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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 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
Diodes and usamons
v_Enhance   47
N 11 minutes ago by EeEeRUT
Source: USA December TST for the 56th IMO, by Linus Hamilton
A physicist encounters $2015$ atoms called usamons. Each usamon either has one electron or zero electrons, and the physicist can't tell the difference. The physicist's only tool is a diode. The physicist may connect the diode from any usamon $A$ to any other usamon $B$. (This connection is directed.) When she does so, if usamon $A$ has an electron and usamon $B$ does not, then the electron jumps from $A$ to $B$. In any other case, nothing happens. In addition, the physicist cannot tell whether an electron jumps during any given step. The physicist's goal is to isolate two usamons that she is sure are currently in the same state. Is there any series of diode usage that makes this possible?

Proposed by Linus Hamilton
47 replies
v_Enhance
Dec 17, 2014
EeEeRUT
11 minutes ago
IMO Genre Predictions
ohiorizzler1434   58
N 25 minutes ago by WLOGQED1729
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
58 replies
1 viewing
ohiorizzler1434
May 3, 2025
WLOGQED1729
25 minutes ago
Centroid, altitudes and medians, and concyclic points
BR1F1SZ   2
N an hour ago by Tkn
Source: Austria National MO Part 1 Problem 2
Let $\triangle{ABC}$ be an acute triangle with $BC > AC$. Let $S$ be the centroid of triangle $ABC$ and let $F$ be the foot of the perpendicular from $C$ to side $AB$. The median $CS$ intersects the circumcircle $\gamma$ of triangle $\triangle{ABC}$ at a second point $P$. Let $M$ be the point where $CS$ intersects $AB$. The line $SF$ intersects the circle $\gamma$ at a point $Q$, such that $F$ lies between $S$ and $Q$. Prove that the points $M,P,Q$ and $F$ lie on a circle.

