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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
+1 w
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
IMO 2008, Question 2
delegat   63
N 5 minutes ago by ezpotd
Source: IMO Shortlist 2008, A2
(a) Prove that
\[\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1\] for all real numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$.

(b) Prove that equality holds above for infinitely many triples of rational numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$.

Author: Walther Janous, Austria
63 replies
delegat
Jul 16, 2008
ezpotd
5 minutes ago
USAMO 2003 Problem 1
MithsApprentice   69
N 12 minutes ago by de-Kirschbaum
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.
69 replies
MithsApprentice
Sep 27, 2005
de-Kirschbaum
12 minutes ago
d(2025^{a_i}-1) divides a_{n+1}
navi_09220114   2
N 35 minutes ago by mickeymouse7133
Source: TASIMO 2025 Day 2 Problem 5
Let $a_n$ be a strictly increasing sequence of positive integers such that for all positive integers $n\ge 1$
\[d(2025^{a_n}-1)|a_{n+1}.\]Show that for any positive real number $c$ there is a positive integers $N_c$ such that $a_n>n^c$ for all $n\geq N_c$.

Note. Here $d(m)$ denotes the number of positive divisors of the positive integer $m$.
2 replies
navi_09220114
Monday at 11:51 AM
mickeymouse7133
35 minutes ago
Funky function
TheUltimate123   22
N an hour ago by jasperE3
Source: CJMO 2022/5 (https://aops.com/community/c594864h2791269p24548889)
Find all functions \(f:\mathbb R\to\mathbb R\) such that for all real numbers \(x\) and \(y\), \[f(f(xy)+y)=(x+1)f(y).\]
Proposed by novus677
22 replies
TheUltimate123
Mar 20, 2022
jasperE3
an hour ago
R+ FE f(f(xy)+y)=(x+1)f(y)
jasperE3   0
an hour ago
Source: p24734470
Find all functions $f:\mathbb R^+\to\mathbb R^+$ such that for all positive real numbers $x$ and $y$:
$$f(f(xy)+y)=(x+1)f(y).$$edit oops sorry I misinterpreted that the original poster had solved it, solvable tho
0 replies
jasperE3
an hour ago
0 replies
Inequality with x+y+z=1.
FrancoGiosefAG   1
N an hour ago by Blackbeam999
Let $x,y,z$ be positive real numbers such that $x+y+z=1$. Show that
\[ \frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leq 0. \]
1 reply
FrancoGiosefAG
5 hours ago
Blackbeam999
an hour ago
Find all numbers
Rushil   10
N an hour ago by SomeonecoolLovesMaths
Source: Indian RMO 1994 Problem 3
Find all 6-digit numbers $a_1a_2a_3a_4a_5a_6$ formed by using the digits $1,2,3,4,5,6$ once each such that the number $a_1a_2a_2\ldots a_k$ is divisible by $k$ for $1 \leq k \leq 6$.
10 replies
Rushil
Oct 25, 2005
SomeonecoolLovesMaths
an hour ago
Some number theory
EeEeRUT   3
N 2 hours ago by MathLuis
Source: Thailand MO 2025 P9
Let $p$ be an odd prime and $S = \{1,2,3,\dots, p\}$
Assume that $U: S \rightarrow S$ is a bijection and $B$ is an integer such that $$B\cdot U(U(a)) - a \: \text{ is a multiple of} \: p \: \text{for all} \: a \in S$$Show that $B^{\frac{p-1}{2}} -1$ is a multiple of $p$.
3 replies
EeEeRUT
May 14, 2025
MathLuis
2 hours ago
Gcd
Rushil   5
N 2 hours ago by SomeonecoolLovesMaths
Source: Indian RMO 1994 problem 5
Let $A$ be a set of $16$ positive integers with the property that the product of any two distinct members of $A$ will not exceed 1994. Show that there are numbers $a$ and $b$ in the set $A$ such that the gcd of $a$ and $b$ is greater than 1.
5 replies
Rushil
Oct 25, 2005
SomeonecoolLovesMaths
2 hours ago
Solve the system
Rushil   20
N 2 hours ago by SomeonecoolLovesMaths
Source: 0
Solve the system of equations for real $x$ and $y$: \begin{eqnarray*} 5x \left( 1 + \frac{1}{x^2 + y^2}\right) &=& 12 \\ 5y \left( 1 - \frac{1}{x^2+y^2} \right) &=& 4 . \end{eqnarray*}
20 replies
Rushil
Oct 25, 2005
SomeonecoolLovesMaths
2 hours ago
Angles made with the median
BBNoDollar   1
N 2 hours ago by Ianis
Determine the measures of the angles of triangle \(ABC\), knowing that the median \(BM\) makes an angle of \(30^\circ\) with side \(AB\) and an angle of \(15^\circ\) with side \(BC\).
1 reply
BBNoDollar
4 hours ago
Ianis
2 hours ago
Find all rationals s.t..
Rushil   12
N 2 hours ago by SomeonecoolLovesMaths
Source: Indian RMO 1994 Problem 7
Find the number of rationals $\frac{m}{n}$ such that

