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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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]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
Mathletes Corner Summer Camp 2025
GP102   0
28 minutes ago
Hello Everybody!
So most of math season has recently come to an end with mathcounts nats having finished over 2 weeks ago.

I'm sure a lot of you are planning to continue preparing this summer not only for competitions like MATHCOUNTS/AMC8 but also some relatively more advanced comps like AMC10/AMC12.
This summer I am planning to host a summer camp to help out with the preparation. I have attached the flyer to the camp below.

Credentials
0 replies
GP102
28 minutes ago
0 replies
Serbian selection contest for the IMO 2025 - P5
OgnjenTesic   3
N 35 minutes ago by atdaotlohbh
Source: Serbian selection contest for the IMO 2025
Determine the smallest positive real number $\alpha$ such that there exists a sequence of positive real numbers $(a_n)$, $n \in \mathbb{N}$, with the property that for every $n \in \mathbb{N}$ it holds that:
\[
        a_1 + \cdots + a_{n+1} < \alpha \cdot a_n.
    \]Proposed by Pavle Martinović
3 replies
OgnjenTesic
May 22, 2025
atdaotlohbh
35 minutes ago
Equation in integers with gcd and lcm
skellyrah   1
N 40 minutes ago by frost23
Find all integers \( x \) and \( y \) such that
\[
\frac{1}{\gcd(x, y)} + \frac{3}{xy} + \frac{y}{\operatorname{lcm}(x, y)} = y,
\]where \( \gcd(x, y) \) denotes the greatest common divisor of \( x \) and \( y \), and \( \operatorname{lcm}(x, y) \) denotes their least common multiple.
1 reply
skellyrah
an hour ago
frost23
40 minutes ago
Mathcounts Nationals Written Score Hub
DhruvJha   79
N 41 minutes ago by Elephant12
Put in your estimated score on the written nats comp on Sunday after the comp so we can get a good idea of the cdr quals are
79 replies
DhruvJha
May 10, 2025
Elephant12
41 minutes ago
set construction nt
top1vien   1
N an hour ago by alexheinis
Is there a set of 2025 positive integers $S$ that satisfies: for all different $a,b,c,d\in S$, we have $\gcd(ab+1000,cd+1000)=1$?
1 reply
top1vien
Today at 10:04 AM
alexheinis
an hour ago
9 Prodigy AoPS or Khanacadamy
ZMB038   64
N an hour ago by valenbb
Hey everyone just was wondering what everybody prefers? Try not to fight so this doesn't get locked!
64 replies
ZMB038
May 22, 2025
valenbb
an hour ago
A,P,Q lies on the Radical Axis
MrCriminal   3
N an hour ago by Blackbeam999
Source: Power Of a Point -Yufei Zhao #P8
Hint Needed
Let \(ABC\) be a triangle and let \(D\) and \(E\) be points on the sides \(AB\) and \(AC\), respectively , such that \(DE\) is parallel to \(BC\). Let \(P\) be any point interior to triangle \(ADE\) , and let \(F\) and \(G\) be the intersections of \(DE\) with the lines \(BP\) and \(CP\), respectively. Let \(Q\) be the second intersection points of the circumcircles of triangles \(PDG\) and \(PFE\) . Prove that the points \(A, P, \text{and } Q\) are collinear .
3 replies
MrCriminal
May 15, 2021
Blackbeam999
an hour ago
1434th post
vincentwant   17
N an hour ago by Sedro
This is my 1434th post. Here are some of my favorite (non-1434-related) problems that I wrote for various contests over the past few years. A $\star$ indicates my favorites.

-----

A function $f(x)$ is defined over the positive integers as follows: $f(1)=0$, $f(p^n)=n$ for $p$ prime, and for all relatively prime positive integers $a$ and $b$, $f(ab)=f(a)f(b)+f(a)+f(b)$. If $N$ is the smallest positive integer such that $f(N)=20$, find the units digit of $N$.

$\textbf{(A)} ~0 \qquad\textbf{(B)} ~2 \qquad\textbf{(C)} ~4 \qquad\textbf{(D)} ~6 \qquad\textbf{(E)} ~8$

(2023 VMAMC 10 #23)

-----

$\star$ If convex quadrilateral $ABCD$ satisfies $AB=6$, $\angle CAB=30^{\circ}$, $\angle CDB=60^{\circ}$, $\angle BCD-\angle ABC=30^{\circ}$, and $CD=1$, what is the value of $BC^2$? Express your answer in simplest radical form.