(Karl Czakler)
2 replies
BR1F1SZ
Monday at 9:45 PM
Tkn
an hour ago
3-var inequality
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b\geq  0 ,a^3-ab+b^3=1  $. Prove that
$$  \frac{1}{2}\geq     \frac{a}{a^2+3 }+ \frac{b}{b^2+3}   \geq  \frac{1}{4}$$$$  \frac{1}{2}\geq     \frac{a}{a^3+3 }+ \frac{b}{b^3+3}   \geq  \frac{1}{4}$$$$  \frac{1}{2}\geq \frac{a}{a^2+ab+2}+ \frac{b}{b^2+ ab+2}  \geq  \frac{1}{3}$$$$  \frac{1}{2}\geq \frac{a}{a^3+ab+2}+ \frac{b}{b^3+ ab+2}  \geq  \frac{1}{3}$$Let $ a,b\geq  0 ,a^3+ab+b^3=3  $. Prove that
$$  \frac{1}{2}\geq     \frac{a}{a^2+3 }+ \frac{b}{b^2+3}   \geq  \frac{1}{4}(\frac{1}{\sqrt[3]{3}}+\sqrt[3]{3}-1)$$$$  \frac{1}{2}\geq     \frac{a}{a^3+3 }+ \frac{b}{b^3+3}   \geq  \frac{1}{2\sqrt[3]{9}}$$$$  \frac{1}{2}\geq \frac{a}{a^2+ab+2}+ \frac{b}{b^2+ ab+2}  \geq  \frac{4\sqrt[3]{3}+3\sqrt[3]{9}-6}{17}$$$$  \frac{1}{2}\geq \frac{a}{a^3+ab+2}+ \frac{b}{b^3+ ab+2}  \geq  \frac{\sqrt[3]{3}}{5}$$
1 reply
sqing
an hour ago
sqing
an hour ago
IMO ShortList 2001, combinatorics problem 3
orl   37
N an hour ago by deduck
Source: IMO ShortList 2001, combinatorics problem 3, HK 2009 TST 2 Q.2
Define a $ k$-clique to be a set of $ k$ people such that every pair of them are acquainted with each other. At a certain party, every pair of 3-cliques has at least one person in common, and there are no 5-cliques. Prove that there are two or fewer people at the party whose departure leaves no 3-clique remaining.
37 replies
orl
Sep 30, 2004
deduck
an hour ago
area of O_1O_2O_3O_4 <=1, incenters of right triangles outside a square
parmenides51   2
N an hour ago by Solilin
Source: Thailand Mathematical Olympiad 2012 p4
Let $ABCD$ be a unit square. Points $E, F, G, H$ are chosen outside $ABCD$ so that $\angle AEB =\angle BF C = \angle CGD = \angle DHA = 90^o$ . Let $O_1, O_2, O_3, O_4$, respectively, be the incenters of $\vartriangle ABE, \vartriangle BCF, \vartriangle CDG, \vartriangle DAH$. Show that the area of $O_1O_2O_3O_4$ is at most $1$.
2 replies
parmenides51
Aug 17, 2020
Solilin
an hour ago
Geo metry
TUAN2k8   3
N an hour ago by TUAN2k8
Help me plss!
Given an acute triangle $ABC$. Points $D$ and $E$ lie on segments $AB$ and $AC$, respectively. Lines $BD$ and $CE$ intersect at point $F$. The circumcircles of triangles $BDF$ and $CEF$ intersect at a second point $P$. The circumcircles of triangles $ABC$ and $ADE$ intersect at a second point $Q$. Point $K$ lies on segment $AP$ such that $KQ \perp AQ$. Prove that triangles $\triangle BKD$ and $\triangle CKE$ are similar.
3 replies
TUAN2k8
Yesterday at 10:33 AM
TUAN2k8
an hour ago
Functional equation of nonzero reals
proglote   8
N 2 hours ago by jasperE3
Source: Brazil MO 2013, problem #3
Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^* $ from the non-zero reals to the non-zero reals, such that \[f(x+y) \left(f(x) + f(y)\right) = f(xy)\] for all non-zero reals $x, y$ such that $x+y \neq 0$.
8 replies
proglote
Oct 24, 2013
jasperE3
2 hours ago
Interesting inequalities
sqing   1
N 2 hours ago by sqing
Source: Own
Let $ a,b\geq  0 ,a^2-ab+b^2+a+b=3  $. Prove that
$$  \frac{39+\sqrt{13}}{78}\geq  \frac{1}{a^2+3}+ \frac{1}{b^2+ 3} \geq  \frac{1}{2}$$Let $ a,b\geq  0 ,a^2+ab+b^2+a+b=3  $. Prove that
$$  \frac{19+\sqrt{10}}{39}\geq  \frac{1}{a^2+3}+ \frac{1}{b^2+ 3} \geq    \frac{39+\sqrt{13}}{78}$$Let $ a,b\geq  0 ,a^2+ab+b^2+a+b=5  $. Prove that
$$  \frac{3}{5}> \frac{1}{a^2+3}+ \frac{1}{b^2+ 3} \geq     \frac{185+3\sqrt{21}}{402}$$
1 reply
sqing
2 hours ago
sqing
2 hours ago
4-var inequality
sqing   2
N 3 hours ago by sqing
Source: SXTB (4)2025 Q2837
Let $ a,b,c,d> 0  $. Prove that
$$   \frac{1}{(3a+1)^4}+ \frac{1}{(3b+1)^4}+\frac{1}{(3c+1)^4}+\frac{1}{(3d+1)^4} \geq \frac{1}{16(3abcd+1)}$$
2 replies
sqing
Yesterday at 2:59 PM
sqing
3 hours ago
Inspired by Bet667
sqing   2
N 3 hours ago by sqing
Source: Own
Let $ a,b $ be a real numbers such that $a^2+kab+b^2\ge a^3+b^3.$Prove that$$a+b\leq k+2$$Where $ k\geq 0. $
2 replies
sqing
Yesterday at 2:46 PM
sqing
3 hours ago
find positive n so that exists prime p with p^n-(p-1)^n$ a power of 3
parmenides51   14
N 3 hours ago by MathLuis
Source: JBMO Shortlist 2017 NT5
Find all positive integers $n$ such that there exists a prime number $p$, such that $p^n-(p-1)^n$ is a power of $3$.

Note. A power of $3$ is a number of the form $3^a$ where $a$ is a positive integer.
14 replies
parmenides51
Jul 25, 2018
MathLuis
3 hours ago
GCD of terms in a sequence
BBNoDollar   1
N 3 hours ago by mashumaro
Determine the general term of the sequence of non-zero natural numbers (a_n)n≥1, with the property that gcd(a_m, a_n, a_p) = gcd(m^2 ,n^2 ,p^2), for any distinct non-zero natural numbers m, n, p.

