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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
Basic ideas in junior diophantine equations
Maths_VC   5
N 9 minutes ago by Royal_mhyasd
Source: Serbia JBMO TST 2025, Problem 3
Determine all positive integers $a, b$ and $c$ such that
$2$ $\cdot$ $10^a + 5^b = 2025^c$
5 replies
Maths_VC
May 27, 2025
Royal_mhyasd
9 minutes ago
Find values for a, b ,c
Ferum_2710   2
N 11 minutes ago by Jupiterballs
Source: Romania JBMO tst 2023 day2 p4
Let $M \geq 1$ be a real number. Determine all natural numbers $n$ for which there exist distinct natural numbers $a$, $b$, $c > M$, such that
$n = (a,b) \cdot (b,c) + (b,c) \cdot (c,a) + (c,a) \cdot (a,b)$
(where $(x,y)$ denotes the greatest common divisor of natural numbers $x$ and $y$).
2 replies
Ferum_2710
Apr 30, 2023
Jupiterballs
11 minutes ago
Center lies on circumcircle of other
Philomath_314   41
N 23 minutes ago by Adywastaken
Source: INMO P1
In triangle $ABC$ with $CA=CB$, point $E$ lies on the circumcircle of $ABC$ such that $\angle ECB=90^{\circ}$. The line through $E$ parallel to $CB$ intersects $CA$ in $F$ and $AB$ in $G$. Prove that the center of the circumcircle of triangle $EGB$ lies on the circumcircle of triangle $ECF$.

Proposed by Prithwijit De
41 replies
Philomath_314
Jan 21, 2024
Adywastaken
23 minutes ago
find question
mathematical-forest   7
N 36 minutes ago by whwlqkd
Are there any contest questions that seem simple but are actually difficult? :-D
7 replies
mathematical-forest
Thursday at 10:19 AM
whwlqkd
36 minutes ago
Inspired by a cool result
DoThinh2001   1
N 36 minutes ago by arqady
Source: Old?
Let three real numbers $a,b,c\geq 0$, no two of which are $0$. Prove that:
$$\sqrt{\frac{a^2+bc}{b^2+c^2}}+\sqrt{\frac{b^2+ca}{c^2+a^2}}+\sqrt{\frac{c^2+ab}{a^2+b^2}}\geq 2+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}.$$
Inspiration
1 reply
DoThinh2001
Today at 12:08 AM
arqady
36 minutes ago
Crossing ٍٍChords
matinyousefi   1
N an hour ago by Trenod
Source: Iranian Combinatorics Olympiad 2020 P3
$1399$ points and some chords between them is given.
$a)$ In every step we can take two chords $RS,PQ$ with a common point other than $P,Q,R,S$ and erase exactly one of $RS,PQ$ and draw $PS,PR,QS,QR$ let $s$ be the minimum of chords after some steps. Find the maximum of $s$ over all initial positions.
$b)$ In every step we can take two chords $RS,PQ$ with a common point other than $P,Q,R,S$ and erase both of $RS,PQ$ and draw $PS,PR,QS,QR$ let $s$ be the minimum of chords after some steps. Find the maximum of $s$ over all initial positions.

Proposed by Afrouz Jabalameli, Abolfazl Asadi
1 reply
matinyousefi
Apr 24, 2020
Trenod
an hour ago
Nice NT with powers of two
oVlad   7
N an hour ago by SimplisticFormulas
Source: Romania TST 2024 Day 1 P3
Let $n{}$ be a positive integer and let $a{}$ and $b{}$ be positive integers congruent to 1 modulo 4. Prove that there exists a positive integer $k{}$ such that at least one of the numbers $a^k-b$ and $b^k-a$ is divisible by $2^n.$

