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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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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
Prove the inequality
Butterfly   0
7 minutes ago

Prove
$$x^2+y^2+7\ge 3(x+y)+\frac{9}{xy+2}~~(x,y>0).$$
0 replies
Butterfly
7 minutes ago
0 replies
Kaprekar Number
CSJL   5
N 8 minutes ago by Adywastaken
Source: 2025 Taiwan TST Round 1 Independent Study 2-N
Let $k$ be a positive integer. A positive integer $n$ is called a $k$-good number if it satisfies
the following two conditions:

1. $n$ has exactly $2k$ digits in decimal representation (it cannot have leading zeros).

2. If the first $k$ digits and the last $k$ digits of $n$ are considered as two separate $k$-digit
numbers (which may have leading zeros), the square of their sum is equal to $n$.

For example, $2025$ is a $2$-good number because
\[(20 + 25)^2 = 2025.\]Find all $3$-good numbers.
5 replies
CSJL
Mar 6, 2025
Adywastaken
8 minutes ago
Projective geometry
definite_denny   0
9 minutes ago
Source: IDK
Let ABC be a triangle and let DEF be the tangency point of incircirle with sides BC,CA,AB. Points P,Q are chosen on sides AB,AC such that PQ is parallel to BC and PQ is tangent to the incircle. Let M denote the midpoint of PQ. Let EF intersect BC at T. Prove that TM is tangent to the incircle
0 replies
definite_denny
9 minutes ago
0 replies
Problem 7 of RMO 2006 (Regional Mathematical Olympiad-India)
makar   11
N 15 minutes ago by SomeonecoolLovesMaths
Source: Functional Equation
Let $ X$ be the set of all positive integers greater than or equal to $ 8$ and let $ f: X\rightarrow X$ be a function such that $ f(x+y)=f(xy)$ for all $ x\ge 4, y\ge 4 .$ if $ f(8)=9$, determine $ f(9) .$
11 replies
makar
Sep 13, 2009
SomeonecoolLovesMaths
15 minutes ago
Hardest in ARO 2008
discredit   26
N 22 minutes ago by JARP091
Source: ARO 2008, Problem 11.8
In a chess tournament $ 2n+3$ players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least $ n$ next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.
26 replies
discredit
Jun 11, 2008
JARP091
22 minutes ago
Inequality
Kei0923   2
N an hour ago by Kei0923
Source: Own.
Let $k\leq 1$ be a fixed positive real number. Find the minimum possible value $M$ such that for any positive reals $a$, $b$, $c$, $d$, we have
$$\sqrt{\frac{ab}{(a+b)(b+c)}}+\sqrt{\frac{cd}{(c+d)(d+ka)}}\leq M.$$
2 replies
Kei0923
Jul 25, 2023
Kei0923
an hour ago
PAMO 2023 Problem 2
kerryberry   6
N an hour ago by justaguy_69
Source: 2023 Pan African Mathematics Olympiad Problem 2
Find all positive integers $m$ and $n$ with no common divisor greater than 1 such that $m^3 + n^3$ divides $m^2 + 20mn + n^2$. (Professor Yongjin Song)
6 replies
kerryberry
May 17, 2023
justaguy_69
an hour ago
My Unsolved Problem
ZeltaQN2008   0
an hour ago
Source: IDK
Given a positive integer \( m \) and \( a > 1 \). Prove that there always exists a positive integer \( n \) such that \( m \mid (a^n + n) \).

P/s: I can prove the problem if $m$ is a power of a prime number, but for arbitrary $m$ then well.....
0 replies
ZeltaQN2008
an hour ago
0 replies
Computing functions
BBNoDollar   3
N an hour ago by wh0nix
Let $f : [0, \infty) \to [0, \infty)$, $f(x) = \dfrac{ax + b}{cx + d}$, with $a, d \in (0, \infty)$, $b, c \in [0, \infty)$. Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\](For $n \in \mathbb{N}^*$ and $x \geq 0$, the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )
3 replies
BBNoDollar
May 21, 2025
wh0nix
an hour ago
Computing functions
BBNoDollar   8
N an hour ago by wh0nix
Let $f : [0, \infty) \to [0, \infty)$, $f(x) = \dfrac{ax + b}{cx + d}$, with $a, d \in (0, \infty)$, $b, c \in [0, \infty)$. Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\](For $n \in \mathbb{N}^*$ and $x \geq 0$, the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )
8 replies
BBNoDollar
May 18, 2025
wh0nix
an hour ago
Find the remainder
Jackson0423   1
N 2 hours ago by wh0nix

Find the remainder when
\[
\frac{5^{2000} - 1}{4}
\]is divided by \(64\).
1 reply
Jackson0423
3 hours ago
wh0nix
2 hours ago
IMO 2018 Problem 1
juckter   170
N 2 hours ago by Adywastaken
Let $\Gamma$ be the circumcircle of acute triangle $ABC$. Points $D$ and $E$ are on segments $AB$ and $AC$ respectively such that $AD = AE$. The perpendicular bisectors of $BD$ and $CE$ intersect minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line.