⁡Note that gcd(a,b,c) denotes the greatest common divisor of the natural numbers a,b,c .
1 reply
BBNoDollar
Yesterday at 10:29 PM
mashumaro
3 hours ago
China South East Mathematical Olympiad 2013 problem 2
s372102   3
N 3 hours ago by AGCN
$\triangle ABC$, $AB>AC$. the incircle $I$ of $\triangle ABC$ meet $BC$ at point $D$, $AD$ meet $I$ again at $E$. $EP$ is a tangent of $I$, and $EP$ meet the extension line of $BC$ at $P$. $CF\parallel PE$, $CF\cap AD=F$. the line $BF$ meet $I$ at $M,N$, point $M$ is on the line segment $BF$, the line segment $PM$ meet $I$ again at $Q$. Show that $\angle ENP=\angle ENQ$
3 replies
s372102
Aug 10, 2013
AGCN
3 hours ago
IMO problem 1
iandrei   77
N Apr 23, 2025 by YaoAOPS
Source: IMO ShortList 2003, combinatorics problem 1
Let $A$ be a $101$-element subset of the set $S=\{1,2,\ldots,1000000\}$. Prove that there exist numbers $t_1$, $t_2, \ldots, t_{100}$ in $S$ such that the sets \[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100  \] are pairwise disjoint.
77 replies
iandrei
Jul 14, 2003
YaoAOPS
Apr 23, 2025
IMO problem 1
G H J
Source: IMO ShortList 2003, combinatorics problem 1
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iandrei
138 posts
#1 • 11 Y
Y by Davi-8191, Wizard_32, aops5234, Adventure10, Sprites, THEfmigm, megarnie, Mango247, and 3 other users
Let $A$ be a $101$-element subset of the set $S=\{1,2,\ldots,1000000\}$. Prove that there exist numbers $t_1$, $t_2, \ldots, t_{100}$ in $S$ such that the sets \[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100  \] are pairwise disjoint.
Attachments:
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Anonymous
334 posts
#2 • 8 Y
Y by ValidName, lahmacun, Adventure10, myh2910, Mango247, and 3 other users
Assume that we have already found t_1, t_2, ..., t_k (k<=99) and are searching for t_{k+1}. We can not take t_{k+1} only of form
t_i+a_j-a_l, where 1<=i<=k and a_j and a_l are elements of A.
So, we have k*101*100 forbidden values of t_{k+1}, which correspond to a_j<>a_l, and k forbidden values which correspond to a_j=a_l. So, at most 99*101*100+99=1000000-1 forbidden values, and at least 1 admissible.
Z K
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Alison
264 posts
#3 • 6 Y
Y by A_Math_Lover, biomathematics, Adventure10, Mango247, and 2 other users
Notice that if instead of taking an arbitrary t_{k+1}, you always take the smallest t_{k+1} that is not forbidden, you will only have to make sure that t_{k+1} is distinct from all t_i+a_j-a_l with a_j>a_l. This is true because the values with a_j<a_l are all <=t_k, and so have been forbidden at some earlier step.

This gives you ceiling(1000000/5051) = 198 t_i's.

However, Fedor's proof also shows that you can choose 100 t_i's such that the sets are parwise disjoint even when taken mod 10^6.
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Anonymous
334 posts
#4 • 6 Y
Y by ValidName, Adventure10, Mango247, and 3 other users
Let A = {a1 < a2 < ... < a101}.

Draw an undirected graph and put a vertex between i and j iff the sets (A+i) and (A+j) are disjoint . The graph will have 10^6 vertices . For an arbitrary vertex x to be joined with y , x-y must not be one of the numbers ai - aj (i<>j) , which are 101*100 numbers . So the vertex x has degree at least 10^6 - 101*100 .

But this means that the graph has at least 10^6(10^6 - 101*100)/2 edges. The problem asks to prove that there is an 100-clique in the graph. But by Turan's theorem , there is a k-clique in a graph with n vertices iff the number of edges is strictly greater than :

M(n,k) = (k-2)/(k-1) * (n^2 - r^2)/2 + r*(r-1)/2

where we have taken r to be the remainder of n when divided by k-1.

