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Questions tagged [alternative-proof]

If you already have a proof for some result but want to ask for a different proof (using different methods).

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Simple Direct Proof of Nakayama's Lemma

Is there a simple direct proof of (this version of) Nakayama's Lemma? By simplicity, I am mostly referring to the concepts required to understand the proof, not how terse the proof is. I have no doubt ...
Greg Nisbet's user avatar
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-1 votes
1 answer
71 views

Detailed Proof of Proposition 2.12 a) from Diamond, Darmon, Taylor, "Fermat's Last Theorem."

I seek a highly detailed proof of this statement in the split multiplicative reduction case. The result can be found on page 57 here. That is, if $E/\mathbb Q$ has split multiplicative reduction at $p$...
Johnny Apple's user avatar
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3 votes
1 answer
180 views

Is there a brute force proof that the first-order characterization of a Jacobson radical is an ideal?

Let rings be commutative and unital. Is there a brute force proof that the first-order characterization of a Jacobson radical is an ideal? Basically, for context, I'm having some trouble wrapping my ...
Greg Nisbet's user avatar
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2 votes
2 answers
64 views

Every countably infinite linear order has a copy of $\omega$ or $\omega^{op}$

Every countably infinite linear order $L$ has a copy of $\omega$ or $\omega^{op}$. I'm interested in different kinds of proofs of this fact. One I came up with is: pick $x_0 \in L$. Wlog $[x_0, +\...
Carla_'s user avatar
  • 349
3 votes
1 answer
109 views

Understanding how group cohomology classifies extensions using the derived functor point of view

I am rereading some material about group extensions, in particular because I needed to recall the formula $$H^2(G;A)\cong \mathcal{E}(G;A).$$ We have that $G$ is some group acting on an abelian group $...
DevVorb's user avatar
  • 1,495
3 votes
0 answers
57 views

Problem 237 "Mathematical Quickies:270 Stimulating Problems with Solutions" Particle Movement

I was solving "Mathematical Quickies:270 Stimulating Problems with Solutions" when I came across a very peculiar question (Problem 237): A particle moves in a straight line starting from ...
Cognoscenti's user avatar
3 votes
1 answer
97 views

Real roots of $x^4+ax^3+bx^2+cx+1=0,$ when $a,b,c$ are real and $b\ge\frac{a^2+c^2}{4}$

For real $a,b,c$ and $$b \ge \frac{a^2+c^2}{4}\tag{*}$$ the given polynomial equation $$f(x)=x^4+ax^3+bx^2+cx+1=0\tag{**}$$ can be re-written as $$f(x)=(x^2+ax/2)^2+(b-a^2/4-c^2/4)x^2+(cx/2+1)^2\ge 0\...
Z Ahmed's user avatar
  • 43.6k
0 votes
1 answer
48 views

Understanding a hint from Coxeter

The following problem posed in Coxeter and Greitzer's Geometry Revisited is readily proved by angle chasing (stick angles at $A,B$, chase). I am seeking the proof indicated by the author's hint on ...
RobinSparrow's user avatar
  • 2,042
9 votes
2 answers
106 views

Proving that $b+\frac{1}{a(b-a)}\ge 3$ , if $b>a>0$

Prove that $b+\frac{1}{a(b-a)}\ge 3$ , if $b>a>0$ My Attempt : We can see that $b+\frac{1}{a(b-a)}\ge 3$ if $b>a>0$ By using AM-GM inequality of 3 variables as $$\frac{a+(b-a)+\frac{1}{a(b-...
Z Ahmed's user avatar
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2 votes
1 answer
32 views

Is the following method for proving density of irrational numbers in real numbers without using rational numbers density in real numbers rigorous?

