# Tagged Questions

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### Proving a number can be chosen to a multiply a $\mathbb{R}^2 \rightarrow \mathbb{R}$ function so that it never exceeds another function

Let $p,q,r$ be real numbers with $p,r,pr-q^2 > 0$. I am trying to prove $\exists\gamma> 0$ s.t. $\forall(x,y) \in \mathbb{R}^2 . px^2 + 2qxy + ry^2 \geq \gamma(x^2+y^2).$
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### Differentiability of a function

How can I prove that the function $x^{n+\frac{1}{2}}$ is differentiable? Do I split it up into $x^n$ and $x^{\frac{1}{2}}$? Any suggestions would be great. Thanks
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### Basic inequality proof construction

Given: $x \le y$ and $z \le 0$ Use the result "if $x \le y$, then $-x \ge -y$" To prove that $zy \le zx$
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### One more question about the prime factors of binomial coefficient…

Could someone explain me one more thing: why if $p$ is greater than $4n/5$, but less or equal to $n$, then $p$ does not divide $\binom{4n}{3n}$? thank you in advance!
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### How to “prove” the following very simple vector identities

For my upcoming exam, I have to be able to prove the following four statements (among a bank of others). Though they are pretty obviously true, I'm struggling to come up with a way to prove them ...
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### What is wrong with this proof about the maximizer of a quadratic equation?

Working through Daniel Solow's "How to read and do proofs", I have been stuck at the following problem. Problem: What is wrong with the proof below for the statement. If a, b, and c are real ...
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### Simple formula for triangle numbers

Triangular numbers are defined as Tn:=1+2+...+n for n>= 1. Find a simple formula for Tn+Tn-1 and prove it. I know Tn=n(n+1)/2 so would Tn-1=n(n-1)/2? And then would I prove it by just adding ...
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### Heptagon (septagon) diagonal intersection

Diagonals TV and UW of regular heptagon TUVWXYZ meet at A. Prove that TU+TA=TW (Source: AoPS ItG). My observations: TUVW is an isosceles trapezoid Triangle TUA is congruent to triangle WVA