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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 A Letter to MSM
Arr0w   23
N Sep 19, 2022 by scannose
Greetings.

I have seen many posts talking about commonly asked questions, such as finding the value of $0^0$, $\frac{1}{0}$,$\frac{0}{0}$, $\frac{\infty}{\infty}$, why $0.999...=1$ or even expressions of those terms combined as if that would make them defined. I have made this post to answer these questions once and for all, and I politely ask everyone to link this post to threads that are talking about this issue.
[list]
[*]Firstly, the case of $0^0$. It is usually regarded that $0^0=1$, not because this works numerically but because it is convenient to define it this way. You will see the convenience of defining other undefined things later on in this post.

[*]What about $\frac{\infty}{\infty}$? The issue here is that $\infty$ isn't even rigorously defined in this expression. What exactly do we mean by $\infty$? Unless the example in question is put in context in a formal manner, then we say that $\frac{\infty}{\infty}$ is meaningless.

[*]What about $\frac{1}{0}$? Suppose that $x=\frac{1}{0}$. Then we would have $x\cdot 0=0=1$, absurd. A more rigorous treatment of the idea is that $\lim_{x\to0}\frac{1}{x}$ does not exist in the first place, although you will see why in a calculus course. So the point is that $\frac{1}{0}$ is undefined.

[*]What about if $0.99999...=1$? An article from brilliant has a good explanation. Alternatively, you can just use a geometric series. Notice that
\begin{align*}
\sum_{n=1}^{\infty} \frac{9}{10^n}&=9\sum_{n=1}^{\infty}\frac{1}{10^n}=9\sum_{n=1}^{\infty}\biggr(\frac{1}{10}\biggr)^n=9\biggr(\frac{\frac{1}{10}}{1-\frac{1}{10}}\biggr)=9\biggr(\frac{\frac{1}{10}}{\frac{9}{10}}\biggr)=9\biggr(\frac{1}{9}\biggr)=\boxed{1}
\end{align*}
[*]What about $\frac{0}{0}$? Usually this is considered to be an indeterminate form, but I would also wager that this is also undefined.
[/list]
Hopefully all of these issues and their corollaries are finally put to rest. Cheers.

2nd EDIT (6/14/22): Since I originally posted this, it has since blown up so I will try to add additional information per the request of users in the thread below.

INDETERMINATE VS UNDEFINED

What makes something indeterminate? As you can see above, there are many things that are indeterminate. While definitions might vary slightly, it is the consensus that the following definition holds: A mathematical expression is be said to be indeterminate if it is not definitively or precisely determined. So how does this make, say, something like $0/0$ indeterminate? In analysis (the theory behind calculus and beyond), limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits. However, if the expression obtained after this substitution does not provide sufficient information to determine the original limit, then the expression is called an indeterminate form. For example, we could say that $0/0$ is an indeterminate form.

But we need to more specific, this is still ambiguous. An indeterminate form is a mathematical expression involving at most two of $0$, $1$ or $\infty$, obtained by applying the algebraic limit theorem (a theorem in analysis, look this up for details) in the process of attempting to determine a limit, which fails to restrict that limit to one specific value or infinity, and thus does not determine the limit being calculated. This is why it is called indeterminate. Some examples of indeterminate forms are
\[0/0, \infty/\infty, \infty-\infty, \infty \times 0\]etc etc. So what makes something undefined? In the broader scope, something being undefined refers to an expression which is not assigned an interpretation or a value. A function is said to be undefined for points outside its domain. For example, the function $f:\mathbb{R}^{+}\cup\{0\}\rightarrow\mathbb{R}$ given by the mapping $x\mapsto \sqrt{x}$ is undefined for $x<0$. On the other hand, $1/0$ is undefined because dividing by $0$ is not defined in arithmetic by definition. In other words, something is undefined when it is not defined in some mathematical context.

WHEN THE WATERS GET MUDDIED

So with this notion of indeterminate and undefined, things get convoluted. First of all, just because something is indeterminate does not mean it is not undefined. For example $0/0$ is considered both indeterminate and undefined (but in the context of a limit then it is considered in indeterminate form). Additionally, this notion of something being undefined also means that we can define it in some way. To rephrase, this means that technically, we can make something that is undefined to something that is defined as long as we define it. I'll show you what I mean.