(2024 STNUOCHTAM Sprint #30)

-----

Let $N=1!\cdot2!\cdot4!\cdot8!\cdots(2^{1000})!$ and $d$ be the greatest odd divisor of $N$. Let $f(n)$ for even $n$ denote the product of every odd positive integer less than $n$. If $d=f(a_1)^{b_1}f(a_2)^{b_2}f(a_3)^{b_3}\cdots f(a_k)^{b_k}$ for positive integers $a_1,a_2,\dots,a_k$ and $b_1,b_2,\dots,b_k$ where $k$ is minimized, find the number of divisors of $a_1a_2a_3\cdots a_k{}$.

(2024 STNUOCHTAM Sprint #29)

-----

$\star$ There exists exactly one positive real number $k$ such that the graph of the equation $\frac{x^3+y^3}{xy-k}=k$ consists of a line and a point not on the line. The distance from the point to the line can be expressed as $\frac{a}{\sqrt{b}}$, where $a$ and $b$ are positive integers and $b$ is not divisible by any square greater than $1$. Find $a+b$.

(2023-2024 WOOT AIME 3 #12)

-----

Let $a\odot b=\frac{4ab}{a+b+2\sqrt{ab}}$. If $x,y,z,k$ are positive real numbers such that $x\odot y=4z$, $x\odot z=\frac{9}{4}y$, and $y\odot z=kx$, find $k$. Express your answer as a common fraction.

(2024 STNUOCHTAM Sprint #26)

-----

$\star$ Let $ABCDE$ be a convex pentagon satisfying $AB = BC = CD = DE$, $\angle ABC = \angle CDE$, $\angle EAB = \angle AED = \frac{1}{2}\angle BCD$. Let $X$ be the intersection of lines $AB$ and $CD$. If $\triangle BCX$ has a perimeter of $18$ and an area of $11$, find the area of $ABCDE$.

$\textbf{(A)} ~136 \qquad\textbf{(B)} ~137 \qquad\textbf{(C)} ~138 \qquad\textbf{(D)} ~139 \qquad\textbf{(E)} ~140$

(2024 TMC AMC 10 #25)

-----

$\star$ Let $\triangle{ABC}$ be an acute scalene triangle with longest side $AC$. Let $O$ be the circumcenter of $\triangle{ABC}$. Points $X$ and $Y$ are chosen on $AC$ such that $OX\perp BC$ and $OY\perp AB$. If $AX=7$, $XY=8$, and $YC=9$, the area of the circumcircle of $\triangle{ABC}$ can be expressed as $k\pi$. Find $k$.

$\textbf{(A)} ~145\qquad \textbf{(B)} ~148\qquad \textbf{(C)} ~153\qquad \textbf{(D)} ~157\qquad \textbf{(E)} ~162\qquad$

(2024 XCMC 10 #23)

-----

Find the sum of the digits of the unique prime number $p\geq 31$ such that $$\binom{p^2-1}{846}+\binom{p^2-2}{846}$$is divisible by $p$.

$\textbf{(A)} ~7\qquad \textbf{(B)} ~8\qquad \textbf{(C)} ~10\qquad \textbf{(D)} ~11\qquad \textbf{(E)} ~13\qquad$

(2024 XCMC 10 #24)

-----

$\star$ Alex has a $4$ by $4$ grid of squares. Let $N$ be the number of ways that Alex can fill out each square with one of the letters $A$, $B$, $C$, or $D$ such that in every row and column, the number of $A$'s and $B$'s are the same, and the number of $C$'s and $D$'s are the same. (For example, a row with squares labeled $BDAC$ or $DCCD$ is valid, while a row with squares labeled $ACDA$ or $CBCB$ is not valid.) Find the remainder when $N$ is divided by $7$.

$\textbf{(A)} ~0\qquad \textbf{(B)} ~1\qquad \textbf{(C)} ~3\qquad \textbf{(D)} ~4\qquad \textbf{(E)} ~6\qquad$

(2024 XCMC 10 #25)

-----

How many ways are there to divide a $4$ by $4$ grid of squares along the gridlines into two or more pieces such that if three pieces meet at a point $P$, then there are actually four pieces with a vertex at $P$? An example is shown below.