The motivation for this question is: I told my friend to use: $\forall x_{1}, x_{2} \in \mathbb{R}, x_{1} < x_{2}, \exists r \in \mathbb{Q}: x_{1} < r <x_{2}.$ To prove: $\forall x_{1}, x_{2} ...
Mahmoud albahar's user avatar
6 votes
3 answers
186 views

Prove that any sequence of five distinct integers must contain a 3-chain

This task is from MIT OpenCourse Mathematics for CS 2010 course, problem set 2, exercise 1(d). I am aware that this question has already been asked several times previously on this platform. Yet, the ...
Lina's user avatar
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1 vote
1 answer
79 views

Prove $f'(r_{+}) + f'(r_{-}) < 0$ for roots of $1 - \frac{M}{r^2} + \frac{Q}{r^4} - r^2 = 0$

Problem. Let $M, Q > 0$ be given. Let $$f(r) := 1 - \frac{M}{r^2} + \frac{Q}{r^4} - r^2, \quad r > 0.$$ If $f$ has three distinct positive real roots $r_c > r_{+} > r_{-} > 0$, prove ...
River Li's user avatar
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5 votes
1 answer
105 views

Is this proof correct for Putnam 1986 B4?

I have been practicing on some old Putnam questions, and I attempted to solve this 1986 Putnam B4: For a positive real number $r$ define $G(r)$ to be the minimum value of $|r-\sqrt{m^2+n^2}|$ for all ...
Riccardo Caiulo's user avatar
0 votes
2 answers
48 views

Existence of certain function satisfying certain conditions

I want to show that there does not exist $f ∈ C([0, 1], R)$ satisfying the following two conditions: (i) $\int_{0}^{1} f(x) dx = 1$; (ii) $\lim_{n\to\infty} \int_{0}^{1}f(x)^n dx = 0.$ Suppose there ...
Ricci Ten's user avatar
  • 470
0 votes
2 answers
155 views

$2^x + 3^x + 6^x = x^2$, no. Of solutions. (Without differentiation) [closed]

I know this problem can be approached and even be solved by differentiation. The differentiation is always positive (for $x<0$, i also proved $x$ needs to be negative) so strictly increasing and ...
Arham Shah's user avatar
-1 votes
0 answers
80 views

Alternative solution to showing that $\langle x^2 +1, y\rangle$ is a maximal ideal and its possible generalization?

The following is taken from the text An Introduction to Grobner Bases by: Ralf Froberg, and the following Notes: $\langle x^2 +1, y\rangle$ is maximal, pg.3 Question (5a) Background Notation 1: $\...
Seth's user avatar
  • 3,641
2 votes
2 answers
112 views

Find the angle $ECA$

I found this puzzle some time ago on the Internet. Triangle $ABC$ is like on the picture: where $AD$ is an altitude. One need to find angle $ECA$. How to solve it? Insane attempt: $\left|\angle EBD\...
Piotr Wasilewicz's user avatar
0 votes
0 answers
35 views

If $f_n(x)\rightrightarrows f(x), ~(f_n(x)-f_n(x\frac{n}{n+1}))n\rightrightarrows g$ then $xf'(x)=g(x)$

I want to prove following statement: Let $f_n:[0,1]\to\mathbb{R}$ be sequence of continuous functions uniformly convergent to $f\left(x\right)$. If $\left(f_n\left(x\right)-f_n\left(x\frac{n}{n+1}\...
Jakub Pawlak's user avatar
0 votes
0 answers
42 views

Shorter proof of an example of a $\sigma$-algebra

In almost every book on probability and measure theory, when you see $\sigma$-algebras and measurable space, you can find the next example Example. Let $\Omega$ be a set, let $S$ be a subset of $\...
RataMágica's user avatar
0 votes
1 answer
47 views

Prove that $L(X) \subseteq L(X \cup Y)$.