Proposed by Silouanos Brazitikos, Evangelos Psychas and Michael Sarantis, Greece
170 replies
juckter
Jul 9, 2018
Adywastaken
2 hours ago
Nice "if and only if" function problem
ICE_CNME_4   2
N 2 hours ago by wh0nix
Let $f : [0, \infty) \to [0, \infty)$, $f(x) = \dfrac{ax + b}{cx + d}$, with $a, d \in (0, \infty)$, $b, c \in [0, \infty)$. Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\](For $n \in \mathbb{N}^*$ and $x \geq 0$, the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )

Please do it at 9th grade level. Thank you!
2 replies
ICE_CNME_4
Yesterday at 7:23 PM
wh0nix
2 hours ago
Minimum of this fuction
persamaankuadrat   1
N 2 hours ago by alexheinis
Source: KTOM January 2020
If $x$ is a positive real number, find the minimum of the following expression

$$\lfloor x \rfloor + \frac{500}{\lceil x\rceil^{2}}$$
1 reply
persamaankuadrat
4 hours ago
alexheinis
2 hours ago
problem 5
termas   74
N May 14, 2025 by maromex
Source: IMO 2016
The equation
$$(x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016)$$is written on the board, with $2016$ linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these $4032$ linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?
74 replies
termas
Jul 12, 2016
maromex
May 14, 2025
problem 5
G H J
G H BBookmark kLocked kLocked NReply
Source: IMO 2016
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pad
1671 posts
#72 • 1 Y
Y by tigerzhang
We claim the minimum is $2016$. First note $k\ge 2016$ since erasing any $2015$ (or less) factors will leave a common linear factor on both sides.

We claim the following construction works, by having $p(x)$ have all the $0,1\mod 4$ roots and $q(x)$ the $2,3\mod 4$ roots:
\[
\begin{array}{cccc}
    p(x) := (x-1)(x-4)&(x-5)(x-8)\ \cdots \ (x-2013)(x-2016) \\
    q(x) := (x-2)(x-3)&(x-6)(x-7)\ \cdots \ (x-2014)(x-2015). 
\end{array}
\]We need to check that $p(x)-q(x)=0$ has no real solutions. We claim $p(x)<q(x)$ for all $x\in \mathbb{R}$.

Make cases on which of the following intervals $x$ lies in:
\[
\begin{array}{c|cccc|cccc|c|c}
 (-\infty,1) & (1,2)&(2,3) & (3,4) & (4,5)&(5,6)&(6,7)&(7,8)&(8,9)&\cdots &(2016,\infty) \\ 
 &a & b & c & d & a & b & c &d &\cdots 
\end{array}
\]It suffices to use only open intervals since on the boundaries, exactly one of $p(x)$, $q(x)$ is zero, so their difference is nonzero.

Case 1: $x\in (-\infty,1)$ or $x\in (2016,\infty)$. Note
\begin{align*}
    (x-1)(x-4) \quad &< \quad (x-2)(x-3) \\
    (x-5)(x-8) \quad &< \quad (x-6)(x-7) \\
    (x-9)(x-12) \quad &< \quad (x-10)(x-11) \\
    (x-13)(x-16) \quad &< \quad (x-14)(x-15) \\
    &\  \vdots
\end{align*}and that for $x<1$ and $x>2016$, both sides are positive for each row. Therefore, we can multiply the inequalities, and this implies $p(x) < q(x)$.

Case 2: $x\in$ interval of type $a$ or type $c$. Then an odd number of terms of $p(x)$ are negative, and an even number of terms of $q(x)$ are negative. (For example, when $x\in (5,6)$, $3$ of $p(x)$ and $2$ of $q(x)$ are negative.) Hence $p(x)<0<q(x)$.

Case 3: $x\in$ interval of type $b$ or type $d$. Note
\begin{align*}
    x-1 \quad &> \quad x-2 \\
    (x-4)(x-5) \quad &> \quad (x-3)(x-6) \\
    (x-8)(x-9) \quad &> \quad (x-7)(x-10) \\
    (x-12)(x-13) \quad &> \quad (x-11)(x-14) \\
    &\  \vdots \\ 
    (x-2012)(x-2013) \quad &> \quad (x-2011)(x-2014) \\
    2016-x \quad &> \quad 2015-x. 
\end{align*}The key is that for each row, both sides are positive. So multiplying gives $-p(x) > -q(x)$, i.e. $p(x)<q(x)$.

Remarks
This post has been edited 1 time. Last edited by pad, Dec 3, 2020, 8:10 PM
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megarnie
5610 posts
#73
Y by
We claim the answer is $\boxed{k=2016}$.

First part: Show that $k<2016$ doesn't work.
Suppose that $k<2016$. Then there is some integer $1\le a\le2016$ such that $x-a$ is on both sides of the equation after erasing, which implies that $a$ is a real solution.

Second part: Find a valid construction for $k=2016$.
The construction is \[(x-1)(x-4)(x-5)(x-8)(x-9)(x-12)\cdots(x-2013)(x-2016)=(x-2)(x-3)(x-6)(x-7)(x-10)(x-11)\cdots (x-2014)(x-2015).\]
Obviously there are no solutions for integers $1\le x\le 2016$.