IMAGE

(2025 ELMOCOUNTS CDR #19)

-----

How many ways are there to label each cell of a 4-by-4 grid of squares with either 1, 2, 3, or 4 such that no two adjacent cells have the same label and no two adjacent cells have labels that sum to 5?

(2025 ELMOCOUNTS Sprint #20)

-----

Let $a,b,c,d,e,f$ be real numbers satisfying the system of equations
$$\begin{cases}
a+b+c+d+e+f=1 \\
a+2b+3c+4d+5e+6f=2 \\
a+3b+6c+10d+15e+21f=4 \\
a+4b+10c+20d+35e+56f=8 \\
a+5b+15c+35d+70e+126f=16 \\
a+6b+21c+56d+126e+252f=32. \\
\end{cases}$$What is the value of $a+3b+9c+27d+81e+243f$?

(2025 ELMOCOUNTS Sprint #26)

-----

There are seven students at a camp. There are seven classes available and each student chooses some of the classes to take. Every student must choose at least two classes. How many ways are there for the students to choose the classes such that each pair of classes has exactly one student in common?

(2025 ELMOCOUNTS Team #8)

-----

$\star$ In $\triangle{ABC}$, the incircle is tangent to $\overline{BC}$ at $D$, and $E$ is the reflection of $D$ across the midpoint of $\overline{BC}$. Suppose that the inradii of $\triangle ABE$ and $\triangle ACE$ are $4$ and $11$ respectively, and the distance between their incenters is $25$. What is the inradius of $\triangle{ABC}$? Express your answer as a common fraction.

(2025 ELMOCOUNTS Team #10)

-----

Let $n$ be a positive integer and let $S$ be the set of all $n$-tuples of $0$'s and $1$'s. Two elements of $S$ are said to be neighboring if and only if they differ in only one coordinate. Bob colors the elements of $S$ red and blue such that each blue $n$-tuple is neighboring to exactly two red $n$-tuples and no two red $n$-tuples neighbor each other. If $n>100$, find the least possible value of $n$.

(2025 ELMOCOUNTS Target #6)
17 replies
1 viewing
vincentwant
Sunday at 5:09 PM
Sedro
an hour ago
Hellopoo
Bet667   8
N an hour ago by Aiden-1089
In how many ways can you tile a $3 \cdot n$ rectangle with $2 \cdot 1$ dominoes?
8 replies
Bet667
Yesterday at 11:22 AM
Aiden-1089
an hour ago
Clifford's chain of circles, concurrent Simson lines
kosmonauten3114   0
an hour ago
Source: My own, but most likely already known
Let $C_1$, $C_2$, $C_3$, $C_4$ be circles having a common point $P$. Denote by $P_{ij}$ the intersection point, other than $P$, of $C_i$ and $C_j$ ($\{i,j\} \in \{1,2,3,4\},i<j$). Let $Q$ be the common point of the 4 circles $\odot(P_{23}P_{24}P_{34})$, $\odot(P_{13}P_{14}P_{34})$, $\odot(P_{12}P_{14}P_{24})$, $\odot(P_{12}P_{13}P_{23})$.
Let $\ell_1$ be the Simson line of $P$ with respect to $\triangle{P_{12}P_{13}P_{14}}$. Define $\ell_2$, $\ell_3$, $\ell_4$ cyclically.
Let $\ell_1'$ be the Simson line of $Q$ with respect to $\triangle{P_{23}P_{24}P_{34}}$. Define $\ell_2'$, $\ell_3'$, $\ell_4'$ cyclically.
Prove that if $\ell_1$, $\ell_2$, $\ell_3$, $\ell_4$ are concurrent, then, $\ell_1'$, $\ell_2'$, $\ell_3'$, $\ell_4'$ are also concurrent.
0 replies
kosmonauten3114
an hour ago
0 replies
Inequality
srnjbr   4
N an hour ago by sqing
For real numbers a, b, c and d that a+d=b+c prove the following:
(a-b)(c-d)+(a-c)(b-d)+(d-a)(b-c)>=0
4 replies
srnjbr
Oct 30, 2024
sqing
an hour ago
Solution of an interesting inequality
imnotgoodatmathsorry   1
N an hour ago by imnotgoodatmathsorry
Source: @Alphabetamath on Facebook
$\text{The problem:}$
1 reply
imnotgoodatmathsorry
an hour ago
imnotgoodatmathsorry
an hour ago
Similarity of two triangles!
ariopro1387   1
N an hour ago by Mahdi_Mashayekhi
Source: Iran Team selection test 2025 - P5
In a scalene triangle $ABC$, $D$ is the point of tangency of the incircle with the side $BC$. Points $T_B$ and $T_C$ are the intersections of the angle bisectors of $\angle ABC$ and $\angle ACB$ with the circumcircle of $ABC$, respectively. Let $X_B$ be the antipodal point of $A$ in the circumcircle of $ACD$, and let $X_C$ be the antipodal point of $A$ in the circumcircle of $ABD$.
Prove that triangles $B T_C X_C$ and $C T_B X_B$ are similar.
1 reply
ariopro1387
Today at 10:34 AM
Mahdi_Mashayekhi
an hour ago
trigonometric inequality
MATH1945   7
N 2 hours ago by sqing
Source: ?
In triangle $ABC$, prove that $$sin^2(A)+sin^2(B)+sin^2(C) \leq \frac{9}{4}$$
7 replies
MATH1945
May 26, 2016
sqing
2 hours ago
two solutions
τρικλινο   10
N Apr 14, 2025 by Safal
in a book:CORE MATHS for A-LEVEL ON PAGE 41 i found the following