One example of making something undefined into something defined is the extended real number line, which we define as
\[\overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,+\infty\}.\]So instead of treating infinity as an idea, we define infinity (positively and negatively, mind you) as actual numbers in the reals. The advantage of doing this is for two reasons. The first is because we can turn this thing into a totally ordered set. Specifically, we can let $-\infty\le a\le \infty$ for each $a\in\overline{\mathbb{R}}$ which means that via this order topology each subset has an infimum and supremum and $\overline{\mathbb{R}}$ is therefore compact. While this is nice from an analytic standpoint, extending the reals in this way can allow for interesting arithmetic! In $\overline{\mathbb{R}}$ it is perfectly OK to say that,
\begin{align*}
a + \infty = \infty + a & = \infty, & a & \neq -\infty \\
a - \infty = -\infty + a & = -\infty, & a & \neq \infty \\
a \cdot (\pm\infty) = \pm\infty \cdot a & = \pm\infty, & a & \in (0, +\infty] \\
a \cdot (\pm\infty) = \pm\infty \cdot a & = \mp\infty, & a & \in [-\infty, 0) \\
\frac{a}{\pm\infty} & = 0, & a & \in \mathbb{R} \\
\frac{\pm\infty}{a} & = \pm\infty, & a & \in (0, +\infty) \\
\frac{\pm\infty}{a} & = \mp\infty, & a & \in (-\infty, 0).
\end{align*}So addition, multiplication, and division are all defined nicely. However, notice that we have some indeterminate forms here which are also undefined,
\[\infty-\infty,\frac{\pm\infty}{\pm\infty},\frac{\pm\infty}{0},0\cdot \pm\infty.\]So while we define certain things, we also left others undefined/indeterminate in the process! However, in the context of measure theory it is common to define $\infty \times 0=0$ as greenturtle3141 noted below. I encourage to reread what he wrote, it's great stuff! As you may notice, though, dividing by $0$ is undefined still! Is there a place where it isn't? Kind of. To do this, we can extend the complex numbers! More formally, we can define this extension as
\[\mathbb{C}^*=\mathbb{C}\cup\{\tilde{\infty}\}\]which we call the Riemann Sphere (it actually forms a sphere, pretty cool right?). As a note, $\tilde{\infty}$ means complex infinity, since we are in the complex plane now. Here's the catch: division by $0$ is allowed here! In fact, we have
\[\frac{z}{0}=\tilde{\infty},\frac{z}{\tilde{\infty}}=0.\]where $\tilde{\infty}/\tilde{\infty}$ and $0/0$ are left undefined. We also have
\begin{align*}
z+\tilde{\infty}=\tilde{\infty}, \forall z\ne -\infty\\
z\times \tilde{\infty}=\tilde{\infty}, \forall z\ne 0
\end{align*}Furthermore, we actually have some nice properties with multiplication that we didn't have before. In $\mathbb{C}^*$ it holds that
\[\tilde{\infty}\times \tilde{\infty}=\tilde{\infty}\]but $\tilde{\infty}-\tilde{\infty}$ and $0\times \tilde{\infty}$ are left as undefined (unless there is an explicit need to change that somehow). One could define the projectively extended reals as we did with $\mathbb{C}^*$, by defining them as
\[{\widehat {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}.\]They behave in a similar way to the Riemann Sphere, with division by $0$ also being allowed with the same indeterminate forms (in addition to some other ones).
23 replies
Arr0w
Feb 11, 2022
scannose
Sep 19, 2022
k i Marathon Threads
LauraZed   0
Jul 2, 2019
Due to excessive spam and inappropriate posts, we have locked the Prealgebra and Beginning Algebra threads.

We will either unlock these threads once we've cleaned them up or start new ones, but for now, do not start new marathon threads for these subjects. Any new marathon threads started while this announcement is up will be immediately deleted.
0 replies
LauraZed
Jul 2, 2019
0 replies
k i Basic Forum Rules and Info (Read before posting)
jellymoop   368
N May 16, 2018 by harry1234
f (Reminder: Do not post Alcumus or class homework questions on this forum. Instructions below.) f
Welcome to the Middle School Math Forum! Please take a moment to familiarize yourself with the rules.

Overview:
[list]
[*] When you're posting a new topic with a math problem, give the topic a detailed title that includes the subject of the problem (not just "easy problem" or "nice problem")
[*] Stay on topic and be courteous.
[*] Hide solutions!
[*] If you see an inappropriate post in this forum, simply report the post and a moderator will deal with it. Don't make your own post telling people they're not following the rules - that usually just makes the issue worse.
[*] When you post a question that you need help solving, post what you've attempted so far and not just the question. We are here to learn from each other, not to do your homework. :P
[*] Avoid making posts just to thank someone - you can use the upvote function instead
[*] Don't make a new reply just to repeat yourself or comment on the quality of others' posts; instead, post when you have a new insight or question. You can also edit your post if it's the most recent and you want to add more information.
[*] Avoid bumping old posts.
[*] Use GameBot to post alcumus questions.
[*] If you need general MATHCOUNTS/math competition advice, check out the threads below.
[*] Don't post other users' real names.
[*] Advertisements are not allowed. You can advertise your forum on your profile with a link, on your blog, and on user-created forums that permit forum advertisements.
[/list]