Claim: There are no solutions for $x<1$ or $x>2016$.
Proof: We have \[(x-1)(x-4)=x^2-5x+4<(x-2)(x-3)=x^2-5x+6, (x-5)(x-8)<(x-6)(x-7),\]and so on until $(x-2013)(x-2016)<(x-2014)(x-2015)$.

Note that $(x-1)(x-4), (x-2)(x-3), (x-5)(x-8),\ldots$ are all positive, which implies the right-hand side is strictly greater than the left-hand side.


Now we only focus on $x$ such that $1<x<2016$.
Claim: $x$ such that $4j+1<x<4j+2$ don't have any solutions, where $j$ is a non-negative integer.
Proof: I will show that the sign (positive or negative) of the LHS is different from the sign of the RHS.
We can do this by cancelling any two terms such that they have the same sign and are on different sides. So we can cancel everything except $x-(4j+1)$ and $x-(4j+2)$, which obviously have a different sign.

Claim: $x$ such that $4j+3<x<4j+4$ don't have any solutions, where $j$ is a non-negative integer.
We proceed similarly as above and we get that the LHS and RHS have different signs (by cancelling out everything except $x-(4j+3)$ and $x-(4j+4)$).

Claim: $x$ such that $4j<x<4j+1$ don't have any solutions, where $j$ is a non-negative integer.
Proof: Note that $(x-1)(x-4), (x-2)(x-3), (x-5)(x-8),\ldots$ are all positive (because for each of them, both of the factors are either $\ge x-4j$ or $\le x-(4j+1)$), which implies that LHS<RHS.

Claim: $x$ such that $4j+2<x<4j+3$ don't have any solutions, where $j$ is a non-negative integer. This finishes the proof.
Remove the factors $(x-1), (x-2), (x-2015), (x-2016)$ (we will put them back later). So the equation becomes \[(x-4)(x-5)(x-8)(x-9)(x-12)(x-13)\cdots(x-2012)(x-2013)=(x-3)(x-6)(x-7)(x-10)\cdots(x-2011)(x-2014).\]
We have $(x-4)(x-5)>(x-3)(x-6), (x-8)(x-9)> (x-7)(x-10), \ldots, (x-2012)(x-2013)>(x-2011)(x-2014)$. All of these terms are positive (proof is similar to the above claim), which implies \[(x-4)(x-5)(x-8)(x-9)(x-12)(x-13)\cdots(x-2012)(x-2013)>(x-3)(x-6)(x-7)(x-10)\cdots(x-2011)(x-2014)>0.\]
Now, $(x-1)>(x-2)>0$ and $(x-2016)<(x-2015)<0$, which implies $(x-1)(x-2016)<(x-2)(x-2015)<0$.

So LHS=(small negative number) $\cdot$ (large positive number) = small negative number and RHS=(large negative number) $\cdot$ (small positive number)=large negative number, so LHS<RHS.
This post has been edited 2 times. Last edited by megarnie, Oct 25, 2023, 11:38 AM
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JAnatolGT_00
559 posts
#74
Y by
We claim the answer $k=2016.$ As the bound it's suffice to note, that no common factors should left in different sides.

As an example, we erase $1008$ factors from each side in such way, that exactly polynomials $$P(x)=\prod_{i=1}^{504}(x-4i+3)(x-4i),$$$$Q(x)=\prod_{i=1}^{504}(x-4i+1)(x-4i+2)$$remains on different sides of equality. We claim, that $\forall x\in \mathbb{R}:P(x)\neq Q(x),$ which suffices.

Case 1. $x\in (-\infty;1)\cup (1;+\infty)$
Notice that $0<(x-4k+3)(x-4k)<(x-4k+1)(x-4k+2)$ for $k=1,2,\dots,504.$ By multiplying we are done.

Case 2. $x\in \left \{ 1,2,\dots,2016 \right \}$
In that case exactly one of $P(x),Q(x)$ is zero, hence the conclusion.

In case $3$ we assume, that $x\in [1;2016]\backslash \mathbb{Z}.$ Here we consider two additional cases.

Case 3.1 $\lfloor x \rfloor$ is odd.
We easily obtain, that number of negative factors in $P(x),Q(x)$ are respectively odd and even, therefore $P(x)<0<Q(x).$

Case 3.2 $\lfloor x \rfloor$ is even.
For $k=1,2,\dots,503$ we have $(x-4k)(x-4k+3)>(x-4k+1)(x-4k+2)>0.$ Moreover $(x-1)>(x-2)>0$ and $2016-x>2015-x>0,$ so multiplying altogether we obtain $-P(x)>-Q(x).$
This post has been edited 3 times. Last edited by JAnatolGT_00, Oct 4, 2023, 3:24 AM
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AngeloChu
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#75
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First, let us prove the minimum $k$ is $k = 2016$.
If we want to remove terms so $(x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016)$ will have no solution, there obviously must be no repeats on the 2 sides.
If $k \le 2015$, there must be at least one overlapping term for the two sides, and all $k<2016$ don't work.
For $k=2016$, we can take $\prod_{a=1}^{504}(x-(4a-3))(x-4a)$ for the LHS and $\prod_{a=1}^{504}(x-(4a-2))(x-(4a-1))$ for the RHS.
If $x$ is not in the range of $[1, 2, 3, \dotsm, 2016]$, the RHS is always larger than the LHS
If $x$ is in the range of $[1, 2, 3, \dotsm, 2016]$, at most 1 of $\prod_{a=1}^{504}(x-(4a-3))(x-4a)$ and $\prod_{a=1}^{504}(x-(4a-2))(x-(4a-1))$ is 0. If one of the 2 equations is zero, it is obvious it is not equal. Otherwise, the LHS is always smaller than the RHS.
Therefore, the minimum $k$ that satisfies the requirements is $k=2016$
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awesomeming327.
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#76
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The answer is $2016$. Note that if we only remove $2015$ factors, then there exists a common factor on each side $x-a$, which implies that $a$ is a solution to the equation.