(i) $0 < \frac{m}{n} < 1$;

(ii) $m$ and $n$ are relatively prime;

(iii) $mn = 25!$.
12 replies
Rushil
Oct 25, 2005
SomeonecoolLovesMaths
2 hours ago
An inequality
Rushil   11
N 3 hours ago by SomeonecoolLovesMaths
Source: Indian RMO 1994 Problem 8
If $a,b,c$ are positive real numbers such that $a+b+c = 1$, prove that \[ (1+a)(1+b)(1+c) \geq 8 (1-a)(1-b)(1-c) . \]
11 replies
Rushil
Oct 25, 2005
SomeonecoolLovesMaths
3 hours ago
Minimum moves to reach 25
lkason   0
3 hours ago
Source: Final of the XXI Polish Championship in Mathematical and Logical Games
Mateusz plays a game of erasing-writing on a large board. The board is initially empty.

In each move, he can either:
-- Write two numbers equal to $1$ on the board.
-- Erase two numbers equal to $n$ and write instead the numbers $n-1$ and $n+1$.

What is the minimal number of moves Mateusz needs to make for the number 25 to appear on the board?

Note: Numbers on the board retain their values; their digits cannot be combined or split.

Spoiler, answer:
Click to reveal hidden text
0 replies
lkason
3 hours ago
0 replies
IMO ShortList 2001, combinatorics problem 4
orl   12
N Apr 11, 2025 by Maximilian113
Source: IMO ShortList 2001, combinatorics problem 4
A set of three nonnegative integers $\{x,y,z\}$ with $x < y < z$ is called historic if $\{z-y,y-x\} = \{1776,2001\}$. Show that the set of all nonnegative integers can be written as the union of pairwise disjoint historic sets.
12 replies
orl
Sep 30, 2004
Maximilian113
Apr 11, 2025
IMO ShortList 2001, combinatorics problem 4
G H J
Source: IMO ShortList 2001, combinatorics problem 4
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orl
3647 posts
#1 • 5 Y
Y by narutomath96, Adventure10, ImSh95, Mango247, ohhh
A set of three nonnegative integers $\{x,y,z\}$ with $x < y < z$ is called historic if $\{z-y,y-x\} = \{1776,2001\}$. Show that the set of all nonnegative integers can be written as the union of pairwise disjoint historic sets.
Attachments:
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orl
3647 posts
#2 • 3 Y
Y by Adventure10, ImSh95, Mango247
Please post your solutions. This is just a solution template to write up your solutions in a nice way and formatted in LaTeX. But maybe your solution is so well written that this is not required finally. For more information and instructions regarding the ISL/ILL problems please look here: introduction for the IMO ShortList/LongList project and regardingsolutions :)
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jgao
82 posts
#3 • 3 Y
Y by Adventure10, ImSh95, Mango247
Anyone with a solution
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Ilthigore
315 posts
#4 • 5 Y
Y by narutomath96, DCMaths, ImSh95, Adventure10, Mango247
Just do it.