Cătălin Liviu Gherghe
7 replies
oVlad
Jul 31, 2024
SimplisticFormulas
an hour ago
Inequality in triangle
Nguyenhuyen_AG   0
2 hours ago
Let $a,b,c$ be the lengths of the sides of a triangle. Prove that
\[\frac{1}{(a-4b)^2}+\frac{1}{(b-4c)^2}+\frac{1}{(c-4a)^2} \geqslant \frac{1}{ab+bc+ca}.\]
0 replies
Nguyenhuyen_AG
2 hours ago
0 replies
D,E,F are collinear.
TUAN2k8   2
N 2 hours ago by TUAN2k8
Source: Own
Help me with this:
2 replies
TUAN2k8
May 28, 2025
TUAN2k8
2 hours ago
Combinatorial identity
MehdiGolafshan   4
N 2 hours ago by watery
Let $n$ is a positive integer. Prove that
$$\sum_{k=0}^{n-1}\frac{1}{k+1}\binom{n-1}{k} = \frac{2^n-1}{n}.$$
4 replies
MehdiGolafshan
Jan 16, 2023
watery
2 hours ago
JBMO Shortlist 2023 G7
Orestis_Lignos   7
N 3 hours ago by tilya_TASh
Source: JBMO Shortlist 2023, G7
Let $D$ and $E$ be arbitrary points on the sides $BC$ and $AC$ of triangle $ABC$, respectively. The circumcircle of $\triangle ADC$ meets for the second time the circumcircle of $\triangle BCE$ at point $F$. Line $FE$ meets line $AD$ at point $G$, while line $FD$ meets line $BE$ at point $H$. Prove that lines $CF, AH$ and $BG$ pass through the same point.
7 replies
Orestis_Lignos
Jun 28, 2024
tilya_TASh
3 hours ago
Reflected point lies on radical axis
Mahdi_Mashayekhi   5
N 3 hours ago by Mahdi_Mashayekhi
Source: Iran 2025 second round P4
Given is an acute and scalene triangle $ABC$ with circumcenter $O$. $BO$ and $CO$ intersect the altitude from $A$ to $BC$ at points $P$ and $Q$ respectively. $X$ is the circumcenter of triangle $OPQ$ and $O'$ is the reflection of $O$ over $BC$. $Y$ is the second intersection of circumcircles of triangles $BXP$ and $CXQ$. Show that $X,Y,O'$ are collinear.
5 replies
Mahdi_Mashayekhi
Apr 19, 2025
Mahdi_Mashayekhi
3 hours ago
Find the value
sqing   18
N 3 hours ago by Yiyj
Source: 2024 China Fujian High School Mathematics Competition
Let $f(x)=a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0,$ $a_i\in\{-1,1\} ,i=0,1,2,\cdots,6 $ and $f(2)=-53 .$ Find the value of $f(1).$
18 replies
sqing
Jun 22, 2024
Yiyj
3 hours ago
Number Theory
fasttrust_12-mn   14
N 3 hours ago by Namisgood
Source: Pan African Mathematics Olympiad P1
Find all positive intgers $a,b$ and $c$ such that $\frac{a+b}{a+c}=\frac{b+c}{b+a}$ and $ab+bc+ca$ is a prime number
14 replies
fasttrust_12-mn
Aug 15, 2024
Namisgood
3 hours ago
Composite sum
rohitsingh0812   39
N Apr 23, 2025 by YaoAOPS
Source: INDIA IMOTC-2006 TST1 PROBLEM-2; IMO Shortlist 2005 problem N3
Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$.
Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.
39 replies
rohitsingh0812
Jun 3, 2006
YaoAOPS
Apr 23, 2025
Composite sum
G H J
Source: INDIA IMOTC-2006 TST1 PROBLEM-2; IMO Shortlist 2005 problem N3
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rohitsingh0812
105 posts
#1 • 8 Y
Y by yayitsme, centslordm, Adventure10, HWenslawski, megarnie, SatisfiedMagma, and 2 other users
Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$.
Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.
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Davron
484 posts
#2 • 41 Y
Y by mathbuzz, mjuk, HarvardMit, div5252, Swag00, jt314, Anar24, A_Math_Lover, Wizard_32, richrow12, Pluto1708, green_leaf, rashah76, Toinfinity, Illuzion, mathleticguyyy, yayitsme, Wizard0001, centslordm, hakN, Adventure10, HWenslawski, megarnie, W.R.O.N.G, arinastronomy, Mango247, rstenetbg, Ab_Rin, Stuffybear, kiyoras_2001, aidan0626, Sedro, and 9 other users
All the coefficients of $f(x)=(x+a)(x+b)(x+c)-(x-d)(x-e)(x-f)=$ $Sx^2+(ab+bc+ca-de-ef-fd)x+(abc+def)$ are multiples of $S$. Evaluating $f$ at $d$ we get that $f(d)=(a+d)(b+d)(c+d)$ is a multiple of $S$.
So this implies that $S$ is composite, since $a+d,b+d,c+d$ are all strictly less than $S$.

davron
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spider_boy
210 posts
#3 • 2 Y
Y by centslordm, Adventure10
My solution (some kind of minimality idea).

Suppose $S$ is prime.
We observe that if $(a, b, c, d, e ,f)$ is a solution then $(a-1, b-1, c-1, d+1, e+1, f+1)$ is also a solution. Note that $S$ does not change. Denote $X=\min\{a,b,c\}$. We can claim that $(a-X, b-X, c-X, d+X, e+X, f+X)$ also satisfies the conditions.
So we get $S \mid (d+X)(e+X)(f+X)$. But this is impossible, since $d+X,e+X, f+X$ are all $<S$. :)
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Particle
179 posts
#4 • 2 Y
Y by centslordm, Adventure10
Hidden due to length
This post has been edited 1 time. Last edited by Particle, May 21, 2013, 2:29 AM
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subham1729
1479 posts
#5 • 3 Y
Y by centslordm, Adventure10, Mango247
Well, note $(a+d)(b+d)(c+d)=A+dB+d^2S$ now so we get $S|(a+d)(b+d)(c+d)$ , suppose $S$ is prime then it must be less than one of $a+d,b+d,c+d$ but as $S=a+b+c+d$ so absurd and done.
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ksun48
1514 posts
#6 • 4 Y
Y by centslordm, Adventure10, Mango247, and 1 other user
Assume $S$ is a prime. Then $(x-a)(x-b)(x-c)$ and $(x+d)(x+e)(x+f)$ are the same polynomial, mod $S$, so by Lagrange's theorem $\{a,b,c\} = \{-d,-e,-f\}$. Thus $a+b+c+d+e+f \ge 3S > S$.
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AMN300
563 posts
#7 • 3 Y
Y by centslordm, Adventure10, Mango247
Solution
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RayThroughSpace
426 posts
#8 • 3 Y
Y by centslordm, Adventure10, Mango247
For the sake of contradiction, let $S$ be prime. Note $S$ divides $(x-a)(x-b)(x-c) - (x+d)(x+e)(x+f)$. Substitute $x= a,b,c,-d,-e,-f,$ we get $S$ divides $(d+a)(d+b)(d+c) , (e+a)(e+b)(e+c), (f+a)(f+b)(f+c)$ and