$~$
Now, we'll show that \[\prod_{k=1}^{504}{(x-(4k-3))(x-4k)}=\prod_{k=1}^{504}{(x-(4k-2))(x-(4k-1))}\]has no real solutions. Clearly, there are no integer solutions between $1$ and $2016$. The key idea is that \[(x-(4k-3))(x-4k)=(x-(4k-2))(x-(4k-1))-2 < (x-(4k-2))(x-(4k-1)).\]Suppose $x<1$, $x>2016$, or $x\in (4k,4k+1)$ then every factor on the LHS is positive and less than every factor on the RHS, so the LHS is less than the RHS. If $x\in (4k-3,4k-2)\cup (4k-1,4k)$ for some integer $k\in \{1,\dots ,504\}$ then every factor on the LHS is positive except one, so it is negative. Every factor on the RHS is still positive, so it is positive. Again LHS $<$ RHS.

$~$
It remains to see about $x\in (4k-2,4k-1)$. Note that $(x-(4k-3))(x-4k)$ and $(x-(4k-2))(x-(4k-1))$ are the only negative terms on each side. We have $(4k-x)(x-(4k-3))\ge 9 ((4k-1)-x)(x-(4k-2))$.

$~$
Now, let $s=(x-(4(k-i)-2))$ be from $4i$ to $4i+1$ then \[\frac{(x-(4(k-i)-2))(x-(4(k-i)-1))(x-(4(k+i)-2))(x-(4(k+i)-1))}{(x-(4(k-i)-3))(x-4(k-i))(x-(4(k+i)-3))(x-4(k+i))}=\frac{s^2}{s^2-4}\]and it is easy to see from here that no amount of this being multiplied together with distinct positive $i$'s will ever exceed $9$.
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Mogmog8
1080 posts
#77 • 1 Y
Y by centslordm
We claim the answer is $2016$. If we remove at most $2015$ factors, we will have a factor repeated on both sides by Pidgeonhole, which clearly yields a real solution to the equation.

For a construction, let the LHS only contain factors of the form $(x-a)$ where $a\equiv 1,4\pmod{4}$ and let the RHS contain the rest of the factors. Our equation is then \[\prod_{k=0}^{503}\left(x-(4k+1)\right)\left(x-(4k+4)\right)=\prod_{k=0}^{503}\left(x-(4k+2)\right)\left(x-(4k+3)\right)\]Notice \[\left(x-(4k+1)\right)\left(x-(4k+4)\right)+2=\left(x-(4k+2)\right)\left(x-(4k+3)\right)\quad(*)\]Hence,
  • If $x<1$ or $x>2016$ one side is always larger than the other.
  • If $x\in\{1,2,\dots,2016\}$ then one side is zero and the other side nonzero.
  • If $x\in (4k+1,4k+2)\cup (4k+3,4k+4)$ for any $k\in \{0,1,\dots,503\}$ then one side is negative and the other is positive.

We have one case remaining: suppose $x\in(4j+2,4j+3)$, ie $x=4j+2+\epsilon$ for $\epsilon\in(0,1)$, for some $j\in \{0,1,\dots,503\}$ is a solution to the equation. Consider \[P=\prod_{0\le k\le 503,k\neq j}\frac{\left(x-(4k+1)\right)\left(x-(4k+4)\right)}{\left(x-(4k+2)\right)\left(x-(4k+3)\right)}=\frac{\left(x-(4j+2)\right)\left(x-(4j+3)\right)}{\left(x-(4j+1)\right)\left(x-(4j+4)\right)}=\frac{\epsilon^2-\epsilon}{\epsilon^2-\epsilon-2}<\frac{1}{9}\]Let $f(k)$ be a term in $P$. Notice \[f(j-\ell)\ge \frac{(4\ell)\cdot (4\ell-1)-2}{(4\ell)\cdot (4\ell-1)}\]and \[f(j+\ell)\ge \frac{(4\ell)\cdot (4\ell-1)-2}{(4\ell)\cdot (4\ell-1)}\]by $(*)$. Notice $f$ increases as we get further from $j$. Thus,
\begin{align*}
P&=[f(j-1)f(j-2)f(j-3)\dots f(j-1008)][f(j+1)f(j+2)\dots f(j+1008)]
\\&\ge\left(\prod_{\ell=1}^{1008}\frac{(4\ell)\cdot (4\ell-1)-2}{(4\ell)\cdot (4\ell-1)}\right)^2
\end{align*}It suffices to show \[\prod_{\ell=1}^{1008}\left(1+\frac{2}{(4\ell)(4\ell-1)-2}\right)<3\]in order to obtain a contradiction. We know $\ln(x+1)\le x$ so
\begin{align*}
\ln\left(\prod_{\ell=1}^{1008}\left(1+\frac{2}{(4\ell)(4\ell-1)-2}\right)\right)&\le\sum_{\ell=1}^{1008}\frac{2}{(4\ell)(4\ell-1)-2}
\\&\le\sum_{\ell=1}^{1008}\frac{2}{(4\ell-1)^2}
\\&=\frac{2}{9}+2\sum_{\ell=1}^{1008}\frac{1}{(4\ell+3)^2}
\\&\le\frac{2}{9}+2\sum_{\ell=1}^{1008}\frac{1}{(4\ell)^2}
\\&\le\frac{2}{9}+2\frac{1}{16}\cdot\frac{\pi^2}{6}
\\&<\ln(3)
\end{align*}as desired. $\square$
This post has been edited 1 time. Last edited by Mogmog8, Sep 1, 2023, 1:55 PM
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signifance
140 posts
#79
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This proof seems shorter than others, can someone verify?

