Singular and eigen values properties... 
Let $A\in\mathcal {M}_n(\mathbb{R})$, we will denote $\lambda_{\max}(A)$ the biggest eigenvalue of $A$ in absolute value, as for $B\in\mathcal M_{m,n}(\mathbb{R})$ we will denote $\sigma_{\max}(B)$ the highest singular value of $B$.

I have $3$ things to show from which I have shown 1 and for the last one I only have an idea.
$1$. with $A$ a symetric matrix show that $\lambda_{\max}(A)= \max_{\|v\|=1}v^tAv$
$$A=A^t\implies \max_{\|v\|=1}(v^tAv)
=\max_{\|v\|=1}(v^tA^tv)
=\max_{\|v\|=1}((Av)^tv)
=\max_{\|v\|=1}((\lambda v)^tv)
=\max_{\|v\|=1}(\lambda v^tv)
=\max_{\|v\|=1}(\lambda \|v\|^2)
=\max_{\|v\|=1}(\lambda)
=\max(\lambda).$$
$2$. Show that $\sigma_{\max}(B) = \sqrt{\lambda_{\max}(B^tB)}$.
$3$. with $C\in\mathcal M_n(\mathbb{R})\implies\lambda_{\max}(\frac{C+C^t}{2})\leq\sigma_{\max}(C)$.
Here I think is something relating with AM-GM Mean right?
 A: *

*There are a few errors in your attempt. When you use $Av = \lambda v$ you are assuming $v$ is an eigenvector of $A$, even though the maximum is over all unit norm $v$ which may include non-eigenvectors. Also your attempt does not really use symmetry; note that you could have proceeded as $v^t A v = v^t (\lambda v) = \lambda \|v\|^2$ directly (disregarding the eigenvalue issue mentioned above). Symmetry is important here because of the spectral theorem. This may help you fix your proof.

*I suppose you are defining singular values from the SVD? Then write $B=U\Sigma V^t$ and note $B^t B = V \Sigma^t \Sigma V^t$. Since $\Sigma^t \Sigma$ is a diagonal matrix, this is a diagonalization of $B^t B$, so you can write down the eigenvalues of $B^t B$ in terms of the singular values of $B$.

*Let $U\Sigma V^t$ be the SVD of $C$. Note $(C+C^t)/2$ is symmetric. Using the first part, $$\lambda_{\max}(\frac{C+C^t}{2}) = \max_{\|v\|=1} v^t C v = \max_{\|v\|=1} v^t U \Sigma V^t v \le \max_{\|x\|=\|y\|=1} x^t \Sigma y = \sigma_{\max}(C),$$
where the inequality comes from the change of variables $x=U^t v$ and $y = V^t v$ and noting that orthogonal matrices are norm-preserving (i.e. $\|x\|=\|v\|$).

Edit:
As I mentioned in my comment, I think $\lambda_{\max}$ should be the largest eigenvalue, not the largest in absolute value.
Response to your comment:


*

*Let $UDU^t$ be the eigendecomposition of $A$. Then $\max_{\|v\|=1} v^t A v = \max_{\|v\|=1} v^t UDU^t v = \max_{\|w\| = 1} w^t D w = \lambda_{\max}(A)$, where we have made the change of variables $w = U^t v$ and noted that orthogonal matrices are norm-preserving.

*I already told you that the eigenvalues of $B^t B$ are the diagonal entries of the diagonal matrix $\Sigma^t \Sigma$.