$S|(a+d)(a+e)(a+f)$
$S|(b+d)(b+e)(b+f)$
$S|(c+d)(c+e)(c+f)$

From the first set of 3 conditions, WLOG, we can let $S|(a+d)$, $S|(b+e)$ and $S|(c+f)$. However, each of $(a+d),(b+e), (c+f)$ are smaller than $S$, a contradiction.
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v_Enhance
6882 posts
#9 • 15 Y
Y by Wizard_32, RAMUGAUSS, Gems98, rashah76, Cindy.tw, v4913, Robokop, SerdarBozdag, centslordm, hakN, mathleticguyyy, lahmacun, HamstPan38825, Adventure10, Mango247
Let $p$ be a fixed prime. We say a $6$-tuple $(a,b,c,d,e,f)$ of integers is good if $a+b+c+d+e+f = p$ and $p \mid abc+def$ and $p \mid ab+bc+ca-de-ef-fd$.

Claim: If $(a,b,c,d,e,f)$ is good then so is $(a+1, b+1, c+1, d-1, e-1, f-1)$.

Proof. Direct computation. $\blacksquare$

We claim there exists no good $6$-tuple of positive integers. If not, WLOG $f$ is the smallest of the six numbers. Then by repeating the claim, the tuple $(a+f, b+f, c+f, d-f, e-f, 0)$ is good. But now $p \mid (a+f)(b+f)(c+f)$ and $p > \max(a+f, b+f, c+f)$, contradiction.
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yayups
1614 posts
#10 • 3 Y
Y by centslordm, Adventure10, Mango247
We see that
\begin{align*}
a+b+c&\equiv -(d+e+f)\pmod{S} \\
ab+bc+ca&\equiv -(de+ef+fd)\pmod{S} \\
abc &\equiv -def\pmod{S}.
\end{align*}Now, assuming that $S$ is prime, we see that
\[(X-a)(X-b)(X-c)=(X+d)(X+e)(X+f)\]over $\mathbb{F}_S[X]$, so because of unique factorization, we have WLOG that $a\equiv -d\pmod{S}$, $b\equiv -e\pmod{S}$, and $c\equiv -f\pmod{S}$. In particular, this means that $S\mid a+d$. However, $S>a+d$, so this isn't possible. Therefore, $S$ couldn't have been prime to start with. $\blacksquare$
This post has been edited 1 time. Last edited by yayups, Apr 5, 2019, 9:30 PM
Reason: latex align fix
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Zorger74
760 posts
#11 • 1 Y
Y by centslordm
Solution
This post has been edited 1 time. Last edited by Zorger74, Jan 23, 2021, 12:41 AM
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Spacesam
596 posts
#12 • 1 Y
Y by centslordm
AFSOC $S = p$ for some prime $p$. Define \begin{align*}
    P(x) &= (x + a)(x + b)(x + c) = x^3 + x^2(a + b + c) + x(ab + bc + ca) + abc \\
    Q(x) &= (x - d)(x - e)(x - f) = x^3 - x^2(d + e + f) + x(de + ef + fd) - def,
\end{align*}and consider \begin{align*}
    R(x) &= P(x) - Q(x) \\
    &= x^2 \cdot p + x(ab + bc + ca - de - ef - fd) + abc + def.
\end{align*}Taking mod $p$, we find that $P(x) \equiv Q(x) \pmod p$.

