## To prove CD is parallel to EF

ABC is an equilateral triangle. CD is the median. F is ACTUALLY and E is on AB, and CF=AE is one third of the side of the triangle. To prove CD is parallel to FE

## Prove $8^n = Θ(4^n)$

how would I prove

$$8^n = Θ(4^n)$$ is either true or false.

I so far have attempted to prove big O but cant find the value of C1

## How to prove “if every subset of a set is a CFL, then the set must be regular.”

If every subset of a set is a CFL, then the set must be regular.

I want to prove it, could anyone please give me some hints?

## Prove that $\forall \ A_k\subset [3n]$, with $|A_k|=2n$, $\exists\ a_i,\ a_j\in A_k\$ s.t. $\ a_i-a_j=n-1$

Let $$n\geq2$$ be an integer. Prove that every subset $$A_k\subset [3n]$$, with $$|A_k|=2n$$, contains two elements $$a_i,\ a_j\$$ s.t. $$\ a_i-a_j=n-1$$, and prove that $$\nexists\ a_i,\ a_j \in A_k\ \forall k\$$ s.t. $$\ a_i – a_j =n$$.

It seems obvious that I will have to apply the Pigeonhole Principle, but I have no clear idea about how to start. Could you give me some hints? Thanks in advance!

## Let f be a function defined on a nbhd of 0 w/ the property that |f(x)| ≤ x^2 for all x. Prove that f is differentiable at 0 and calculate f'(0).

Let f be a function defined on a neighborhood of 0 with the property that |f(x)| ≤ x^2 for all x. Prove that f is differentiable at 0 and calculate f'(0).

I have started this problem by first letting I be a nbhd of 0. Then I let x be an element of I/{0}. I am pretty sure I then must show the limit exists and evaluate it at 0. However, I am not sure how to show it is defined at 0 with just the fact that |f(x)| ≤ x^2 for all x.

## Prove that $\frac{x+y}{1+xy}$ is an Abelian Group

Let $$I=\left]-1,\ 1\right[$$ be an interval, and $$\left(I,\ \star\right)$$ be a magma such that:

$$\left(\forall\ \left(x,\ y \right) \in I^2\right)\ x \star y=\frac{x+y}{1+xy}$$

I need to prove that $$\left(I,\ \star\right)$$ is an Abelian group.

A simple way to prove it is by checking that the magma $$\left(I,\ \star\right)$$ satisfies the group axioms including commutativity.

Based on the following statement I would like to approach that problem:

Let $$f$$ be a homomorphism from $$\left(X,\ \perp \right)$$ to $$\left(I,\ \star\right)$$, then the algebraic structure of $$\left(I,\ \star\right)$$ is exactly the algebraic structure of $$\left(X,\ \perp \right)$$

So I would like to find out a usual Abelian group – ex. $$\left(\mathbb{R},\ +\right)$$ – and a homomorphism $$f$$ from that group to $$\left(I,\ \star\right)$$.

Assume that $$f$$ is a homomorphism from $$\left(X,\ \perp \right)$$ to $$\left(I,\ \star\right)$$, then

$$\left(\forall\ \left(x,\ y \right) \in X^2\right)\ f\left(x \perp y \right)= f\left(x\right) \star f\left(y \right)$$

$$\Leftrightarrow f\left(x \perp y \right)= \frac{f\left(x\right) + f\left(y \right)}{1+f\left(x\right) \cdot f\left(y \right)}$$

I got stuck on that. Could anyone support me with some hints how to get it done.

## If $f\geq0 , f(0)=f(1)=0$,and $\int_0^1f”/f dx$ exists,how to prove $\int_0^1f”/f dx\geq \pi^2$

If $$f\geq0 , f(0)=f(1)=0$$,and $$\int_0^1f”/f dx$$ exists,how to prove $$\int_0^1\frac{f”}{f}dx\geq \pi^2$$

The lower bound $$\pi^2$$ was guessed by myself, if it was not true ,then how to find this lower bound ?

My way is to have an analytic extension with $$T=2$$, and use the Fourier Series to have an evaluation .

But it seems like that I was wrong.

## Prove that if string_is_prefix returns true, then the string has a length as big as the prefix

I’m trying to create a version of string_is_prefix that prove that a string is longer than the prefix if it returns true. Here’s the code I have so far:

#include "share/atspre_staload.hats"  dataprop StringLen(bool) =     | LessThan(false)     | {m,n:nat | m >= n} GreaterThanEqual(true)  fun string_is_prefix {m, n: nat} (string_prefix: string(m), string: string(n)): [b:bool] (StringLen(b) | bool(b)) =     let val prefix_len = string_length(string_prefix)         val string_len = string_length(string)         fun is_prefix {i: nat | m <= n; i < m} (index: size_t(i)): [b:bool] bool(b) =             let val equal = g1ofg0(string_get_at(string, index) = string_get_at(string_prefix, index))             in                 if index + 1 >= prefix_len then                     equal                 else                     equal * is_prefix(index + 1)             end     in         if prefix_len > string_len then             (LessThan() | false)         else if prefix_len > 0 then             let val result = is_prefix(i2sz(0))             in                 if result then                     (GreaterThanEqual{n, m}() | result)                 else                     (LessThan() | result)             end         else             (GreaterThanEqual{n, m}() | true)     end  fun func(string: stringGte(1)) =     ()  implement main0(argc, argv) = {     val prefix_string: string(1) = "t"     val string: [n:nat] string(n) = g1ofg0(argv[0])     val (pf | is_prefix) = string_is_prefix(prefix_string, string)     val () =         if is_prefix then             let prval GreaterThanEqual{n, m}() = pf             in                 func(string)             end } 

Unfortunately, the call to func(string) does not compile as if this proof was not working:

unsolved constraint: C3NSTRprop(C3TKmain(); S2Eapp(S2Ecst(>=); S2EVar(5277->S2Evar(n$8654(14309))), S2Eintinf(1)))  How can I make this proof work? ## Use Weirstrass’s Inequality to prove that following inequality Use Weirstrass’s Inequality to prove that $$\sum_{i=1}^n \frac{1}{\sqrt{i}}\le \frac{1}{\sqrt{n!}} \prod_{i = 2}^n (\sqrt{i-1}) + 2 \left(\sum_{i = 2}^n \frac{1}{\sqrt{i}}\right)$$ Using Weirstrass’s Inequality, I get $$1 + \sum_{i=1}^n \frac{1}{\sqrt{i}} \le \frac{1}{\sqrt{n!}} \left(\prod_{i = 1}^n (1 + \sqrt{i})\right)$$ ## How to prove that the time complexity of this algorithm is O($\sqrt{N}\$)?

  int n;   cin >> n;   int sum = 0;   for (int i = 1; sum <= n; i++) {       sum += i;   } 

If I assumed that $$N = 100$$, the loop will run $$13$$ steps, which is almost the square root of $$N$$, if $$N = 10000$$, the loop will run $$141$$ steps, which is almost the square root of $$N$$, but I don’t know how to prove that, I only know it by intuition