If the two sides share a factor they share a 0 so it doesn't work, so $k\ge2016$. Now take LHS as $\prod_{k=1}^{504}(x-(4k-3))(x-4k)$ and RHS as $\prod_{k=1}^{504}(x-(4k-2))(x-(4k-1))$. Comparing each respective term it's easily computed that $(x-(4k-3))(x-4k)<(x-(4k-2))(x-(4k-1))$ (the key being for ANY x), so if both sides are negative LHS>RHS, if both sides are positive LHS<RHS, and both sides can't be zero, done.

@below oh sorry ill try to get it fixed whoops my brain went out the window
This post has been edited 2 times. Last edited by signifance, Oct 4, 2023, 3:42 AM
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IAmTheHazard
5003 posts
#80 • 1 Y
Y by centslordm
signifance wrote:
Comparing each respective term it's easily computed that $(x-(4k-3))(x-4k)<(x-(4k-2))(x-(4k-1))$ (the key being for ANY x), so if both sides are negative LHS>RHS, if both sides are positive LHS<RHS, and both sides can't be zero, done.

$-2<-1$, $1<2$, but $-2\cdot 1=-1\cdot 2$.
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thdnder
198 posts
#81
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Here is a terrible solution I was able to find.

Answer: $2016$.

Note that for every $k$, one of $(x-k)$ must be erased, otherwise, $k$ is the root of the remaining equation. Thus we have to erase at least $2016$ linear factor. Now for $0 \le k \le 503$, erase factors $(x - (4k+2)), (x - (4k+3))$ from RHS and erase factors $(x - (4k+1)), (x - (4k+4))$ from LHS. Then we will prove that the equation $\prod_{k=0}^{503}(x - (4k+1))(x - (4k+4)) = \prod_{k=0}^{503}(x - (4k+2))(x - (4k+3))$ has no real solution.

If $x \not\in [1, 2016]$, then $(x - (4k+2))(x - (4k+3)) > (x - (4k+1))(x - (4k+4)) > 0$, hence $\prod_{k=0}^{503}(x - (4k+1))(x - (4k+4)) < \prod_{k=0}^{503}(x - (4k+2))(x - (4k+3))$. Thus we can assume $x \in [1, 2016]$. Also note that we may assume $x$ is not an integer.

Now we'll split 3 cases.

Case 1: $x \in [4s, 4s+1]$ for some $0 \le s \le 503$.

In this case, we have $(x - (4s+2))(x - (4s+3)) > (x - (4s+1))(x - (4s+4)) > 0$ for all $0 \le k \le 503$, so $\prod_{k=0}^{503}(x - (4k+1))(x - (4k+4)) < \prod_{k=0}^{503}(x - (4k+2))(x - (4k+3))$. $\blacksquare$

Case 2: $x \in [4s+1, 4s+2]$ or $x \in [4s+3, 4s+4]$ for some $0 \le s \le 503$.

Note that $(x - (4s+2))(x - (4s+3)) > 0 > (x - (4s+1))(x - (4s+4))$ and for all $s \neq k$, we have $(x - (4k+2))(x - (4k+3)) > (x - (4k+1))(x - (4k+4)) > 0$, thus $\prod_{k=0}^{503}(x - (4k+1))(x - (4k+4)) < 0 < \prod_{k=0}^{503}(x - (4k+2))(x - (4k+3))$. $\blacksquare$

Case 3: $x \in [4s+2, 4s+3]$ for some $0 \le s \le 503$.

For the sake of contradiction, assume $\prod_{k=0}^{503}(x - (4k+1))(x - (4k+4)) = \prod_{k=0}^{503}(x - (4k+2))(x - (4k+3))$. Then we get $\frac{(x - (4s+1))(x - (4s+4))}{(x - (4s+2))(x - (4s+3))} = \prod_{k \neq s}(\frac{(x - (4s+2))(x - (4s+3))}{(x - (4s+1))(x - (4s+4))})$, or equivalently, $(1 + \frac{2}{(x - (4s+2))((4s+3) - x)}) = \prod_{k \neq s} (1 + \frac{2}{(x - (4k+1))(x - (4k+4))})$. Taking $\log$ from 2 sides and using the inequality $x \ge \log(x+1)$, we get $\log(1 + \frac{2}{(x - (4s+2))((4s+3) - x)}) < \sum_{k \neq s} (\frac{2}{(x - (4k+1))(x - (4k+4))})$. Combined with the fact that $9 \le 1 + \frac{2}{(x - (4s+2))((4s+3) - x)}$, we get $\log(3) < \sum_{k \neq s} (\frac{1}{(x - (4k+1))(x - (4k+4))})$.