However, plug in $x = d$. We know $Q$ has a root at $d$, so \begin{align*}
    P(d) = (d + a)(d + b)(d + c) \equiv 0\pmod p,
\end{align*}but $p > d + a, d + b, d + c$ and so we are done.
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Mano
46 posts
#13 • 1 Y
Y by centslordm
$S\geq6$, so assume for the sake of contradiction that $S$ is prime. Let $P(x)\coloneqq (x-a)(x-b)(x-c)$ and $Q(x)\coloneqq (x+a)(x+b)(x+c)$. Multiplying out, we get that $P(x)\equiv Q(x)\;(\text{mod}\,S)$ for all integers $x$. In particular, we get that for an integer $x$, $S$ divides $P(x)$ if and only if $S$ divides $Q(x)$, which, because $S$ is assumed to be a prime, means that $\{a, b, c\}$ is just some permutation of $\{-d, -e, -f\}$ modulo $S$. WLOG, assume that $a\equiv -d$, $b\equiv -e$ and $c\equiv -f$ modulo $S$. But this means that $a+d$, $b+e$ and $c+f$ are all multiples of $S$, which is impossible, because they are positive integers which sum up to $S$.
This post has been edited 1 time. Last edited by Mano, Apr 28, 2021, 9:35 AM
Reason: 6, not 1
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bever209
1522 posts
#14 • 1 Y
Y by centslordm
Note that $(x+a)(x+b)(x+c)-(x-d)(x-e)(x-f)$ has all of its coefficients divisible by $S$. This implies $S|(d+a)(d+b)(d+c)$. FTSOC assume $S$ is prime. Then it must divide either $d+a,d+b,d+c$, all of which are smaller than $S$, a contradiction, so we are done.
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bora_olmez
277 posts
#15
Y by
Assume for the sake of contradiction that $S$ is prime. Notice that $$S \mid (a+k)(b+k)(c+k)+(d-k)(e-k)(f-k)$$for all $k \in \mathbb{Z}$.
Then taking $k$ to be $-a,-b,-c$ we have that $\{d,e,f\} = \{-a,-b,-c\}$ in $\mathbb{F}_S$ using that $S$ is prime, yet, adding any such values of $a,b,c,d,e,f$, we have that $$S = a+b+c+d+e+f \geq 3S$$which is a contradiction. $\blacksquare$
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Sprites
478 posts
#16
Y by
Suppose that $S$ is a prime and let $x=-d,y=-e,z=-f$,so that $abc \equiv xyz \pmod S$
Now consider the polynomial $ (a+k)(b+k)(c+k)+(d-k)(e-k)(f-k)$ and clearly inputting $x,y,z$ implies $S|(a+d)(a+e)(a+f)$,which implies that $S$ divides one of $(a+d),(a+e),(a+f)$,a contradiction.
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AwesomeYRY
579 posts
#17 • 1 Y
Y by trying_to_solve_br
Note that the polynomial $f(x) = (x-a)(x-b)(x-c)-(x+d)(x+e)(x+f) = (ab+ac+bc-de-ef-df)x-(abc+def)$ is divisible by $S$ for all integers $x$. Thus,
\[S\mid f(a) = -(a+d)(a+e)(a+f)\]To finish, $S$ cannot be prime because $a+d,a+e,a+f<a+b+c+d+e+f = S$, so we're done. $\blacksquare$.
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HoRI_DA_GRe8
599 posts
#18
Y by
Sol
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Mogmog8
1080 posts
#19 • 1 Y
Y by centslordm
Notice \begin{align*}S\mid f(x)&=Sx^2+(ab+bc+ca-de-df-fd)x+(abc+def)\\&=(x+a)(x+b)(x+c)-(x-d)(x-e)(x-f)\end{align*}Letting $x=d,$ we have $S\mid (d+a)(d+b)(d+c),$ which is absurd if $S$ is prime as $S>d+a,d+b,d+c.$ $\square$
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awesomeming327.
1741 posts
#20
Y by
Assume for the sake of contradiction $S$ is prime, and take polynomials $P(x)=(x-a)(x-b)(x-c)$ and $Q(x)=(x+d)(x+e)(x+f).$ Note that \[P(x)\equiv x^3 - (a+b+c)x^2 + (ab+bc+ca)x-abc \pmod S\]Note that on the other hand, \[Q(x)\equiv x^3 + (d+e+f)x^2 + (de+ef+fd)x+def \pmod S\]Now, $Q(x)-P(x)=Sx^2+(de+ef+fd-ab-bc-ca)x+abc+def.$ Note that this is divisible by $S.$ In particular, $Q(a)-P(a)=(a+d)(a+e)(a+f)$ is divisible by $S.$ Note that each of those factors are less than $S$ but $S$ is prime, which is a contradiction.
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HamstPan38825
8869 posts
#21
Y by
Observe that we have the system of congruences
\begin{align*}
a+b+c &\equiv -d-e-f \pmod S \\
ab+bc+ca &\equiv de+ef+fd \pmod S \\
abc &\equiv -def \pmod S.
\end{align*}This means that $$(x-a)(x-b)(x-c) \equiv (x+d)(x+e)(x+f) \pmod S$$for all $x \in \mathbb Z$ as the resulting polynomials are equivalent under modulo $S$. Letting $x=a$, $$S \mid (a+d)(a+e)(a+f),$$but $a+d, a+e, a+f < S$, and thus $S$ must be composite.
This post has been edited 1 time. Last edited by HamstPan38825, Jul 20, 2022, 12:02 AM
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Kimchiks926
256 posts
#22 • 2 Y
Y by mkomisarova, Mango247
Suppose otherwise that $S$ is prime $p$. Then we have the following equations.
\begin{align*} 
a+b+c \equiv -(d+e+f) \pmod p \\ 
ab+bc+ca \equiv de+ef+df \pmod p \\ 
abc \equiv -def \pmod p 
\end{align*}We will construct numbers $a_1,b_1, c_1, d_1, e_1, f_1$ such that:
\begin{align*} 
a_1+b_1+c_1 \equiv -(d_1+e_1+f_1) \pmod p \\ 
a_1b_1+b_1c_1+c_1a_1 \equiv d_1e_1+e_1f_1+d_1f_1 \pmod p \\ 
a_1b_1c_1 \equiv -d_1e_1f_1 \pmod p 
\end{align*}In fact, this is not so hard to do. It is just enough to take $a_1=a+1, b_1=b+1, c_1 =c_1$ and $d_1 =d-1, e_1 =e-1, f_1 =f-1$. It is easy to see that they satisfy the first congruence. Note that:
\begin{align*} 
(a+1)(b+1)+(b+1)(c+1)+(c+1)(a+1) \equiv (d-1)(e-1)+(e-1)(f-1)+(f-1)(d-1) \pmod p\\ 
ab+bc+ca +2(a+b+c)+3 \equiv de+ef+fd -2(d+e+f)+3 \pmod p
\end{align*}We see that the second desired congruence is also satisfied by given congruences. Also, observe that:
\begin{align*} 
(a+1)(b+1)(c+1) \equiv -(d-1)(e-1)(f-1)  \pmod p \\
a+b+c+(ab+bc+ca)+abc+1 \equiv -((d+e+f)-(de+ef+fd)+def-1) \pmod p \\
(a+b+c+de+f) +(ab+bc+ca-de-ef-fd)+(abc+def) \equiv 0 \pmod p
\end{align*}We see that the third desired congruence is also satisfied by given congruences.