Given that $L(X)$ is a subspace generated by $X$, any vector in $L(X)$ is a linear combination of the vectors in $X$. That is: $$v=L(X)\rightarrow v=\sum_{i=1}^{m}a_i x_i: (x_i\in X, a_i \in R, \...
Antonius Anonymous's user avatar
2 votes
3 answers
82 views

Solution Checking: given positive $a,b,c,x,y,z$ with $a+x=b+y=c+z = S$, show that $ay+bz+cx<S^2$

This question is from an Olympiad training website (question 5). I am aware there are multiple solutions given in this post for the same problem. However, my solution seems to be fairly simple, and ...
rosemary 2.0's user avatar
0 votes
0 answers
40 views

Proof for the existence of some parameters that make functions' values linearly independent

I have a set of functions $\{F_1(x,s_1),F_2(x,s_2),\cdots, F_n(x,s_n)\}$ where each $F_i(x)$ is a nonlinear function of $s_i \in \mathbb R$ and $F_i \neq F_j$, $i \neq j$. I was wondering how to prove ...
Johny's user avatar
  • 27
1 vote
2 answers
92 views

Prove $GH \parallel BC$ in olympiad geometry problem

Earlier today a question was asked but the poster showed no work and the question was downvoted and quickly deleted. The situation posed was as in the left diagram: a triangle $ABC$ has equal segments ...
RobinSparrow's user avatar
  • 2,042
0 votes
1 answer
49 views

Image of connected set by local diffeomorphism

Suppose $F:U\subset \mathbb{R}^n\rightarrow V\subset \mathbb{R}^n$ is $C^1$ and has $Df(u)$ an inversible operator for every $u\in U$. Assume $C\subset U$ is open, connnected, bounded with $\overline{...
Kadmos's user avatar
  • 2,352
2 votes
1 answer
55 views

$\int^0_{-\log(2)}|f(e^x)\cdot\frac{1-x}{e^x}|dx$ Where $f$ is given

$$\int^0_{-\log(2)}|f(e^x)\cdot\frac{1-x}{e^x}|dx$$ Where $f(x)=\frac{\log(x)+x}{x}$ Firstly we find the roots of this function , $f(e^x)=0 $ or $x=1\notin(-\log(2),0) \iff e^x=x_0\iff x=\log(x_0)$. ...
Antony Theo.'s user avatar
  • 1,408
0 votes
1 answer
66 views

Prove that $T_n=2^{2^n+1}+1$ is composite for $n \in \mathbb{N}$ [duplicate]

I was asked to prove for $n \in \mathbb{N}$ that $T_n=2^{2^n+1}+1$ is composite. One solution by induction is as follows: Base case $T_1$: $2^{2^n+1}+1=2^3+1=9=3^2$, so $T_1$ is composite with a ...
RobinSparrow's user avatar
  • 2,042
2 votes
2 answers
91 views

Proving (p → r) ∨ (q → r) ≡ (p ∧ q) → r

So I do understand that this could be much more easily proven using basic logical equivalences as follows: (p → r) ∨ (q → r) ≡ (¬p ∨ r) ∨ (¬q ∨ r) ≡ (¬p ∨ ¬q) ∨ r ≡ ¬(p ∧ q) ∨ r ≡ (p ∧ q) → r However, ...
Bob Marley's user avatar
0 votes
0 answers
12 views

Finding general expression for symmetric trace-free tensors (STFs)

Given a symmetric $n$-tensor $I_{\alpha_1 ... \alpha_n}$, its tracefree version is given by $$Q_{\alpha_1 ... \alpha_n}=\sum_{k=0}^{\left\lfloor{\frac{n}{2}}\right\rfloor} (-1)^k \frac{ \binom{n}{k} \...
Sanjana's user avatar
  • 265
0 votes
0 answers
61 views

Can we view $\frac{\gcd(m,n)}{n} \binom{n}{m}$ as the size of a finite group where $n \geqslant m\geqslant 1$?