Note that $(\frac{1}{(x - (4k+1))(x - (4k+4))}) = \frac{1}{3}(\frac{1}{(x - (4k+4))} - \frac{1}{(x - (4k+1))})$, so $3\log(3) < \sum_{k \neq s} (\frac{1}{(x - (4k+4))} - \frac{1}{(x - (4k+1))}) < \sum_{k \neq s} (\frac{1}{(x - (4k+4))} - \frac{1}{(x - 4k)}) = \frac{1}{x - 4s} + \frac{1}{x - 2016} - \frac{1}{x} - \frac{1}{(x - (4s + 4))}) \le 3 < 3\log(3)$.

Thus we get an evident contradiction. $\blacksquare$

Hence is all case, the equation $\prod_{k=0}^{503}(x - (4k+1))(x - (4k+4)) = \prod_{k=0}^{503}(x - (4k+2))(x - (4k+3))$ has no real solution, as needed. $\blacksquare$
This post has been edited 1 time. Last edited by thdnder, May 4, 2024, 11:03 AM
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Martin2001
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#83
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The answer is $\boxed{2016}.$ We show that removing all with constant either $0,1 \pmod 4$ from the left side and $2,3 \pmod 4$ from the right side works. Note that for all $0 \leq k,$ we have that
$$(x-(4k+1))(x-(4k+4))<(x-(4k+2))(x-(4k+3)).$$We can simply multiply all the inequalities of this form together whenever $\lfloor x \rfloor \not\equiv 2 \pmod 4.$ However, when $\lfloor x \rfloor \equiv 2 \pmod 4,$ we have that for exactly one inequality the left and right sides are both negative, where the left is maximum the product of $9$ and the right side. Therefore we show that the the $$\frac{\prod \text{all right sides}}{\prod \text{all left sides}}<9.$$Note that $$\prod_{k \geq 0} \frac{(4k+2)(4k+3)}{(4k+1)(4k+4)}=\frac21 (\frac{3}{4} \cdot \frac{6}{5})\cdot (\frac{7}{8} \cdot \frac{10}{9}) <2.$$Note that we have two of these products because $$\frac{(4k+2)(4k+3)}{(4k+1)(4k+4)}<\frac{(4k+2+n)(4k+3+n)}{(4k+1+n)(4k+4+n)},$$so because $2^2<9,$ we're done$.\blacksquare$
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Mr.Sharkman
501 posts
#84
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The answer is $\boxed{2016}.$ Obviously, if we remove less than $2016$ terms, there will still be common factors on both sides.


For $2016,$ we make the following construction:
$$(x-1)(x-4)(x-5)(x-8) \cdots =  (x-2)(x-3)(x-6)(x-7) \cdots.$$If $x \ge 2016,$ or $x \le 0,$ then $(x-(4k+1))(x-(4k+4)) < (x-(4k+2))(x-(4k+3)),$ and all of these pairs of products are positive, so the left hand side is clear less than the right hand side. If $x \in (4k+1, 4k+4),$ but $x \not \in (4k+2, 4k+3),$ for some integer $0 \le k  \le 503,$ then the LHS is less than $0,$ while the right hand side is not.


Finally, if $x \in (4k+2, 4k+3),$ then
$$\frac{(x-(4k+2))(x-(4k+3))}{(x-(4k+1))(x-(4k+4))} \le \frac{1}{9}.$$Now, we get that everything else is at most
$$\log\left(\prod_{i=0}^{\max(2015-k, k-1)} \frac{(4i+2)(4i+3)}{(4i+1)(4i+4)}\right) = 2 \cdot \sum_{i=0}^{k} \log\left(1+\frac{2}{(4i+1)(4i+4)}\right)$$$$ \le \sum_{i=0}^{k} \frac{2}{(4i+1)(4i+4)} \le \frac{1}{8} \cdot \frac{\pi^{2}}{6} \le e^{2} < 9.$$Thus, this ratio cannot be $1.$
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InterLoop
279 posts
#85
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solution
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HamstPan38825
8868 posts
#86
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The answer is $k=2016$. Clearly we cannot have $k<2016$, otherwise at least one factor $x-k$ appears on both sides.

To see that $k=2016$ is achievable, consider the polynomial \[P(x) = \prod_{i=0}^{503}(x-4i-1)(x-4i-4) - \prod_{i=0}^{503} (x-4i-2)(x-4i-3).\]We will prove that $P(x) > 0$ for all real numbers $x$. Let $Q(x)$ and $R(x)$ denote the two products, such that $P(x) = Q(x) - R(x)$.

Claim: $Q(x) > R(x)$ when $x < 0$ or $x > 2016$.