Since $a,b,c, d,e,f$ are positive integers and their sum it $p$, then they all are less than $p$. WLOG $a$ is the largest among $a,b,c$. With each construction we increase $a,b,c$ by $1$ and decrease $d,e,f$ by $1$. We can keep doing it until $a$ becomes $p$. Suppose that we needed for that $k$ constructions, in other words $a+k =p=a_k$. Consider congruence:
$$ a_kb_kc_k \equiv d_ke_kf_k \equiv 0 \pmod p $$We see that it implies that $d_ke_kf_k \equiv 0 \pmod p$. WLOG assume that $d_k = d-k$ is the one which is divisible by $p$. Then $0 \equiv a_k + d_k = a+k+d-k = a+ d \pmod p$, which is contradiction, since it forces $b,c,e,f$ to be $0$.

Our initial assumption was wrong, therefore $S$ composite as desired.
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SatisfiedMagma
461 posts
#23 • 1 Y
Y by kamatadu
I'm not smart as others to see that you can really form a polynomial... Anyways here is a solution by induction which kind of makes the polynomial in a different way which I feel is intuitive too...

Solution. Assume on the contrary that $S$ is prime. We will make use of these handy identities in the solution.
\begin{align}
        (a+1)(b+1)(c+1) &= abc + (ab+bc+ca) + (a+b+c) + 1 \\
        (d-k)(e-k)(f-k) &= def - (de+ef+df) + (d+e+f) - 1
    \end{align}The following claim is the crux of the problem.

Claim: $S \mid (a+x)(b+x)(c+x) + (d-x)(e-x)(f-x)$ for all $x \in \mathbb{Z}_{\ge 0}$.

Proof: The proof is by induction with the base case being trivial. As an induction hypothesis, assume that $S \mid (a+k)(b+k)(c+k) + (d-k)(e-k)(f-k)$. For ease of notation label $a+k = A$, $d - k = D$ and so on. Now finally for the inductive step, we will just work backward.
\begin{align*}
            S \mid (A+1)(B+1)(C+1) &+ (D-1)(E-1)(F-1) \\
            \iff S \mid (ABC + DEF) + (AB+BC+CA) &- (DE+EF+FA) + (A+B+C+\ldots)
        \end{align*}By the induction hypothesis, we will cancel of $ABC + DEF$ term. It is not hard to see that $S \mid (A+B+C+D+E+F)$ as well.
\begin{align*}
            \iff S \mid AB + BC + CA - DE - EF - FA
        \end{align*}It isn't hard to expand this out and realize its a tautology. So, the induction is complete.$\square$
Finally choose $x = d$. This would give
\[S = a + b + c + d + e + f \mid (a + d)(b + d)(c+d).\]Since $S$ is a prime, it must divide one of the factors. But since all the factors are positive and less than $S$, its the desired contradiction. $\blacksquare$
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sman96
136 posts
#24 • 1 Y
Y by lian_the_noob12
Let's allow $0$ as value of $d,e,f$.
FTSOC, let's assume that $S = p$ is a prime.
Here, \begin{align*}
S &= a+b+c+d+e+f\\ &= (a+1) + (b+1)+(c+1) + (d-1)+(e-1)+(f-1)
\end{align*}
Now, $p\mid a+b+c+d+e+f$, $p\mid abc+def$ and $p\mid ab+bc+ca-de-ef-df$. Summing these up gives, $$p\mid (a+1)(b+1)(c+1) + (d-1)(e-1)(f-1)$$Also, \begin{align*}
&\sum_{cyc}(a+1)(b+1) - \sum_{cyc}(d-1)(e-1)\\
=& ab+bc+ca-de-ef-df +2(a+b+c+d+e+f)
\end{align*}
Therefore, $\displaystyle p\mid \sum_{cyc}(a+1)(b+1) - \sum_{cyc}(d-1)(e-1)$.
So, if $(a,b,c,d,e,f)$ is a solution then, $(a+1, b+1, c+1, d-1,e-1,f-1)$ is also a solution with $S= p$.

