang2.71
Jul 15, 2025
pie854
an hour ago
Next term is sum of three largest proper divisors
vsamc   24
N an hour ago by pie854
Source: 2025 IMO P4
A proper divisor of a positive integer $N$ is a positive divisor of $N$ other than $N$ itself.

The infinite sequence $a_1, a_2, \cdots$ consists of positive integers, each of which has at least three proper divisors. For each $n\geq 1$, the integer $a_{n+1}$ is the sum of the three largest proper divisors of $a_n$.

Determine all possible values of $a_1$.

Proposed by Paulius Aleknavičius, Lithuania
24 replies
vsamc
Jul 16, 2025
pie854
an hour ago
Infinite number of sets with an intersection property
Drytime   8
N May 31, 2025 by math90
Source: Romania TST 2013 Test 2 Problem 4
Let $k$ be a positive integer larger than $1$. Build an infinite set $\mathcal{A}$ of subsets of $\mathbb{N}$ having the following properties:

(a) any $k$ distinct sets of $\mathcal{A}$ have exactly one common element;
(b) any $k+1$ distinct sets of $\mathcal{A}$ have void intersection.
8 replies
Drytime
Apr 26, 2013
math90
May 31, 2025
Infinite number of sets with an intersection property
G H J
Source: Romania TST 2013 Test 2 Problem 4
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Drytime
88 posts
#1 • 3 Y
Y by nicegeo, doxuanlong15052000, Adventure10
Let $k$ be a positive integer larger than $1$. Build an infinite set $\mathcal{A}$ of subsets of $\mathbb{N}$ having the following properties:

(a) any $k$ distinct sets of $\mathcal{A}$ have exactly one common element;
(b) any $k+1$ distinct sets of $\mathcal{A}$ have void intersection.
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tenniskidperson3
2376 posts
#2 • 2 Y
Y by dgrozev, Adventure10
Enumerate the $k$-tuples; $(a_1, a_2, \ldots a_k)\rightarrow \binom{a_1}{1}+\binom{a_2-1}{2}+\binom{a_3-1}{3}+\ldots+\binom{a_k-1}{k}$ gives one method, where $a_1<a_2<a_3<\ldots<a_k$. Put $n$ into each of the $k$ sets with indices $a_i$ correlating to $n$. Then each $k$-tuple contains a unique element, and no number is contained in $k+1$ sets.
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dgrozev
2483 posts
#3 • 4 Y
Y by nghiepdu-socap, Adventure10, Mango247, and 1 other user
tenniskidperson3 wrote:
Enumerate the $k$-tuples; $(a_1, a_2, \ldots a_k)\rightarrow \binom{a_1}{1}+\binom{a_2-1}{2}+\binom{a_3-1}{3}+\ldots+\binom{a_k-1}{k}$ gives one method, where $a_1<a_2<a_3<\ldots<a_k$. Put $n$ into each of the $k$ sets with indices $a_i$ correlating to $n$. Then each $k$-tuple contains a unique element, and no number is contained in $k+1$ sets.
I really don't understand the above construction but it is not possible that a set in the family $ \mathcal{A} $ consists of finite number of elements.

A construction of family $ \mathcal{A} $:
When $k=2$ consider infinite many lines in a plane in general positions. When $k=3$ - infinite many planes in space in general positions will do the job. Of course a line(plane) consists of points with real coordinates, not in $\mathbb{N}$, but it just gives as a motivation.

Let $\{a_i\}_{i=1}^{\infty}$ be be a sequence of real numbers which are pairwise different and $a_i \neq 0$.
Let denote:

$P_j = \left\{ (x_1,x_2,\ldots,x_{k}) \mid x_i\in \mathbb{R}, \sum_{i=1}^{k} a_j^i x_i -1 =0  \right\} $
$j=1,2,\ldots$.

Now lets see that every $k$ different hyperplanes $P_{j_{\ell}},\, \ell=1,2,\ldots,k$ have exactly one common point $(x_1,x_2,\ldots,x_{k})$. This common point will satisfy the system

$ \sum_{i=1}^{k} a_{j_{\ell}}^i x_i  = 1 \,,\, \ell=1,2,\ldots, k $

But the determinant of the system is exactly the Vandermonde determinant $V=\prod_{t=1}^{k} a_{j_{t}}\prod_{1 \leq  s < t \leq k} ( a_{j_{t}}-a_{j_{s}} ) \neq 0 $. So the above system has an unique solution.
To see that every $k+1$ different planes have void intersection, assume on the contrary the hyperplanes $P_{j_{\ell}},\, \ell=1,2,\ldots,k+1$ have a common point $(x_1,x_2,\ldots,x_{k})$
Then the polynomial: $P(x)= x_1 x^1+x_2x^2+\ldots+x_k x^k-1$ will have $k+1$ different roots $a_{j_{\ell}},\, \ell=1,2,\ldots, k+1$, which means $P(x)$ is identically zero which is impossible.