We partition the nonnegative integers into three sets X, Y and Z by taking disjoint historic triples {x,y,z} x<y<z and putting x into X, y into Y and z into Z according to a simple algorithm. We also keep track of a set S which is the set of nonnegative integers not yet in X,Y or Z. The algorithm is as follows:

i) Take the least element of S, x, and put it into X (so remove it from S). We also put x+1776+2001 into Z.
ii) If x+1776 is in S, we put it into Y. If x+1776 is no longer in S, we put x+2001 into Y instead.
iii) Return to step (i), having removed another historic triple from S.

Since we keep taking the least remaining element of S, it is clear that if such an algorithm succeeds, we have put every integer into precisely one of X,Y and Z once, which is equivalent to having partitioned the integers into historic triples.

Step (i) will clearly succeed, because x is in S by definition, and x+1776+2001 must be in S, as it cannot yet be in X (being bigger than x), or Y (being bigger than x+2001) and can only be put into Z by this value of x.

Step (ii) must also succeed. Suppose it doesn't (noting that for x<1776 it will obviously always work). Then x+2001 must be in Z. But this implies that x-1776 was placed into at some point X. At the time this happened, x should have been placed into Y, but by virtue of its being in S, it isn't already in Y, so this is a contradiction. Thus step (ii) succeeds also, and the algorithm works.
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ABCDE
1963 posts
#5 • 3 Y
Y by ImSh95, Adventure10, Mango247
We use the following algorithm: Let $a$ be the smallest integer that has not been put in a set. If $a+1776$ has not been put in a set make the set $\{a,a+1776,a+3777\}$ and otherwise make the set $\{a,a+2001,a+3777\}$. Suppose that this fails at some point and we can make neither of those sets. This would mean that $a+2001$ was already in a set. It cannot be the least element of a set because otherwise we would have made a set with $a$ as the least element instead when we first put $a+2001$ in a set. If $a+2001$ is the middle element of a set then the least element has to be $a+225$ since $a$ is not in a set yet, but this would be the same contradiction as $a+2001$ being the least element. This means that $a+2001$ is the largest element of its set, so the least element would be $a-1776$. But then we would have made our middle element $a$, a contradiction. Hence, this algorithm works and we are done.
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EulersTurban
386 posts
#6 • 1 Y
Y by ImSh95
So obviously we are going to have two types of historic sets:
$$\{x,1776+x,3777+x\} \; (I)$$$$\{x,2001+x,3777+x\} \; (II)$$so obviously we don't want to end up in a situation such that all the points except one in the set of $\mathbb{Z}_{\geq 0}$ are covered at all.So we need to devise an algorithm such that we don't get into that position.

Algorithm
The algorithm, takes the smallest number now covered by any of the sets. Use type $I$ if possible, if not then use $II$.
To prove that this works we use contradiction.
Suppose this fails, so let $x_i$ be the $i$-th turn.We say that the algorithm has failed on the $(n+1)$-th turn. Where it failed must have been at $x_{n+1}+2001$ or $x_{n+1}+1776$, since $x_{n+1}+3777$ hasn't been covered at all.Furthermore $x_{n+1}+b$ must have been the largest members of their respected set.So by backtracking we have that:
$$x_{n+1}+2001-3777=x_{n+1}-1776 \; - \; \text{minimal element of set}$$But notice how then $x_{n+1}$ would have been covered by our algorithm.Thus a contradiction is taking place here.
For the case of $x_{n+1}+1776$, we also get a contradiction.
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mathleticguyyy
3217 posts
#7 • 2 Y
Y by centslordm, ImSh95
Needless to say, the flavortext didn't age well.

Call an element clogged if it's in a historic set and unclogged otherwise. We use a greedy algorithm, where on the $i$th step, we construct a new historic set $H_i$ with $x$ as the smallest unclogged element, $y$ as $x+1776$ if it's unclogged and $x+2001$ otherwise, and $z=x+3777$. I claim that it will not terminate.