1st solution


$x^2-5x=0$



$ x(x-5)=0$



hence x=0 or x=5



2nd solution



$x^2-5x=0$

$x-5=0$ dividing by x



hence the solution x=0 has been lost



is the above correct?
10 replies
τρικλινο
Apr 12, 2025
Safal
Apr 14, 2025
two solutions
G H J
G H BBookmark kLocked kLocked NReply
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τρικλινο
502 posts
#1
Y by
in a book:CORE MATHS for A-LEVEL ON PAGE 41 i found the following


1st solution


$x^2-5x=0$



$ x(x-5)=0$



hence x=0 or x=5



2nd solution



$x^2-5x=0$

$x-5=0$ dividing by x



hence the solution x=0 has been lost



is the above correct?
Z K Y
The post below has been deleted. Click to close.
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Safal
171 posts
#2
Y by
2nd Solution is basically wrong. Why? Here is the explanation.

$$x(x-5)=0$$Then there are two cases either $x=0$ or $x\neq 0$.When we are admitting the case $x=0$ we cannot divide by $0$. So, in the case we apply divison by $x$ then $x\neq 0$ is a solid prerequisite to do so.Thus, $x-5=0$ from $x(x-5)=0$ we must take the assumption in hand that $x\neq 0$. For example take the extension of the same problem in $\mathbb{F}_5$ then the same problem reads $$x^2=0$$,implying only one solution with optimistic repetation of root $0$, two times that is the multiplicity of $0$ in $x^2$. Thankfully, we are lucky enough that we are in the field of $\text{char}$ $0$.The reason is that, the book you mention was a book for below 10std (as far as I remember it is below 10th std) students, where prerequisite assumption is that ,we should work on field of $\text{char}$ $0$.
This post has been edited 10 times. Last edited by Safal, Apr 12, 2025, 7:52 PM
Z K Y
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τρικλινο
502 posts
#3
Y by
Safal wrote:
2nd Solution is basically wrong. Why? Here is the explanation.

$$x(x-5)=0$$Then there are two cases either $x=0$ or $x\neq 0$.When we are admitting the case $x=0$ we cannot divide by $0$. So, in the case we apply divison by $x$ then $x\neq 0$ is a solid prerequisite to do so.Thus, $x-5=0$ from $x(x-5)=0$ we must take the assumption in hand that $x\neq 0$. For example take the extension of the same problem in $\mathbb{F}_5$ then the same problem reads $$x^2=0$$,implying only one solution with optimistic repetation of root $0$, two times that is the multiplicity of $0$ in $x^2$. Thankfully, we are lucky enough that we are in the field of $\text{char}$ $0$.The reason is that, the book you mention was a book for below 10std (as far as I remember it is below 10th std) students, where prerequisite assumption is that ,we should work on field of $\text{char}$ $0$.

so how do we get x=0 or x=5 ,since we assumed $x\neq 0$.
Z K Y
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maxamc
585 posts
#4
Y by
Move this to MSM, reported
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Safal
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#5
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τρικλινο wrote:
Safal wrote:
2nd Solution is basically wrong. Why? Here is the explanation.

