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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Alternative proof of Apollonius's identity
How does one prove Apollonius's identity without coordinates or the law of cosines (or similar trigonometric laws)? Is there a known rearrangement proof similar to the many such proofs for the ...
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Proofs with the "Purified Pigeonhole Principle".
In his EWD980 and EWD1094, Dijkstra provides a formulation of the Pigeonhole Principle as such:
"For a non-empty, finite bag of numbers, the maximum value is at least the average (and the ...
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A trigonometric maximum problem involving trigonometric constraints
Let $a,b,c,\alpha,\beta\in\mathbb{R}^+$ and $\alpha+\beta<2\pi$. Prove that if and only if
$$\frac{\sin\alpha}{a\sqrt{b^2+c^2-2bc\cos\alpha}}=\frac {\sin\beta}{b\sqrt{a^2+c^2-2ac\cos\beta}}=-\frac{\...
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Simple Direct Proof of Nakayama's Lemma
Is there a simple direct proof of (this version of) Nakayama's Lemma? By simplicity, I am mostly referring to the concepts required to understand the proof, not how terse the proof is. I have no doubt ...
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Detailed Proof of Proposition 2.12 a) from Diamond, Darmon, Taylor, "Fermat's Last Theorem."
I seek a highly detailed proof of this statement in the split multiplicative reduction case. The result can be found on page 57 here.
That is, if $E/\mathbb Q$ has split multiplicative reduction at $p$...
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Is there a brute force proof that the first-order characterization of a Jacobson radical is an ideal?
Let rings be commutative and unital.
Is there a brute force proof that the first-order characterization of a Jacobson radical is an ideal?
Basically, for context, I'm having some trouble wrapping my ...
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Every countably infinite linear order has a copy of $\omega$ or $\omega^{op}$
Every countably infinite linear order $L$ has a copy of $\omega$ or $\omega^{op}$. I'm interested in different kinds of proofs of this fact.
One I came up with is: pick $x_0 \in L$. Wlog $[x_0, +\...
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Understanding how group cohomology classifies extensions using the derived functor point of view
I am rereading some material about group extensions, in particular because I needed to recall the formula
$$H^2(G;A)\cong \mathcal{E}(G;A).$$
We have that $G$ is some group acting on an abelian group $...
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Problem 237 "Mathematical Quickies:270 Stimulating Problems with Solutions" Particle Movement
I was solving "Mathematical Quickies:270 Stimulating Problems with Solutions" when I came across a very peculiar question (Problem 237):
A particle moves in a straight line starting from ...
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Real roots of $x^4+ax^3+bx^2+cx+1=0,$ when $a,b,c$ are real and $b\ge\frac{a^2+c^2}{4}$
For real $a,b,c$ and
$$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\...
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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 ...
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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
$$\frac{a+(b-a)+\frac{1}{a(b-...
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Is the following method for proving density of irrational numbers in real numbers without using rational numbers density in real numbers rigorous?
The motivation for this question is:
I told my friend to use:
$\forall x_{1}, x_{2} \in \mathbb{R}, x_{1} < x_{2}, \exists r \in \mathbb{Q}: x_{1} < r <x_{2}.$
To prove:
$\forall x_{1}, x_{2} ...
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Prove that any sequence of five distinct integers must contain a 3-chain
This task is from MIT OpenCourse Mathematics for CS 2010 course, problem set 2, exercise 1(d). I am aware that this question has already been asked several times previously on this platform. Yet, the ...
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Prove $f'(r_{+}) + f'(r_{-}) < 0$ for roots of $1 - \frac{M}{r^2} + \frac{Q}{r^4} - r^2 = 0$
Problem. Let $M, Q > 0$ be given. Let
$$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 ...
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