Suppose FTSOC that on some turn, both $x+1776$ and $x+2001$ are clogged. Note that $x+2001$ must be a $z$ element by size, which means that either $x$ or $x+2021-1776$ is clogged as an $y$ element. The former is obviously impossible, and in the latter case, $x$ must have been clogged in the first place.
This post has been edited 2 times. Last edited by mathleticguyyy, Dec 13, 2021, 2:06 AM
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awesomeming327.
1727 posts
#8 • 1 Y
Y by ImSh95
Let $1776=a,2001=b$. Consider the following algorithm: consider a set $S$. Let $m$ be the smallest nonnegative integer not in $S$, then let $S$ be $S\cup \{m,m+a,m+a+b\}$ whenever possible, and let $S$ be $S\cup\{m,m+b,m+a+b\}$ if not possible, and we call $m$ an operator.

We claim that every point, one of the two historic sets above will be available. First, we take care of $m+a+b.$ Note that if $m+a+b$ is in the set, then there must've been an operator that is at least $m$. However, that raises a contradiction because if a number larger than $m$ is used as an operator, then by then $m$ must have been in $S$ already, but now that we are using $m$ as an operator, $m$ must not have been in $S$.

Now, if $m+a$ is unavailable, then either $m+a-b$ was an operator before or $m-b$ was an operator before. If $m+b$ is unavailable, then $m-a$ was an operator before. Since $m$ is currently being used as an operator, the historic set associated with $m-a$ is $\{m-a,m+b-a,m+b\}$, but the set with $m+b-a$ is only used when $m$ is unavailable! Obviously this cannot be the case, so we're done.
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cj13609517288
1922 posts
#9 • 2 Y
Y by ImSh95, Mango247
We provide an algorithm that generates sets that cover any nonnegative integer eventually. Call a historic set an $\alpha$ set if $y-x=1776$ and $z-y=2001$. Similarly, call a historic set a $\beta$ set if $y-x=2001$ and $z-y=1776$.

Our algorithm first covers $0$ with an $\alpha$ set, then repeatedly tries to cover the lowest uncovered number with the same type of set until the $x$ of the new set has to be more than one greater than the $x$ of the previous set, then it switches to the other type of set. If at one point, the lowest uncovered number is greater than any covered number, restart by covering the lowest uncovered number with an $\alpha$ set.

Now, we will prove that our algorithm works. When the first $1776$ $\alpha$ sets are made, there will be a gap between the largest $y$ and the smallest $z$. When we start to use the $\beta$ sets, note how the previously largest $y$ and the previously largest $z$ are now one-upped by the $x$ and $y$ in the new $\beta$ set. This continues until a new gap is formed between the largest $y$ and the smallest $z$, and the algorithm is continued. If at any point the gap doesn't exist anymore, we can simply restart the algorithm. Now, we note that the $x$ and $z$ cannot possibly clash($z$ is supposed to always be $1776+2001$ above $x$ which increases every time), and $y$ cannot clash because it one-ups the previous largest $z$. QED.
This post has been edited 2 times. Last edited by cj13609517288, Nov 16, 2022, 6:35 PM
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programmeruser
2455 posts
#10 • 1 Y
Y by ImSh95
Note that a historic set is either in the form $\{x, x+1776, x+1776+2001\}$ (which we will call a 1776 set) or $\{x, x+2001, x+1776+2001\}$ (a 2001 set).

We can use the following algorithm to achieve this task:
We let $x$ be the smallest number not yet covered. If none of the numbers in the 1776 set have been covered already, then we use a 1776 set, otherwise we use a 2001 set.

Now we prove that this algorithm works. We prove that if the 1776 set does not work, then the 2001 set will always work.
Assume for the sake of contradiction that the 2001 set does not work.
Clearly the $x$ element can't have already been covered.
If the $x+1776+2001$ element has been covered, then that means that on some previous step, we either used the 1776 set on $x+2001$ or the 2001 set on $x+1776$. However, this is impossible as $x < x+1776 < x+2001$.
If the $x+2001$ has already been covered, then either we used the 1776 set on $x+2001-1776$, or we used any set on $x-1776$. Clearly the first one is impossible as $x+2001-1776>x$. Now we consider the second case. If we used the 1776 set on $x-1776$, then it would imply that we would have already covered $x$, a contradiction. If we used the 2001 set, then that means that some element in the 1776 set was already covered. If $x+2001$ was already covered, then this implies that $x$ was already covered, contradiction. Otherwise, this implies that $x$ was already covered, another contradiction.