In our case n=10^6 , k=100 and r=1. A simple calculation shows that the number of vertices is greater than M(10^6,100)+1 and thus we are done.

P.S. : in an IMO paper , should one prove Turan's theorem or not ? I guess so ..
Z K
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Anonymous
334 posts
#5 • 3 Y
Y by Adventure10, Mango247, and 1 other user
Fedor Petrov wrote:
Assume that we have already found t_1, t_2, ..., t_k (k<=99) and are searching for t_{k+1}. We can not take t_{k+1} only of form
t_i+a_j-a_l, where 1<=i<=k and a_j and a_l are elements of A.
So, we have k*101*100 forbidden values of t_{k+1}, which correspond to a_j<>a_l, and k forbidden values which correspond to a_j=a_l. So, at most 99*101*100+99=1000000-1 forbidden values, and at least 1 admissible.
so it is right to assume that for every K there exists a number such that \lim \lambda
Z K
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me@home
2349 posts
#6 • 4 Y
Y by Adventure10, Mango247, and 2 other users
Fedor Petrov wrote:
Assume that we have already found $\{t_k\}_{1=k}^{99}$ and are searching for $t_{k+1}$. We can not take $t_{k+1}$ only of form
$t_i+a_j-a_l \ | \ 1\leq i\leq k \ \ a_j, a_l \in A$.
So, we have $k*101*100$ forbidden values of $t_{k+1}$, which correspond to $a_j<>a_l$, and $k$ forbidden values which correspond to $a_j=a_l$. So, at most $99*101*100+99=1000000-1$ forbidden values, and at least $1$ admissible.
Alison wrote:
Notice that if instead of taking an arbitrary $t_{k+1}$, you always take the smallest $t_{k+1}$ that is not forbidden, you will only have to make sure that $t_{k+1}$ is distinct from all $t_i+a_j-a_l \ | \ a_j>a_l$. This is true because the values with $a_j<a_l$ are all $\leq t_k$, and so have been forbidden at some earlier step.

This gives you $\lceil 1000000/5051 \rceil = 198 t_i's$.

However, Fedor's proof also shows that you can choose $100 t_i's$ such that the sets are parwise disjoint even when taken $mod 10^6$.
Guest wrote:
Let $A = {a_1 < a_2 < ... < a_{101}}$.

Draw an undirected graph and put a vertex between $i, j$ iff the sets $(A+i)$ and $(A+j)$ are disjoint . The graph will have $10^6$ vertices . For an arbitrary vertex $x$ to be joined with $y$ , $x-y$ must not be one of the numbers $ai - aj (i<>j)$ , which are $101*100$ numbers . So the vertex $x$ has degree at least $10^6 - 101*100$ .

But this means that the graph has at least $10^6(10^6 - 101*100)/2$ edges. The problem asks to prove that there is an $100-clique$ in the graph. But by Turan's theorem , there is a $k-clique$ in a graph with $n$ vertices iff the number of edges is strictly greater than :

$M(n,k) = (k-2)/(k-1) * (n^2 - r^2)/2 + r*(r-1)/2$

where we have taken $r$ to be the remainder of $n$ when divided by $k-1$.

In our case $n=10^6 , k=100 , r=1$. A simple calculation shows that the number of vertices is greater than $M(10^6,100)+1$ and thus we are done.

P.S. : in an IMO paper , should one prove Turan's theorem or not ? I guess so ..
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msecco
154 posts
#7 • 4 Y
Y by Adventure10, Mango247, and 2 other users
This problem has been proposed by Carlos Gustavo Tamm de Araujo Moreira, from Brazil.
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Bugi
1857 posts
#8 • 3 Y
Y by Adventure10 and 2 other users
http://www.artofproblemsolving.com/Wiki/index.php/IMO_Problems_and_Solutions

It already says so in the Wiki. If you know an author which isn't listed there, please contribute!
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Dragonboy
38 posts
#9 • 3 Y
Y by Adventure10, Mango247, and 1 other user
May be my solution is wrong but it seems to me that we just need $|S|\geq 99\binom{101}{2}+100$. Please help me if you find any bug in my solution
SOLUTION
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Mahi
52 posts
#10 • 4 Y
Y by Adventure10, Mango247, and 2 other users
Dragonboy wrote:
Make a set $S_{i+1}\subset S_i$ such that $S_{i+1}$ doesn't contain $t_{i+1}$ and any element $K$ satisfying $K-t_{i+1}=|x-y|$ for any distinct $x,y\in A$

In here, you are just striking out the elements such that $K-t_{i+1}=|x-y|$, but the case remains where $K-t_{i+1}=-|x-y|$, which can also satisfy $y+t_i=x+t_j$ instead of $y+t_j=x+t_i$. So the limit is (about) doubled and it reaches near $10^6$, which can be achieved with a little more strict bounding.