Prove that the expression $$\frac{\gcd(m,n)}{n} \binom{n}{m} \tag{1}$$ is an integer for all pairs of integers $n \geqslant m \geqslant 1.$ This problem appeared in the famous William Lowell Putnam ...
stoic-santiago's user avatar
0 votes
0 answers
21 views

Alternate way of proving a relation is not a transitive relation

Here's the question: Let $R$ be a relation on the set of real numbers defined by aRb $\iff ab > -1$. Is $R$ an equivalence relations? I was able to prove that R is symmetric and reflexive. But R ...
Divyansh Singh's user avatar
2 votes
2 answers
102 views

Coincidence with the recurrence and solution for Fibonacci convolution?

When playing around trying to answer a question, I discovered a curious recurrence and solution for the Fibonacci convolution $$a_n=\sum_{i+j=n} f_i f_j$$ where $f_i$ is ith element the Fibonacci ...
Aaron's user avatar
  • 24.6k
0 votes
1 answer
60 views

Proving $o(H\cap K)=\gcd(o(H),o(K))$. [closed]

Let $H$ and $K$ be subgroups of a finite cyclic group $G,$ then $|H \cap K| = \gcd(|H|,|K|)$.*The same question is already asked in this plateform here. So this is indeed a duplicate question but i am ...
user avatar
2 votes
3 answers
161 views

A constructive proof of this innocent set theoretic proposition?

I was reading Freiwald's An Introduction to Set Theory and Topology, and I came across the following exercise from Chapter 1: E8. Suppose $A$, $B$, $C$, and $D$ are sets with $A\ne\emptyset$ and $B\...
Atom's user avatar
  • 4,057
3 votes
3 answers
120 views

Show that $\frac{x^3+y^3}{x^2+y^2}$ is not differentiable at $(0,0)$

Let be $f:\mathbb{R}^2\to\mathbb{R}$ where $$ f(x,y):=\begin{cases} \frac{x^3+y^3}{x^2+y^2},&(x,y)\neq (0,0)\\0,&(x,y)=(0,0).\end{cases} $$ Show that $f$ is not differentiable at $(0,0)$. My ...
Philipp's user avatar
  • 4,564
2 votes
2 answers
158 views

Using Generating Functions To Count Things

At the theater, the movie X is currently showing. However, there are only $7$ tickets left in row $F$, $8$ tickets left in row $G$, and $9$ tickets left in row $H$. A group of students enters the ...
Math_fun2006's user avatar
2 votes
2 answers
59 views

Alternative proof $g(u)=6 + 5 \sin u + \sin(2 u)- \cos u - \cos(2 u) \ge 0$ for $u\in\left[-\frac \pi 2, \frac \pi 2\right]$

The given inequality $$g(u)=6 + 5 \sin u + \sin(2 u)- \cos u - \cos(2 u) \ge 0$$ for $u\in\left[-\frac \pi 2, \frac \pi 2\right]$, comes out from an answer given to this other recent question. The ...
user's user avatar
  • 156k
1 vote
1 answer
68 views

How to find an expression for the $n$th partial derivatives of $1/r$?

From Pirani's lectures on General Relativity I got the following identity which he asks the reader to prove by induction which is easy $$\partial_{\alpha_1} \partial_{\alpha_2} ... \partial_{\alpha_n} ...
Sanjana's user avatar
  • 265
1 vote
1 answer
59 views

To prove or disprove $MM^t=\alpha I_{m\times m}$,where $M$ is an $m\times n$ matrix of rank $m$ with $m<n$.

