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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
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0 replies
jwelsh
Jul 1, 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
Numbers in set can be colored red and blue
orl   27
N 6 minutes ago by eg4334
Source: IMO Shortlist 2007, C3, AIMO 2008, TST 2, P2
Find all positive integers $ n$ for which the numbers in the set $ S = \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that:

(i) the numbers $ x$, $ y$, $ z$ are of the same color,
and
(ii) the number $ x + y + z$ is divisible by $ n$.

Author: Gerhard Wöginger, Netherlands
27 replies
+1 w
orl
Jul 13, 2008
eg4334
6 minutes ago
British Math Olympiad Round 1 2005, Q6
maths_dodo   0
12 minutes ago
Hello, I'm a bit stuck on this little problem...

Let T be a set of 2005 coplanar points with no three collinear. Show that, for any of the 2005 points, the number of triangles it lies strictly within, whose vertices are points in T, is even.

I tried for some small cases like 4 or 5 points, and I'm trying to use these to help me with the larger case.

(British Math Olympiad Round 1 2005, Problem 6)
0 replies
maths_dodo
12 minutes ago
0 replies
The inekoalaty game
sarjinius   32
N 15 minutes ago by Baimukh
Source: 2025 IMO P5
Alice and Bazza are playing the inekoalaty game, a two‑player game whose rules depend on a positive real number $\lambda$ which is known to both players. On the $n$th turn of the game (starting with $n=1$) the following happens:
[list]
[*] If $n$ is odd, Alice chooses a nonnegative real number $x_n$ such that
\[
    x_1 + x_2 + \cdots + x_n \le \lambda n.
  \][*]If $n$ is even, Bazza chooses a nonnegative real number $x_n$ such that
\[
    x_1^2 + x_2^2 + \cdots + x_n^2 \le n.
  \][/list]
If a player cannot choose a suitable $x_n$, the game ends and the other player wins. If the game goes on forever, neither player wins. All chosen numbers are known to both players.

Determine all values of $\lambda$ for which Alice has a winning strategy and all those for which Bazza has a winning strategy.

Proposed by Massimiliano Foschi and Leonardo Franchi, Italy
32 replies
sarjinius
Jul 16, 2025
Baimukh
15 minutes ago
Function with primes
dangerousliri   43
N 17 minutes ago by TigerOnion
Source: BMO 2019, Problem 1
Let $\mathbb{P}$ be the set of all prime numbers. Find all functions $f:\mathbb{P}\rightarrow\mathbb{P}$ such that:
$$f(p)^{f(q)}+q^p=f(q)^{f(p)}+p^q$$holds for all $p,q\in\mathbb{P}$.

