# Tagged Questions

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 that a function is homogeneous of degree one

Given a (profit) function of the form $$\pi(p) = \sup \{p.y:y \in Y\}$$, where $p \in R_+^k$ is a positive (price) vector and $Y \in R^k$ is a (production-possibility) set. I need to proof that ...
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### Suppose that $a, b$, and $c$ are distinct points in $C$. Suppose that $l$ is the line which bisects $∠bac$ and $m$ is the line which bisects $∠acb$

Suppose that $a, b$, and $c$ are distinct points in $C$. Suppose that $l$ is the line which bisects $∠bac$ and $m$ is the line which bisects $∠acb$. Now let $z$ be the point $l \cap m$. Let $n$ be the ...
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### $10$-digit perfect squares that contain each of the digits $1, 2, 3, 4, 5$ twice

Are there any 10-digit perfect squares that contain each of the digits $1, 2, 3, 4, 5$ twice? Perfect squares belong to the set $\{0, 1, 4, 7\}$ modulo $9$ and any such number will be equal to ...
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### Proof that $e^{-x} \ge 1-x$

My aim is to prove that $e^{-x} \geq 1-x$ for any $x \geq 0$. What I found so far is Bernoulli's inequality, which states that $$1+x\leq\left(1+\frac{x}{n}\right)^n\xrightarrow [n\to\infty]{} e^x$$ ...
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### Alternate Proof for the Cancellation Laws

I will first state the theorem: Given a group $(G,*)$ the following laws apply for $a, b, c \in S$ If $a*b=a*c$ then $b=c$ If $b*a=c*a$ then $b=c$ Attempt at alternate Proof: Consider the ...
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### Possibly not an acceptable proof for uncountablity of countable product of countable sets

Here is a text from the book Topology by Munkres: Theorem 7.7. $\ \ \$ Let $X$ denote the two element set $\{0,1\}.$ Then the set $X^\omega$ is uncountable. Proof. $\ \ \ \$ We show that, ...
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### Find infinitely many pairs of integers $a$ and $b$ with $1 < a < b$, so that $ab$ exactly divides $a^2 +b^2 −1$

Find infinitely many pairs of integers $a$ and $b$ with $1 < a < b$, so that $ab$ exactly divides $a^2 +b^2 −1$. What are the possible values of $$\frac{x^2+y^2-1}{xy}$$? I have discovered ...
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### Simplifying Fraction for Nested Radicals

A while back, a problem asked me to simplify ...
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### Proof in Algebraic Topology without appeal to intuition

My question arose from the proof of proposition 1.26 in Hatchers Algebraic Topology. There he constructs a space $Z$ from a path-connected space $X$ as follows: attach a set of 2-cells $e_\alpha$ ...
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### If $f,g: X\to Y$ be two functions continuous on $X$ then show that $\{x:f(x)=g(x)\}$ is closed in $X$

Problem. Let $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$ be two topological spaces and $f,g:X\to Y$ such that $f$ and $g$ both are continuous on $X$. Show that the set $E:=\{x:f(x)=g(x)\}$ is ...
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### Show $\dbinom{a}{b}\dbinom{b}{c}=\dbinom{a}{a-c}\dbinom{a-c}{a-b}$

I am trying to give a non-algebraic proof for this equality: $$\dbinom{a}{b}\dbinom{b}{c}=\dbinom{a}{a-c}\dbinom{a-c}{a-b}$$ So far, I could only use the identity $\dbinom{x}{y}=\dbinom{x}{x-y}$. ...
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