Although, nice approach with algorithmic way :)
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Dragonboy
38 posts
#11 • 3 Y
Y by Adventure10, Mango247, and 1 other user
Mahi wrote:
Dragonboy wrote:
Make a set $S_{i+1}\subset S_i$ such that $S_{i+1}$ doesn't contain $t_{i+1}$ and any element $K$ satisfying $K-t_{i+1}=|x-y|$ for any distinct $x,y\in A$

In here, you are just striking out the elements such that $K-t_{i+1}=|x-y|$, but the case remains where $K-t_{i+1}=-|x-y|$, which can also satisfy $y+t_i=x+t_j$ instead of $y+t_j=x+t_i$. So the limit is (about) doubled and it reaches near $10^6$, which can be achieved with a little more strict bounding.

Although, nice approach with algorithmic way :)
I think i have mentioned that $t_{i+1}$ is the smallest element in $S_i$ and any other element $K$ in $S_i$ is greater than $t_{i+1}$.So, there is no $K$ satisfying $K-t_{i+1}=-|x-y|$
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Mahi
52 posts
#12 • 4 Y
Y by Adventure10, Mango247, and 2 other users
Yes, if you choose $t_i$'s in increasing sequence, then it reduces to $t_i-t_j=y-x$ where $i>j$ which implies $y>x$ and thus the strategy is optimized by two :)
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Dragonboy
38 posts
#13 • 4 Y
Y by Adventure10, Mango247, and 2 other users
Mahi wrote:
Yes, if you choose $t_i$'s in increasing sequence, then it reduces to $t_i-t_j=y-x$ where $i>j$ which implies $y>x$ and thus the strategy is optimized by two :)
I'm not understanding what you're saying. Let me make it more clear for you (As much as i can)
After choosing all $t_i$ by the algorithm, for the sake of contradiction , let's assume there exists $t_i>t_j$ and $x>y$ such that $t_i-t_j=x-y$.
But It's not possible since we've banished such $t_i$ when we choose $t_{j+1}$ (According to algorithm).
Is it clear now?
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Mahi
52 posts
#14 • 4 Y
Y by Adventure10, Mango247, and 2 other users
In my last post, I just shared my opinion about the algorithm. I understood it earlier. It was clear to me after I noticed the part "greatest in the set $S_i$". Thanks for your concern.
By the way, something similar was also told by Alison in a previous post.
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Wolstenholme
543 posts
#15 • 3 Y
Y by Adventure10 and 2 other users
Let each element of $ S $ be the vertex of a graph where two vertices $ u, v $ are connected by an edge if and only if the sets $ A + u $ and $ A + v $ are disjoint. Consider an arbitrary vertex $ v $. Since the $ |A + v| = 101 $ the maximum number of vertices $ w $ such that sets $ A + v $ and $ A + w $ are not disjoint is $ 100 * 101 $. Therefore every vertex of the graph has degree at least $ 10^6 - 100*101 - 1. $ Therefore the graph has at least $ \frac{10^6(10^6 - 100*101 - 1)}{2} $ edges. It suffices to show that this graph contains $ K_{100} $ as a subgraph.

Now, by Turan's Theorem, the maximum number of edges a graph with $ 10^6 $ vertices that does not contain $ K_{100} $ may contain is obtained when the graph is a complete $ 99 $-partite graph with $ 98 $ independent sets of size $ 10101 $ and $ 1 $ independent set of size $ 10102 $. It is easy to compute that this graph has $ \binom{98}{2}10101^2 + 98 \cdot 10101 \cdot 10102 = 494949494949 $ edges. But since $ \frac{10^6(10^6 - 100*101 - 1)}{2} = 494949500000 > 494949494949 $ we have the desired result.
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