Let $M$ be an $m\times n$ matrix of rank $m$ with $m<n$. If for some non-zero real number $\alpha$, we have $x^tMM^t x=\alpha x^tx$, for all $x\in \Bbb R^m$ then prove or disprove $MM^t=\alpha I_{m\...
Gggg's user avatar
  • 618
1 vote
1 answer
83 views

Proving set all of functions from $\mathbb{N} \to \mathbb{N}$ is uncountable [closed]

I know this is a duplicate question, but I could not find a solution similar to my own on this website, so I am making a post to see potential errors in it. My proof is generally as follows: Let S = {...
Jack Hueson's user avatar
-1 votes
1 answer
164 views

To show a homomorphism defined on a finite group $G$ is an identity homomorphism on $G$. [duplicate]

Let $G$ be a finite group.Let $f:G\rightarrow G$ be a group homomorphism.Define $H=\{g\in G:f(g)=g\}$ such that $o(H)>\frac {o(G)}{2}$.Then $f(x)=x$,$\forall x \in G$. I tried to prove the above ...
Gggg's user avatar
  • 618
0 votes
1 answer
55 views

$-a=(-1)\cdot a,\forall a\in\mathbb{R}$ using axioms

I've been trying to prove $$-a=(-1)\cdot a$$ for every $a\in\mathbb{R}$ using only axioms, however, every demonstration I found use one of these two properties: $$0\cdot a=0,\quad\forall a\in\mathbb{R}...
mvfs314's user avatar
  • 2,084
0 votes
1 answer
44 views

Proving triangle midsegment theorem without quadrilaterals or similarity

I'm here with a question on a basic fact presented to you below: The midline of a triangle is parallel to the third side of that triangle and half its length. There is a known proof listed here on ...
Rusurano's user avatar
  • 844
0 votes
0 answers
50 views

How can you prove directly that two polynomials with only real roots can make a polynomial that has complex roots with addition?

Generally, if a polynomial over the real numbers of degree $n$ has $n$ roots then it means that it only has real roots. I found an example where when I add two of these polynomial, I get one with a ...
Gergely Tóth's user avatar
0 votes
1 answer
67 views

Proving Cosine Addition Identity using Complex numbers

I want to see how one can stumble upon the following using complex numbers $\cos{A} + \cos{B} = 2\cos{\frac{A+B}{2}}\cos\frac{A-B}{2}$ I know getting to angle sum formulas like $\cos{(A+B)}$ and $\sin{...
Maddy's user avatar
  • 45
0 votes
0 answers
24 views

Polar Set of a Polyhedra is a Polytope

I am having trouble to verify Proposition 2 from the following MIT OCW document: Is there a way for us to see the equivalence of $C_1$ and $C_2$ simply through the definition? I do notice there was a ...
Partial T's user avatar
  • 583
0 votes
0 answers
41 views

Can Parameshvara's formula be proven without trigonometry?

Parameshvara's formula connects sides of cyclic quadrilaterals with its circumradius and area. Ptolemy's theorem can be proven by triangle similarity, as shown here. Brahmagupta's formula can also be ...
Rusurano's user avatar
  • 844
4 votes
4 answers
428 views

Looking for alternative proofs of this statement about angles

This is the theorem to prove. Below is my proof that I consider rather long and complex. The given data is on this drawing: Construct $\angle DCE = \angle DCB$. The point $E$ on ray $CE$ is chosen in ...
Rusurano's user avatar
  • 844
0 votes
0 answers
56 views

How to prove the SSS triangle congruence without isosceles triangles or circles?

The SSS triangle congruence is the following theorem from elementary geometry: If three sides of a triangle are equal to the three sides of another triangle, then the two triangles are congruent. ...
Rusurano's user avatar
  • 844
2 votes
0 answers
37 views

Finding the sum of the floor function of $a,(b-1)/2,c$ given two symmetric sums

Problem: Let $a<b<c$ be $3$ real numbers satisfying $a+b+c=6$, $ab+bc+ca=9$. Then, determine the value of $\lfloor{a}\rfloor+\lfloor\frac{b-1}{2}\rfloor+\lfloor{c}\rfloor$. My method of solution:...
Cognoscenti's user avatar
0 votes
0 answers
31 views

How to define congruent angles without geometric transformations or measures?

I'm trying to build a proof of triangle SAS congruence (see the existing proof I know of) that does not base on geometric transformations, but only on basic axioms and definitions. Along my journey to ...
Rusurano's user avatar
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