Now we continue this process until one of the elements becomes $0$. WLOG that be $f$.
Then we will get, $p\mid abc$. But therefore $p$ must divide one of $a,b,c$ which isn't possible as, $p = a+b+c+d+e$. This gives a contradiction. $\blacksquare$
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IAmTheHazard
5005 posts
#25 • 1 Y
Y by centslordm
Suppose that $S$ was prime. Then the polynomials $(x+a)(x+b)(x+c)$ and $(x-d)(x-e)(x-f)$ have equivalent expansions modulo $p$, hence we must have $\{-a,-b,-c\} \equiv \{d,e,f\} \pmod{S} \implies \{S-a,S-b,S-c\}=\{d,e,f\}$. But then $d+e+f=3S-a-b-c \implies S=a+b+c+d+e+f=3S$: contradiction. $\blacksquare$
This post has been edited 1 time. Last edited by IAmTheHazard, Aug 12, 2023, 5:46 PM
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lifeismathematics
1188 posts
#26
Y by
Consider $P(x)=(x+a)(x+b)(x+c)-(x-d)(x-e)(x-f)$

$P(x)=Sx^2+x(ab+bc+ca-ed-ef-df)+(abc+def) \implies S|P(x)$ $\forall x \in \mathbb{Z}$

$P(d)=(a+d)(b+d)(c+d)$ hence as $S|P(d)$. Now , $\mathrm{FTSOC}$ assume $S$ is prime, then as we have $S|(a+d)(b+d)(c+d)$ we have at least one of $(a+d), (b+d) , (c+d)> S$ but since $S=a+b+c+d+e+f$ this is absurd for $a,b,c,d,e, f \in \mathbb{Z}^{+}$ , hence our assumption false and hence $S$ is composite $\blacksquare$.
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sixoneeight
1138 posts
#27 • 1 Y
Y by fuzimiao2013
Solved with xor, xook, and xonk, also known as fuzimiao2013, sixoneeight, popop614.

We can rewrite the condition as
\[
(t-a)(t-b)(t-c) \equiv (t+d)(t+e)(t+f) \pmod{S}
\]for all integer $t$. Suppose for the sake of contradiction that $S$ was prime. Take $t=a$. Then, $S$ divides one of $a+d$, $a+e$, $a+f$. However, since $a,b,c,d,e,f$ are positive integers, that would mean that one of $a+d, a+e, a+f$ is equal to $S$. This is a contradiction as the rest of the variables could not be positive, so $S$ must be composite.
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math_comb01
662 posts
#28
Y by
FTSOC let $S$ be a prime
$a+b+c \equiv -(d+e+f)$
$abc \equiv (-d)(-e)(-f)$
$ab+bc+ca \equiv (-d)(-e)+(-e)(-f)+(-d)(-f)$
In $\mathbb{F}_S$
$(x-a)(x-b)(x-c) \equiv (x+d)(x+e)(x+f)$
therefore $\{a,b,c\} \equiv \{-d,-e,-f\}$ in $\mathbb{F}_S$
$\therefore$ we're done.
This post has been edited 2 times. Last edited by math_comb01, Dec 27, 2023, 11:57 AM
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EpicBird08
1756 posts
#29 • 2 Y
Y by GeoKing, torch
Thanks to torch for suggesting this problem to me!

Let $k$ be an integer. Note that $S$ divides the quantity $$abc + def + k(ab+bc+ca-de-ef-fd) + k^2(a+b+c+d+e+f) + 1 - 1 = (a+k)(b+k)(c+k) - (d-k)(e-k)(f-k).$$Now there are $2$ cases.

Case 1: The smallest of $a,b,c,d,e,f$ is one of $a,b,c.$ WLOG suppose it is $a.$ Letting $k = -a$ gives $S \mid -(d+a)(e+a)(f+a),$ so switching the positive sign yields $$S \mid (d+a)(e+a)(f+a).$$If $S$ was prime, then it would have to divide at least one of these factors, say $d+a$ (the other cases are symmetrical to this). Then $a+b+c+d+e+f \mid d+a,$ which is a contradiction due to size issues.