$$x(x-5)=0$$Then there are two cases either $x=0$ or $x\neq 0$.When we are admitting the case $x=0$ we cannot divide by $0$. So, in the case we apply divison by $x$ then $x\neq 0$ is a solid prerequisite to do so.Thus, $x-5=0$ from $x(x-5)=0$ we must take the assumption in hand that $x\neq 0$. For example take the extension of the same problem in $\mathbb{F}_5$ then the same problem reads $$x^2=0$$,implying only one solution with optimistic repetation of root $0$, two times that is the multiplicity of $0$ in $x^2$. Thankfully, we are lucky enough that we are in the field of $\text{char}$ $0$.The reason is that, the book you mention was a book for below 10std (as far as I remember it is below 10th std) students, where prerequisite assumption is that ,we should work on field of $\text{char}$ $0$.

so how do we get x=0 or x=5 ,since we assumed $x\neq 0$.

If you read carefully I haven't said that we cannot get $x=0$.The assumption whenever $x\neq 0$ we get $x=5$ else we get the case $x=0$.I can explain you more but the fact is I cannot use argument of field theory to explain it in total details.The reason why it's actually the case lies in field theory logics.
This post has been edited 1 time. Last edited by Safal, Apr 13, 2025, 6:23 AM
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τρικλινο
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#6
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what is field theory logic.
IS the logic that suports the development of field theory?
THIS post should not be moved to Middle School Math
Because in the 2nd solution we have the answer : x different than zero this implies x=5
And according to logic this is equivelant to x=0 or x=5.Hence no solution is lost as the book claims
There for it should be removed back to at least college algebra although i doupt if even there anyone knew of that solution
WEmake use of the law of propositional calculus: ¬p implies q this is equivelant to p or q
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Safal
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#7
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"If you judge a fish because it cannot climb a tree , it will be foolish"-Unknown.

I am not commenting further in this post thank you.

Thanks to aops for moving it to MSM and I support it.
This post has been edited 1 time. Last edited by Safal, Apr 13, 2025, 4:51 PM
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SpeedCuber7
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#8
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@triklino dude that's an awesome username i didn't even know greek letters were allowed lol
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sadas123
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Proof: $x^2-5x=0$ Which means that the roots of this equation have to be real so we can use an method that is lost in the darkness called factoring. We can factor out the $x$ from each of the terms on the left hand side and get $x(x-5)=0$ which with more logic we can find that the possible outcomes is that if the Parantheses are 0 or the x=0 so first we can subsitute a value of x into that to make the value 0 so we get that x=5 and we finally get the solutions of $x=5$ and $x=0$ and to wrap up our proof we can prove that factoring is the best way to go because with quadratics you would only find 2 possibiliteis or 1 depending on the plus minus. And the other thing is that if you divide by x and just solve it with algebra then you will only get the solution of 5. Thus, proving that factoring is the best method out of all of them. We can use the remainder theorem to prove this which can be done easily. $\blacksquare$
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τρικλινο
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#10
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please read the previous posts


The question here is not which is the best method to solve the problem,but if we lose a solution if we solve the problem by dividing the equation by a non zero x
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Safal
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#13
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Dear Triklino , I am high yesterday I am sorry for being rude. Can you please explain your Question properly that is what the thing you exactly want to know. According to what I understand, you wanted to know why in 2nd Solution $x=0$ is lost? right. Well forget about field theory and all that, Let me explain it in layman's term what is actually happening. In second solution , the solution $x=0$ is not actually lost. The reason we are getting $x=5$ but not $x=0$ is beacuse when we are dividing by $x$ we making an assumption that $x\neq 0$ and since we are making this assumption the solution $x=0$ is lost. For example when we divide by $x-5$ the solution $x=5$ is lost why $x-5=y(say)$ and we are assuming $y\neq 0$ which is equivalent to $x\neq 5$. Now divison by zero is not possible which is not at all very easy to explain. Now $x=5$ and $x=0$ is not possible at the same time. Thus either $x=0$ or $x=5$.

Now why I am talking about field beacuse $0$ and $5$ can be same when we are in a field of $\text{char} 5$. If you are avoid knowing what is field that's perfectly fine to learn later, but just in layman's term note that $0=5$ is possible in finite fields of charecteristic $5$.

Well I like sour grapes but fox will be happy if he clear your doubt thanks.

I hope it is clear now. If it is not then text me in dm.
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