Proposed by Dorlir Ahmeti, Albania
43 replies
dangerousliri
May 2, 2019
TigerOnion
17 minutes ago
Functional Equation
Etemadi   15
N 42 minutes ago by math-olympiad-clown
Source: Iran MO 2018, second round, day 2, P4
Find all functions $f:\Bbb {R} \rightarrow \Bbb {R} $ such that:
$$f(x+y)f(x^2-xy+y^2)=x^3+y^3$$for all reals $x, y $.
15 replies
Etemadi
Apr 27, 2018
math-olympiad-clown
42 minutes ago
My favorite problems on Olympiad that i solved (for my Bday :3)
MathLuis   58
N 2 hours ago by pingpongmerrily
Source: ISL, TST's on diferent countrys for IMO, ELMO SL, APMO, RMM
Today i'm oficialy 13 years old and for celebrating i shared my history and now i will put my favorite problems that i solved on my Olympiad carrier.
1.- USA TSTST 2020 P2: Let $ABC$ be a scalene triangle with incenter $I$. The incircle of $ABC$ touches $\overline{BC},\overline{CA},\overline{AB}$ at points $D,E,F$, respectively. Let $P$ be the foot of the altitude from $D$ to $\overline{EF}$, and let $M$ be the midpoint of $\overline{BC}$. The rays $AP$ and $IP$ intersect the circumcircle of triangle $ABC$ again at points $G$ and $Q$, respectively. Show that the incenter of triangle $GQM$ coincides with $D$.
2.- ISL 2020 A8: Let $\mathbb R^+$ be the set of positive real numbers. Determine all functions $f: \mathbb R^+$ $\rightarrow$ $\mathbb R^+$ such that for all positive real numbers $x$ and $y$
$f(x+f(xy))+y=f(x)f(y)+1$
3.- ISL 2020 G6: Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other.
4.- ELMO SL 2013 G9: Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\omega$ whose diagonals meet at $F$. Lines $AB$ and $CD$ meet at $E$. Segment $EF$ intersects $\omega$ at $X$. Lines $BX$ and $CD$ meet at $M$, and lines $CX$ and $AB$ meet at $N$. Prove that $MN$ and $BC$ concur with the tangent to $\omega$ at $X$.
5.- RMM 2013 P3: Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$. The lines $AB$ and $CD$ meet at $P$, the lines $AD$ and $BC$ meet at $Q$, and the diagonals $AC$ and $BD$ meet at $R$. Let $M$ be the midpoint of the segment $PQ$, and let $K$ be the common point of the segment $MR$ and the circle $\omega$. Prove that the circumcircle of the triangle $KPQ$ and $\omega$ are tangent to one another.
6.- ELMO 2010 P6: Let $ABC$ be a triangle with circumcircle $\omega$, incenter $I$, and $A$-excenter $I_A$. Let the incircle and the $A$-excircle hit $BC$ at $D$ and $E$, respectively, and let $M$ be the midpoint of arc $BC$ without $A$. Consider the circle tangent to $BC$ at $D$ and arc $BAC$ at $T$. If $TI$ intersects $\omega$ again at $S$, prove that $SI_A$ and $ME$ meet on $\omega$.
7.- ISL 2019 N4: Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.
8.- ISL 2019 N8: Let $a$ and $b$ be two positive integers. Prove that the integer $a^2+\left\lceil\frac{4a^2}b\right\rceil$ is not a square.
9.- IMO 2015 P3: Let $ABC$ be an acute triangle with $AB > AC$. Let $\Gamma $ be its circumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HQA = 90^{\circ}$ and let $K$ be the point on $\Gamma$ such that $\angle HKQ = 90^{\circ}$. Assume that the points $A$, $B$, $C$, $K$ and $Q$ are all different and lie on $\Gamma$ in this order. Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other.
10.- IMO 2015 P5: Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation $f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)$ for all real numbers $x$ and $y$.
11.- ISL 2016 N6: Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$.
12.- Rumanian TST 2014 Day 4 P2: Let $p$ be an odd prime number. Determine all pairs of polynomials $f$ and $g$ from $\mathbb{Z}[X]$ such that
$f(g(X))=\sum_{k=0}^{p-1} X^k = \Phi_p(X)$.
13.- USA TSTST 2019 P5: Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Gamma$. A line through $H$ intersects segments $AB$ and $AC$ at $E$ and $F$, respectively. Let $K$ be the circumcenter of $\triangle AEF$, and suppose line $AK$ intersects $\Gamma$ again at a point $D$. Prove that line $HK$ and the line through $D$ perpendicular to $\overline{BC}$ meet on $\Gamma$.
I will add more problems soon, but feel free to share solutions and ideas, thanks for taking ur time to see this :D
58 replies
MathLuis
Sep 9, 2021
pingpongmerrily
2 hours ago