Case 2: The smallest of $a,b,c,d,e,f$ is one of $d,e,f.$ WLOG suppose it is $d.$ Letting $k = d$ gives $S \mid (a+d)(b+d)(c+d),$ and repeat the size argument from the previous case.

Therefore, in any case, $S$ must be composite, as desired.
This post has been edited 1 time. Last edited by EpicBird08, Jun 21, 2024, 10:31 PM
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PEKKA
1850 posts
#30
Y by
Used a whole bunch of ARCH Hints after 30 min of failing:
FTSOC $S=p$ for some prime $p.$
Let $d_0=-d, e_0=-e, f_0=-f.$ Then we get a vieta-like relation modulo $P.$ This implies that $a,b,c$ and $d_0,e_0,f_0$ are roots to the same polynomial with coefficients modulo $p.$
Now WLOG $a-d_0 \equiv 0\pmod{p},b-e_0 \equiv 0\pmod{p}, c-f_0 \equiv 0\pmod{p}.$
Therefore, $a+d, b+e, c+f$ are multiples of $p$
Therefore, $S=a+d+b+e+c+f=p(k)$ for some integer $k>1$ which violates the condition that $S=p,$ or $S$ is prime. Therefore $S$ must be composite.
This post has been edited 1 time. Last edited by PEKKA, Jun 23, 2024, 7:03 PM
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cursed_tangent1434
651 posts
#31
Y by
Really cute problem. I'm surprised I found the nice solution! We consider ordered pairs of sets $(\{a,b,c\},\{d,e,f\})$ which satisfy the given conditions. We call such a pair a good pair and if $a+b+c+d+e+f=k$ then we say that it is a $k-$good pair. We start off by proving the following nice result about $k-$good pairs.

Claim : For all integers $k$ for which there exists a $k-$good pair $(\{a,b,c\},\{d,e,f\})$, the pairs $(\{a-1,b-1,c-1\},\{d+1,e+1,f+1\})$ and $(\{a+1,b+1,c+1\},\{d-1,e-1,f-1\})$ are also $k-$good.

Proof : It is not hard to see that they must be $k-$good if they are good at all, since the sum $a+b+c+d+e+f$ does not change in either instance. We show that if $(\{a,b,c\},\{d,e,f\})$ is good then the pair $(\{a-1,b-1,c-1\},\{d+1,e+1,f+1\})$ must also be good. This is a mere calculation. Note that,
\[    (a-1)(b-1)(c-1)+(d+1)(e+1)(f+1) = (abc+def) -(ab+bc+ca-de-ef-fd) + (a+b+c+d+e+f)\]where each of the bracketed terms are known to be multiples of $a+b+c+d+e+f$. The proof of the other is entirely similar, so this finishes the proof of the claim.

We wish to show that there exists no $p-$good pairs for any prime $p$. By way of contradiction, say there exists such a prime $p$ and a $p-$good pair $(\{a,b,c\},\{d,e,f\})$. Then, WLOG say the minimum element of the set $\{a,b,c,d,e,f\}$ is $a$. By way of our claim, it follows that the pair $(\{0,b-a,c-a\},\{d+a,e+a,f+a\})$ is also $p-$good. Note that here, $d'=d+a$ , $e'=e+a$ and $f'=f+a$ are all positive integers. Then,
\[p|(0)(b-a)(c-a)+(d+a)(e+a)(f+a)=d'e'f'\]Since $p$ is a prime, this implies that $p$ divides one of $d'$ , $e'$ and $f'$. But, since $p=0 + (b-a) + (c-a) + (d+a) + (e+a) + (f+a)$ where the first 3 terms are non-negative and the last two terms are strictly positive,
\[p = (b-a) + (c-a) + (d+a) + (e+a) + (f+a) \ge d'+e'+f' > d',e',f' \]which is a clear contradiction. Thus, there cannot exist such a prime $p$ which proves the desired result.
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kingu
220 posts
#32
Y by
Same solution as everyone else.
Storage
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john0512
4191 posts
#33
Y by
a bit silly, statement reminded me of usa tst 2021/1

Suppose FTSOC $a+b+c+d+e+f=p$.

Claim: For any integer $t$,
$$(a+t)(b+t)(c+t)+(d-t)(e-t)(f-t)\equiv 0\pmod p.$$
We have
$$(abc+def)+t(ab+ac+bc-de-df-ef)+t^2(a+b+c+d+e+f)+(t^3-t^3)\equiv 0\pmod{p}.$$
In particular, if $t=d$, then $p\mid (a+d)(b+d)(c+d)$. However, $a+d,b+d,c+d$ are all less than $p$, contradiction if $p$ is prime.

code golf because this is funny
This post has been edited 1 time. Last edited by john0512, Nov 15, 2024, 8:46 PM
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emi3.141592
71 posts
#34
Y by
Suppose that \(S\) is prime. Observe that
\[
a + b + c \equiv -d - e - f \pmod{S}
\]\[
abc \equiv -def \pmod{S}
\]\[
ab + bc + ca \equiv de + ef + fd \pmod{S}
\]By Vieta, we have that for every positive integer \(n\), the following relation holds:
\[
(n + a)(n + b)(n + c) \equiv (n - d)(n - e)(n - f) \pmod{S}.
\]By setting \(n = -a\), we obtain that \(S\) divides one of the numbers \(-a-d\), \(-a-e\), or \(-a-f\).