Proof: Multiply the inequalities $(x-4i-1)(x-4i-4) > (x-4i-2)(x-4i-3)$ for each $0 \leq i \leq 503$. We are guaranteed that both products are positive, so this yields the result. $\blacksquare$

Now, observe that for $x \in [4k+1, 4k+2]$ or $x \in [4k+3, 4k+4]$ for any integer $0 \leq k \leq 503$, $Q$ is nonnegative and $R$ is nonpositive, with equality not holding simultaneously. When $x \in (4k, 4k+1)$, we can multiply the two inequalities
\begin{align*}
\frac{(x-2)(x-3) \cdots (x-(4k-2))(x-(4k-3))}{(x-1)(x-4) \cdots (x-(4k-3))(x-4k)} &> 1 \\
\frac{(x-(4k+2))(x-(4k+3)) \cdots (x-2014)(x-2015)}{(x-(4k+1))(x-(4k+4)) \cdots (x-2013)(x-2016)} &> 1
\end{align*}where each inequality follows again by multiplying the same inequalities in the claim.

The dirty part of the proof happens when $x \in (4k+2, 4k+3)$. Here, $Q(x) < 0$ and $R(x) < 0$, so we will prove that $\frac{Q(x)}{R(x)} < 1$ for these values of $x$. In particular, write
\begin{align*}
\log\left(\frac{(x-2)(x-3) \cdots (x-2014)(x-2015)}{(x-1)(x-4) \cdots (x-2013)(x-2016)}\right) &= \sum_{0 \leq i \leq 503, i \neq k} \log\left(\frac{(x-4i-2)(x-4i-3)}{(x-4i-1)(x-4i-4)}\right) + \log\left|(x-(4k+2))(x-(4k+3))\right| - \log\left|(x-(4k+1))(x-(4k+4))\right|\\
&< \sum_{0 \leq i \leq 503, i \neq k} \log\left(1+\frac 2{(x-4i-1)(x-4i-4)}\right) + \log|(x-(4k+2))(x-(4k+3))| \\
&< \sum_{0 \leq i \leq 503, i \neq k} \frac 2{(x-4i-1)(x-4i-4)} - \log 4 \\
&< 4 \sum_{i=0}^\infty \frac 1{(4i+2)(4i+5)} - \log 4 \\
&< 4 \left(\sum_{i=1}^\infty \frac 1{4i(4i+4)} + \frac 1{10}\right) - \log 4 \\
&< \frac{13}{20} - \log 4 < 0.
\end{align*}Hence we get the result.
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zuat.e
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#87
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We claim removing all $(x-i)$, where $i\equiv 2,3 \pmod{4}$ on one side and $i\equiv0,1 \pmod{4}$ on the other side works.

Consider $P(x)=(x-2)(x-3)(x-6)(x-7)\dots(x-2014)(x-2015)-(x-1)(x-4)(x-5)(x-8)\dots (x-2013)(x-2016)$. Cases $x\geq 2016$, $x\leq 1$, $x=1\leq r$ integer $\leq 2016$, $4k<x<4k+1$, $4k+1<x<4k+2$, $4k+3<4(k+1)$ all succumb to both terms being positive or the inequalities $(x-(4k+2))(x-(4k+3))>(x-(4k+1))(x-(4(k+1)))$, therefore we only study $4k+2<x<4k+3$.
Let $Q(x)=(x-2)(x-3)(x-6)(x-7)\dots(x-2014)(x-2015)$ and $R(x)=(x-1)(x-4)(x-5)(x-8)\dots (x-2013)(x-2016)$ and let $a_k=(x-(4k+1))(x-(4(k+1))))$, hence on $4k+2<x<4k+3$: $\mid\frac{Q(x)}{R(x)}\mid=\mid\frac{(x-(4k+2))(x-(4k+3))}{(x-(4k+1))(x-(4k+4))}\prod_{i=1}^{503}(1+\frac{2}{a_i})\mid$
$<\frac{1}{9}(1+\frac{2}{2\cdot 5})(1+\frac{2}{6\cdot 9})\dots(1+\frac{2}{(4k-2)(4k+1)})(1+\frac{2}{2\cdot 5})(1+\frac{2}{6\cdot 9})\dots(1+\frac{2}{(2010-4k)(2013-4k)})<1\longleftrightarrow \log [(1+\frac{2}{2\cdot 5})(1+\frac{2}{6\cdot 9})\dots(1+\frac{2}{(4k-2)(4k+1)})(1+\frac{2}{2\cdot 5})(1+\frac{2}{6\cdot 9})\dots(1+\frac{2}{(2010-4k)(2013-4k)})]<\frac{2}{2\cdot 5}\frac{2}{6\cdot 9}\dots \frac{2}{(4k-2)(4k+1)}\frac{2}{2\cdot 5}\frac{2}{6\cdot 9}\dots \frac{2}{(2010-4k)(2013-4k)}=\frac{2}{3}(\frac{1}{2}-\frac{1}{5}+\dots +\frac{1}{4k-2}-\frac{1}{4k+1}+\frac{1}{2}-\frac{1}{5}+\dots +\frac{1}{2010-4k}-\frac{1}{2013-4k})=\frac{2}{3}(1+(\frac{1}{6}-\frac{1}{5}+ \dots + \frac{1}{4k-2}-\frac{1}{4k-3}+\frac{1}{6}-\frac{1}{5}+ \dots + \frac{1}{2010-4k}-\frac{1}{2009-4k})-\frac{1}{4k+1}-\frac{1}{2013-4k})<\frac{2}{3}<\log 9$, which is clearly true and combined with $Q(x)<0$ and $R(x)>0$ with such $x$, the result follows.