find real f so that f(2x + f(y)) + f(f(y)) = 4x + 8y
parmenides51   5
N 2 hours ago by AshAuktober
Source: Irmo 2018 p2 q6
Find all real-valued functions $f$ satisfying $f(2x + f(y)) + f(f(y)) = 4x + 8y$ for all real numbers $x$ and $y$.
5 replies
parmenides51
Sep 16, 2018
AshAuktober
2 hours ago
Interesting inequality
sqing   1
N 2 hours ago by SunnyEvan
Source: Own
Let $ a,b,c> 0,c<k$ and $ a+b=2. $ Prove that$$\frac{a+b}{2b(k-c)}+\frac{a+c}{abc}\geq \frac{k+4+\sqrt{k(k+8)}}{2k}$$Where $k\in N^+.$
Let $ a,b,c> 0,c<1$ and $ a+b=2. $ Prove that$$\frac{a+b}{2b(1-c)}+\frac{a+c}{abc}\geq 4$$Let $ a,b,c> 0,c<2$ and $ a+b=2. $ Prove that$$\frac{a+b}{2b(2-c)}+\frac{a+c}{abc}\geq \frac{3+\sqrt 5}{2}$$
1 reply
sqing
4 hours ago
SunnyEvan
2 hours ago
Problem 16
SlovEcience   3
N 2 hours ago by nexu
Find the smallest positive integer \( k \) such that the following inequality holds:
\[
x^k y^k z^k (x^3 + y^3 + z^3) \leq 3
\]for all positive real numbers \( x, y, z \) satisfying the condition \( x + y + z = 3 \).
3 replies
SlovEcience
Today at 10:09 AM
nexu
2 hours ago
Functional Equation
AnhQuang_67   0
2 hours ago
Find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying $$f(max\{x,y\}+min\{f(x),f(y)\})=x+y,\forall x, y\in\mathbb{R}$$
0 replies
AnhQuang_67
2 hours ago
0 replies
Komal A 835-Style Problem
zqy648   0
3 hours ago
Source: 2023 October 谜之竞赛-2
Let $a$, $b$, $c$ be complex numbers where $a\neq 0$. Show that for any integer $n\ge 2$, there doesn't exist a function $f:\mathbb C\to\mathbb C$, such that for any complex number $z$,
\[f^{(n)}(z)=az^2+bz+c.\]Created by Site Mu, Beijing No.101 High School
0 replies
zqy648
3 hours ago
0 replies
Elegant Polynomial Problem
EthanWYX2009   0
3 hours ago
Source: 2024 February 谜之竞赛-6
Let \( f(x)\), \(g(x) \) be real-coefficient polynomials such that for all \( x \in [0,1] \), \( f(x) \geq 1 \) and \( g(x) \leq -1 \). Show that for any positive integer \( n \), there exists a unique polynomial \( P(x) \) of degree \( 2n-1 \) satisfying: [list]
[*]For all \( x \in [0,1] \), $g(x) \leq P(x) \leq f(x)$;
[*]There exist $0 \leq x_0 < x_1 < x_2 < \cdots < x_{2n-1} \leq 1$, such that
\[P(x_{2i}) = g(x_{2i}), \quad P(x_{2i+1}) = f(x_{2i+1})\]holds for all $i = 0$, $1$, $\cdots$, $n-1$. [/list]
Created by Cheng Jiang, Tsinghua University
0 replies
EthanWYX2009
3 hours ago
0 replies
EGMO magic square
Lukaluce   18
N 3 hours ago by dgrozev
Source: EGMO 2025 P6
In each cell of a $2025 \times 2025$ board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to $1$, and the sum of the numbers in each column is equal to $1$. Define $r_i$ to be the largest value in row $i$, and let $R = r_1 + r_2 + ... + r_{2025}$. Similarly, define $c_i$ to be the largest value in column $i$, and let $C = c_1 + c_2 + ... + c_{2025}$.
What is the largest possible value of $\frac{R}{C}$?

Proposed by Paulius Aleknavičius, Lithuania, and Anghel David Andrei, Romania
18 replies
Lukaluce
Apr 14, 2025
dgrozev
3 hours ago
Find real numbers
shobber   5
N 5 hours ago by SomeonecoolLovesMaths
Source: China TST 2003
Can we find positive reals $a_1, a_2, \dots, a_{2002}$ such that for any positive integer $k$, with $1 \leq k \leq 2002$, every complex root $z$ of the following polynomial $f(x)$ satisfies the condition $|\text{Im } z| \leq |\text{Re } z|$,
\[f(x)=a_{k+2001}x^{2001}+a_{k+2000}x^{2000}+ \cdots + a_{k+1}x+a_k,\]where $a_{2002+i}=a_i$, for $i=1,2, \dots, 2001$.
5 replies
shobber
Jun 29, 2006
SomeonecoolLovesMaths
5 hours 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
The post below has been deleted. Click to close.
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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.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
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.
Z K Y
The post below has been deleted. Click to close.
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dgrozev
2481 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
2481 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
8 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
1494 posts
#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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