Now, it could have finished the proof but our sets must consist of natural numbers not of real $k$-tuples.
But we can enumerate all possible intersections of $k$ different hyperplanes $P_j$ and let they be the points $\{p_{\ell}\}_{\ell=1}^{\infty}$. Now consider:

$P'_j = \left\{\ell \mid \ell\in \mathbb{N},\, p_{\ell} \in P_j \right\}$ .

These modified discrete sets will also satisfy the requirements (a) and (b).
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Lawasu
212 posts
#4 • 7 Y
Y by siddigss, Wolstenholme, doxuanlong15052000, B.J.W.T, nguyennam_2020, Adventure10, Mango247
My construction:

For a number $n=p_1^{a_1}\cdot p_2^{a_2}\cdot ...\cdot p_m^{a_m}$ (its decomposition in prime factors) take $f(n)=a_1+a_2+...+a_m$.
Now, for each prime $p$ take $A_p=\{px|\ x\in \Bbb{N},\ f(x)=k-1\}$. Now it's trivial to check the given conditions.
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v_Enhance
6906 posts
#5 • 3 Y
Y by dgrozev, Adventure10, Mango247
dgrozev wrote:
I really don't understand the above construction but it is not possible that a set in the family $ \mathcal{A} $ consists of finite number of elements.
Is this an issue? The problem statement didn't seem to require finite sets...
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dgrozev
2483 posts
#6 • 3 Y
Y by Adventure10, Mango247, and 1 other user
What I meant was that it's impossible some set in $\mathcal{A}$ to be finite. It can be easily shown. But, I don't know why I have decided that tenniskidperson3's construction produces finite sets. It was a long time ago. Sorry, apparently it was my fault. Maybe somehow I misunderstood the word "indices". Anyway, I would have put it in this way:

Let $\mathcal{B}$ be the family of all finite subsets of $\mathbb{N}$ with exactly $k$ elements. Since $\mathcal{B}$ is countable, we can construct a bijection $f: \mathcal{B} \to \mathbb{N}$. Now, let us denote $A_k=\{j\in \mathbb{N}\mid k\in f^{-1}(j)\}\,, k\in \mathbb{N}$ and $\mathcal{A}=\{A_k \mid k\in \mathbb{N}\}$. Apparently $\mathcal{A}$ satisfies the problem's requirement.
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JuanOrtiz
366 posts
#7 • 3 Y
Y by nicegeo, Smoothy, Adventure10
Isn't this trivial... Just let $X$ be the set of square-free positive integers that have $k$ distinct prime divisors and let $A_i$ be the subset of $X$ that has the multiples of $p_i$ (the $i$-th prime).

If we intersect $k$ sets, say $A_{a_1}$, ..., $A_{a_k}$ the intersection is $\{ p_{a_1} \times ... \times p_{a_k} \}$ and if we intersect $k+1$ sets the intersection is clearly empty since we would need $k+1$ distinct prime divisors.
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HHGB
10 posts
#8
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We will construct a countable set $\mathcal{A}$. Suppose its elements (which are subsets of $\mathbb{N}$) are $S_1, S_2, S_3, ...$. Let $X=\{B\subset \mathcal{A}\mid |B|=k\}$. Since there is a bijection between $\mathbb{N}$ and $\mathbb{N}^k$ and $X$ (one can consider B ordered) is a subset of $\mathbb{N}^k$ and not having finitely many elements, there is a bijection $f$ between $\mathbb{N}$ and $X$. For each $B \in X$, let $f(B)$ be in all the $k$ elements of $B$. In this construction, any $k$ distinct sets of $\mathcal{A}$ have exactly one common element by the bijection and any $k+1$ distinct sets of $\mathcal{A}$ have void intersection, because each $k$ sets of them have a unique common element.
This post has been edited 4 times. Last edited by HHGB, May 30, 2025, 8:51 PM
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math90
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#9
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Let $B\subset\mathbb N$ be the subset of all positive integers whose binary representation consists of exactly $k$ ones.

We will define subsets $A_1,A_2,\ldots$ in a way $A_n\subset B$ is the subset of all positive integers where $1$ appears in the $n$-th digit from the right. Then
$$\bigcap_{i=1}^n A_{a_i}=\left\{\sum_{i=1}^n2^{a_i-1}\right\}$$and all such singletons are pairwise distinct, hence every $k+1$ sets have an empty intersection.
This post has been edited 1 time. Last edited by math90, May 31, 2025, 1:40 PM
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