Consequently, as $k<2016$, since otherwise some factor $(x-r)$ would repeat in both polynomials $Q$ and $R$ yielding a real solution, $k=2016$ is our solution.
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maromex
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#88
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For all positive integers $i \le 2016$, we have to eliminate at least one of each pair of $(x - i)$, because otherwise $x = i$ is real solution. This narrows down our search to all $k \ge 2016$, now let's prove $k = 2016$ works.

On the left side, let's erase $(x - 4i)$ and $(x - (4i - 3))$ for all positive integers $i \le 504$, and on the right side, let's erase $(x - (4i - 1))$ and $(x - (4i - 2))$ for all positive integers $i \le 504$. Now we have something equivalent to $$\prod\limits_{i=1}^{504} ((x - (4i - 1))(x - (4i - 2))) = \prod\limits_{i=1}^{504} ((x - 4i)(x - (4i - 3)))$$on the board.
We can rewrite these as differences of squares. We want to prove that $$\prod\limits_{i=1}^{504}((x - (4i - 1.5))^2 - 0.25) > \prod\limits_{i=1}^{504} ((x - (4i - 1.5)^2 - 2.25)$$In the case where the LHS is positive, the RHS can be positive or negative. If the RHS is negative, then the inequality is true. If the RHS is positive, notice that $(x - (4i - 1.5))^2 - 0.25 > (x - (4i - 1.5))^2 - 2.25$. Taking the product for all $1 \le i \le 504$ gives the desired.

Now we need to check the case where the LHS is nonpositive. Notice that, for real $r$, we have $r^2 - 0.25 \le 0$ if and only if $-0.5 \le r \le 0.5$. This interval has range $1$. The numbers of the form $x - (4i - 1.5)$ for positive integer $i \le 504$ are a distance of at least $4$ away from each other, so at most one of them can be in the interval $[-0.5, 0.5]$. In our case, exactly one of these numbers is in this interval, and therefore there is exactly one positive integer $j$ such that $$(x - (4j - 1.5))^2 - 0.25 \le 0.$$We define $j$ to be this positive integer, which depends on the value of $x$. It follows that $-0.5 < x - (4j - 1.5) < 0.5$.
The interval $[-1.5, 1.5]$ has size $3 < 4$, so by a similar argument, $j$ is the only positive integer such that $$(x - (4j - 1.5))^2 - 2.25 \le 0.$$This means that, in the expression we are trying to prove, $$\prod\limits_{i=1}^{504}((x - (4i - 1.5))^2 - 0.25) > \prod\limits_{i=1}^{504} ((x - (4i - 1.5)^2 - 2.25),$$the RHS is negative and the LHS is nonpositive. It would be sufficient to prove that $$\prod\limits_{i=1}^{504} \left( \dfrac{(x - (4i - 1.5))^2 - 0.25}{(x - (4i - 1.5))^2 - 2.25}\right) < 1.$$This can be rewritten as $$\prod\limits_{i=1}^{504} \left( 1 + \dfrac{2}{(x - (4i - 1.5))^2 - 2.25}\right) < 1.$$We have previously obtained $-0.5 \le x - (4j - 1.5) \le 0.5$. Let $s = x - (4j - 1.5)$. We can see that $$s^2 - 2.25 \ge -2.25.$$Then, this results in $$1 + \dfrac{2}{s^2 - 2.25} < \frac{1}{9}.$$Our goal is implied by $$\prod\limits_{i=1}^{j-1} \left( 1 + \dfrac{2}{(x - (4i - 1.5))^2 - 2.25}\right)\prod\limits_{i=j+1}^{504} \left(1 + \dfrac{2}{(x - (4i - 1.5))^2 - 2.25}\right) < 9.$$Furthermore, this is equivalent to $$\prod\limits_{i=1}^{j-1} \left(1 + \dfrac{2}{(s + 4i)^2 - 2.25}\right)\prod\limits_{i=1}^{504-j} \left(1 + \dfrac{2}{(s - 4i)^2 - 2.25} \right) < 9.$$Wow, I can't think of a "morally correct" way to prove this. Instead, let's do it this way:
Notice that $|s - 4| \ge 3.5$ and $|s + 4| \ge 3.5$. Using the Wallis product $$\prod\limits_{n=1}^\infty \left(1+ \dfrac{1}{4n^2 - 1}\right) = \frac{\pi}{2},$$we can see that $$\prod\limits_{i=1}^\infty \left( 1 + \dfrac{2}{(s - 4i))^2 - 2.25}\right) < \frac{\pi^2}{4}.$$Thus, our goal is implied by $\frac{\pi^4}{16} < 9$. Approximating $\pi < 3.2$ allows us to see that $(\frac{\pi}{2})^4 < (1.6)^4 = 6.5536 < 9$, and we are done.
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