By symmetry, we can assume without loss of generality that \(S \mid -a-d\). In particular, we have \(-d \equiv a \pmod{S}\), i.e., \(S \mid d+a\).

However, this is a contradiction, since
\[
a + d < a + b + c + d + e + f = S.
\]
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AshAuktober
1013 posts
#35
Y by
Note that by Vieta, we have $S \mid P(x) = (x-a)(x-b)(x-c) - (x+d)(x+e)(x+f)$.
But putting $x = a$ we get $S$ cannot be prime, qed.
This post has been edited 1 time. Last edited by AshAuktober, Dec 20, 2024, 3:11 PM
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Mr.Sharkman
501 posts
#36
Y by
Assume for the sake of contradiction that $a+b+c+d+e+f$ is prime. Then, $a+b+c \equiv -d-e-f,$ $ab+bc+ca \equiv de+df+ef,$ and $abc \equiv -def,$ when taken modulo $a+b+c+d+e+f.$ Let $i_{1} \equiv -i,$ for $i \in \{a,b,c\}.$ It is clear that $a_{1}+b_{1}+c_{1} \equiv d+e+f,$ and $a_{1}b_{1}+a_{1}c_{1}+b_{1}c_{1} \equiv de+ef+df,$ and $a_{1}b_{1}c_{1} \equiv de+ef+df.$ So, $\{a_{1}, a_{2}, a_{3}\}, \{d, e, f\}$ are the solutions of some cubic modulo $p.$ But, by Lagrange's Theorem, this can only have at most $3$ solutions, so these sets are the same, in some order. So, $i+j \equiv 0,$ where $i \in \{a,b,c\},$ and $j \in \{d,e,f\}.$ But, then $i+j < a+b+c+d+e+f,$ so we have a contradiction.
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Perelman1000000
138 posts
#37
Y by
v_Enhance wrote:
Let $p$ be a fixed prime. We say a $6$-tuple $(a,b,c,d,e,f)$ of integers is good if $a+b+c+d+e+f = p$ and $p \mid abc+def$ and $p \mid ab+bc+ca-de-ef-fd$.

Claim: If $(a,b,c,d,e,f)$ is good then so is $(a+1, b+1, c+1, d-1, e-1, f-1)$.

Proof. Direct computation. $\blacksquare$

We claim there exists no good $6$-tuple of positive integers. If not, WLOG $f$ is the smallest of the six numbers. Then by repeating the claim, the tuple $(a+f, b+f, c+f, d-f, e-f, 0)$ is good. But now $p \mid (a+f)(b+f)(c+f)$ and $p > \max(a+f, b+f, c+f)$, contradiction.

When i first saw this problem on math class in 15 minutes i had exactly same solution as you :)
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Ilikeminecraft
674 posts
#38
Y by
Note that this implies that the polynomials $P(x) = (x + d)(x + e)(x + f), Q(x) = (x - a)(x - b)(x - c)$ are congruent modulo $S.$ Thus, $P(x) - Q(x) \equiv 0\pmod S.$ However, $P(a) - Q(a) = (a + d)(a + e)(a + f) \equiv 0 \pmod S.$ However, each of the factors are less than $S,$ which finishes.
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alexanderhamilton124
401 posts
#39
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Suppose, FTSOC, it is a prime. Let $a + b + c + d + e + f = p$. We have $p \mid ab + bc + ca - de - ef - df \implies p \mid abc + c^2(a + b) - dec - efc - dfc \implies p \mid - def + c^2(-c - d - e - f) - dec - efc - dfc \implies p \mid c^3 + c^2d + c^2e + c^2f + dec + efc + dfc + def = (c + d)(c + e)(c + f)$.

So, $p$ must divide one of them but all three are less than $a + b + c + d + e + f$, a contradiction. So, $S$ is composite.
This post has been edited 1 time. Last edited by alexanderhamilton124, Feb 4, 2025, 7:50 PM
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YaoAOPS
1541 posts
#40
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This means that $S$ divides the coefficients of
\[
	(x+a)(x+b)(x+c) - (x-d)(x-e)(x-f).
\]If $S$ was prime, then this means that $\{a, b, c\} \equiv \{-d, -e, -f\} \pmod{S}$, however then $p$ divides one of $a+d, a+e, a+f$, giving a contradiction by size.
This post has been edited 1 time. Last edited by YaoAOPS, Apr 23, 2025, 2:44 PM
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