Therefore, a situation where both sets don't work, and the algorithm always works. $\square$
This post has been edited 1 time. Last edited by programmeruser, Jan 31, 2023, 12:22 AM
Reason: add extra newline
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awesomehuman
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#11 • 1 Y
Y by ImSh95
We claim that using the following algorithm we can cover the postive integers:
-Find the lowest positive integer $x$ that hasn't already been covered.
-Let $y$ be the lowest if $x+1776$ and $x+2001$ that isn't covered.
-Let $z$ be $x+3777$.
-Then, add the triple $\{x, y, z\}$.

Assume towards a contradiction $x+2001$ is already covered. It can't be the 1st number in its triplet, since it's higher than $x$ and the triplets are added in strictly increasing order based on the 1st number. For the same reason, it can't be the 2nd number in its triplet. So, it is the third in its triplet. The first must be 1776. However, then, because $x$ has not yet been covered, when this triplet was created, $x$ would've been made the 2nd number, a contradiction. So, the 2nd step must be possible.

By similar logic, if $x+3777$ is already covered, then it must be the 3rd in its triplet. However, then $x$ would be the first, a contradiction.

So, we can continue this procedure indefinitely and cover the positive integers. We can flip it to cover the nonpositive integers. So, we can cover all integers.
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huashiliao2020
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#12 • 1 Y
Y by ImSh95
dont like bashy stuff like this
silly from the aime

x,y,z=x (call it 1st element),x+{1776 or 2001} (second elements, where 1776 or 2001 is two options), x+1776+2001
Clearly the optimal strategy is to cover the smaller numbers (because why would you skip and go to larger? That just plugs your historic sets possible)
Now we show that the algorithm choosing the smallest number not used and use the set with smaller sum if possible, use the larger if not. We prove that this works. Suppose at x_n you cannot cover it. This means either both of the second elements are covered or the largest is. Clearly the largest is not by our algorithm of doing small first, so both x+1776 and x+2001 must be covered. x+2001 must be the largest number in its set (it can't be n+1776, and can't be n), hence the smallest number in this set must be x+2001-2001-1776. However, x_n is not covered, hence x_n-1776,x_n,x_n+2001 must have been used (smaller set). Hence, contradiction. $\blacksquare$
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Maximilian113
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#13
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Replace $1776$ and $2001$ with any positive integers $a, b$ with $a<b.$ Note that historic sets would be of the form $\{ x, x+a, x+a+b \}$ or $\{x, x+b, x+a+b\}.$ We use the greedy algorithm, where we operate on the smallest nonnegative integer $k$ not yet covered. If $k+a$ is not yet covered, we use the former set, otherwise we use the latter.

It suffices to show that with this algorithm, no number is covered more than once. For the sake of a contradiction, assume that at some point $\ell$ is the smallest number not covered, but $\ell+a, \ell+b$ are already covered. Assume that this is the first occurence of such a situation. Observe that $\ell+a+b$ cannot be covered as it is larger than any $\ell_i+a+b$ covered previously. $\ell+a, \ell+b$ must not have been the smallest numbers in their sets as they are both larger than $\ell,$ so $\ell+a$ must have been covered with $\{\ell-b, \ell+a-b, \ell+a\}, \{\ell+a-b, \ell+a, \ell+2a\},$ while $\ell+b$ was covered along with $\{\ell-a, \ell+b-a, \ell+b\}, \{\ell+b-a, \ell+b, \ell+2b\}.$ For this the latter cannot occur as $\ell+b-a > \ell,$ so $\{\ell-a, \ell+b-a, \ell+b\}$ must occur. But this overlaps with any of the first two historic sets, contradiction. Hence the algorithm works. QED
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