Here are links to more detailed versions of the rules. These are from the older forums, so you can overlook "Classroom math/Competition math only" instructions.
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What belongs on this forum?
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Mathcounts and how to learn

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Marathons!
Relays might be a better way to describe it, but these threads definitely go the distance! One person starts off by posting a problem, and the next person comes up with a solution and a new problem for another user to solve. Here's some of the frequently active marathons running in this forum:
[list][*]Algebra
[*]Prealgebra
[*]Proofs
[*]Factoring
[*]Geometry
[*]Counting & Probability
[*]Number Theory[/list]
Some of these haven't received attention in a while, but these are the main ones for their respective subjects. Rather than starting a new marathon, please give the existing ones a shot first.

You can also view marathons via the Marathon tag.

Think this list is incomplete or needs changes? Let the mods know and we'll take a look.
368 replies
jellymoop
May 8, 2015
harry1234
May 16, 2018
Unique Rational Number Representation
abhisruta03   18
N 5 minutes ago by Reacheddreams
Source: ISI 2021 P3
Prove that every positive rational number can be expressed uniquely as a finite sum of the form $$a_1+\frac{a_2}{2!}+\frac{a_3}{3!}+\dots+\frac{a_n}{n!},$$where $a_n$ are integers such that $0 \leq a_n \leq n-1$ for all $n > 1$.
18 replies
abhisruta03
Jul 18, 2021
Reacheddreams
5 minutes ago
Math solution
Techno0-8   1
N 24 minutes ago by jasperE3
Solution
1 reply
Techno0-8
3 hours ago
jasperE3
24 minutes ago
D1027 : Super Schoof
Dattier   1
N 32 minutes ago by Dattier
Source: les dattes à Dattier
Let $p>11$ a prime number with $a=\text{card}\{(x,y) \in \mathbb Z/ p \mathbb Z: y^2=x^3+1\}$ and $b=\dfrac 1 {((p-1)/2)! \times ((p-1)/3)! \times ((p-1)/6)!} \mod p$ when $p \mod 3=1$.



Is it true that if $p \mod 3=1$ then $a \in \{b,p-b, \min\{b,p-b\}+p\}$ else $A=p$.
1 reply
Dattier
3 hours ago
Dattier
32 minutes ago
minimizing sum
gggzul   0
34 minutes ago
Let $x, y, z$ be real numbers such that $x^2+y^2+z^2=1$. Find
$$min\{12x-4y-3z\}.$$
0 replies
gggzul
34 minutes ago
0 replies
Geo problem
MathWinner121   10
N 2 hours ago by ethan2011
Using analytical geometry, prove that the sum of the squares of a parallelograms diagonals equal the sum of the squares of the side lengths.
10 replies
MathWinner121
Yesterday at 10:15 PM
ethan2011
2 hours ago
cool math probel i dont know how to solve
Soupboy0   6
N 3 hours ago by iwastedmyusername
Define the derangement sum $D(n)$ to be the sum of all permutations of the digits of $n$ such that no digit appears in its original spot. For example, $D(12) = 21$ and $D(123)=231+312=543$. What is $D(12345)$?
6 replies
Soupboy0
3 hours ago
iwastedmyusername
3 hours ago
Mass points question
Wesoar   2
N 4 hours ago by itsjeyanth
So I was working my way through mass points, and I found a rule that basically says:

"If transversal line EF crosses cevian AD in triangle ABC, you must split mass A into Mass ab and Mass ac. Could someone explain to me why this makes sense/why we couldn't just use mass A?
2 replies
Wesoar
Today at 2:27 AM
itsjeyanth
4 hours ago
Doubt on a math problem
AVY2024   16
N 5 hours ago by maxamc
Solve for x and y given that xy=923, x+y=84
16 replies
AVY2024
Apr 8, 2025
maxamc
5 hours ago
problemo
hashbrown2009   3
N 5 hours ago by maxamc
if x/(3^3+4^3) + y/(3^3+6^3) =1

and

x/(5^3+4^3) + y/(5^3+6^3) =1

find the 2 values of x and y.
3 replies
hashbrown2009
Mar 30, 2025
maxamc
5 hours ago
Geometry Help
ILOVECATS127   0
Today at 1:07 PM
Hello, I needed some help understanding this concept from Chapter 12, Geometry:

Points P, Q and R are on circle O such that

Arc PQ = 78°, arc QR = 123°, and arc PQR = 201°.

1. Find ∠PQO
2. Find ∠POR

Please help me understand HOW to solve these 2 problems.
0 replies
ILOVECATS127
Today at 1:07 PM
0 replies
prime numbers
wpdnjs   111
N Today at 10:54 AM by ostriches88
does anyone know how to quickly identify prime numbers?

thanks.
111 replies
wpdnjs
Oct 2, 2024
ostriches88
Today at 10:54 AM
How many ways to spell the word COUNT
yinglinwu   8
N Today at 3:51 AM by yinglinwu
This is the problem 2.771 from the book Competition Math For Middle School. The printed answer is 16, which is obtained by going through the triangle. I have no idea about what's going on there. Can anyone help explain please?

BTW, my understanding about a way to spell COUNT is simply to choose an order of those letters and write down them one by one at their designated locations, like writing down U first at the 3rd location, then O at the 2nd position, N at 4th, T at 5th, and C at 1st. Therefore, in total we should have 5! ways to spell the word.
8 replies
yinglinwu
May 4, 2025
yinglinwu
Today at 3:51 AM
Easy number theory
britishprobe17   31
N Today at 2:20 AM by martianrunner
The number of factors from 2024 that are greater than $\sqrt{2024}$ are
31 replies
britishprobe17
Oct 16, 2024
martianrunner
Today at 2:20 AM
max number of candies
orangefronted   12
N Today at 1:27 AM by iwastedmyusername
A store sells a strawberry flavoured candy for 1 dollar each. The store offers a promo where every 4 candy wrappers can be exchanged for one candy. If there is no limit to how many times you can exchange candy wrappers for candies, what is the maximum number of candies I can obtain with 100 dollars?
12 replies
orangefronted
Apr 3, 2025
iwastedmyusername
Today at 1:27 AM
problem 5
termas   73
N Apr 14, 2025 by zuat.e
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?
73 replies
termas
Jul 12, 2016
zuat.e
Apr 14, 2025
problem 5
G H J
G H BBookmark kLocked kLocked NReply
Source: IMO 2016
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jj_ca888
2726 posts
#71
Y by
I claim that our answer is $k = 2016$. Suppose we erase $\leq 2015$ factors. There will be $\geq 2017$ left, say $a$ and $b$ on the left and right hand side respectively. Since $a + b = 2017 > 2016$, there must be a factor $x - c$ on both sides of the equation, and this equation would then have root $c$, which is bad.

Then, consider the following construction for $k = 2016$, where all LHS terms are $0, 1 \pmod 4$ and all RHS terms are $2, 3 \pmod 4$:\[(x - 1)(x - 4)\ldots(x - 2013)(x - 2016) = (x - 2)(x - 3)\ldots (x - 2014)(x - 2015)\]which we claim to have no real roots. Note that for all $x$ in intervals of the form $[4i + 1, 4i + 2]$ and $[4i + 3, 4i + 4]$, the two sides much have different signs, so no such roots can exist in these intervals. Furthermore, if $x \leq 1$ or $x \geq 2016$ both sides of the equality are positive but the inequality $(x - (4j + 1))(x - (4j + 4)) < (x - (4j + 2))(x - (4j + 3))$ holds for all $0 \leq j \leq 503$ so taking the product of all these yields LHS $<$ RHS. Hence it remains to consider $x$ in intervals of the form $[4i + 2, 4i + 3]$ and $[4i, 4i + 1]$.

For these intervals $[4i + 2, 4i + 3]$ and $[4i, 4i + 1]$, we note the important identity that $(x - 4j)(x - (4j + 1)) > (x - (4j-1))(x - (4j + 2))$ for all $x \not \in$ intervals of the form $[4i + 2, 4i + 3]$ and $[4i, 4i + 1]$ and $j$. Along with noting that $0 < x - 2 < x - 1$ and $0 > x - 2015 > x - 2016$ we multiply and get LHS $\neq$ RHS, and we are done. We have considered all intervals covering all real numbers, and there are no real roots to this equation, as desired. $\blacksquare$
This post has been edited 1 time. Last edited by jj_ca888, Jul 26, 2020, 2:37 AM
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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
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megarnie
5606 posts
#73
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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.
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JAnatolGT_00
559 posts
#74
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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
470 posts
#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.
1712 posts
#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
5001 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. $\blacksquare$
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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$.\blacksquare$
This post has been edited 1 time. Last edited by Martin2001, Aug 9, 2024, 10:23 PM
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Mr.Sharkman
498 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
280 posts
#85
Y by
solution
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HamstPan38825
8859 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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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.
This post has been edited 1 time. Last edited by zuat.e, Apr 14, 2025, 12:07 AM
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