Let $V$ be a 10-dimensional real vector space and $U_1$ and $U_2$ two linear subspaces such that $U_1 \subseteq U_2$, $\dim_{\mathbb{R}} U_1 = 3$ and $\dim_{\mathbb{R}} U_2 = 6$. Let $\mathcal{E}$ be the set of all linear maps $T : V \longrightarrow V$ which have $U_1$ and $U_2$ as invariant subspaces (i.e., $T(U_1) \subseteq U_1$ and $T(U_2) \subseteq U_2$). Calculate the dimension of $\mathcal{E}$ as a real vector space.

Prove that the following proposition holds for $n = 3$ (5 points) and $n = 5$ (7 points), and does not hold for $n = 4$ (8 points).

“For any permutation $\pi_1$ of $\{1, 2, \dots, n\}$ different from the identity there is a permutation $\pi_2$ such that any permutation $\pi$ can be obtained from $\pi_1$ and $\pi_2$ using only compositions (for example, $\pi = \pi_1 \circ \pi_1 \circ \pi_2 \circ \pi_1$).”

Let $f(x) = 2x(1-x)$, $x \in \mathbb{R}$. Define $$f_n = \underbrace{f \circ \dots \circ f}_{n}.$$

a) (10 points) Find $\lim\limits_{n \to \infty} \int_0^1 f_n(x)\,dx$.

b) (10 points) Compute $\int_0^1 f_n(x)\,dx$ for $n = 1, 2, \dots$.

The function $f : \mathbb{R} \to \mathbb{R}$ is twice differentiable and satisfies $f(0) = 2$, $f'(0) = -2$ and $f(1) = 1$. Prove that there exists a real number $\xi \in (0,1)$ for which $$f(\xi) \cdot f'(\xi) + f''(\xi) = 0.$$

Let $P$ be an algebraic polynomial of degree $n$ having only real zeros and real coefficients.

a) (15 points) Prove that for every real $x$ the following inequality holds: $$\tag{2} (n-1) (P'(x))^2 \ge n P(x) P''(x).$$

b) (5 points) Examine the cases of equality.

Let $f : [0,1] \to \mathbb{R}$ be a continuous function with the property that for any $x$ and $y$ in the interval, $$x f(y) + y f(x) \le 1.$$

a) (15 points) Show that $$\int_0^1 f(x)\,dx \le \frac{\pi}{4}.$$

b) (5 points) Find a function, satisfying the condition, for which there is equality.

Let $V$ be a real vector space, and let $f, f_1, f_2, \dots, f_k$ be linear maps from $V$ to $\mathbb{R}$. Suppose that $f(x) = 0$ whenever $f_1(x) = f_2(x) = \dots = f_k(x) = 0$. Prove that $f$ is a linear combination of $f_1, f_2, \dots, f_k$.

Let $$P = \left\{ f : f(x) = \sum_{k=0}^{3} a_k x^k,\ a_k \in \mathbb{R},\ |f(\pm 1)| \le 1,\ \left| f\left(\pm\tfrac{1}{2}\right) \right| \le 1 \right\}.$$ Evaluate $$\sup_{f \in P} \max_{-1 \le x \le 1} |f''(x)|$$ and find all polynomials $f \in P$ for which the above “sup” is attained.

Let $0 < c < 1$ and $$f(x) = \begin{cases} \dfrac{x}{c} & \text{for } x \in [0, c], \\[2mm] \dfrac{1-x}{1-c} & \text{for } x \in [c, 1]. \end{cases}$$ We say that $p$ is an $n$-periodic point if $$\underbrace{f(f(\dots f}_{n}(p))) = p$$ and $n$ is the smallest number with this property. Prove that for every $n \ge 1$ the set of $n$-periodic points is non-empty and finite.

Let $A_n = \{1, 2, \dots, n\}$, where $n \ge 3$. Let $F$ be the family of all non-constant functions $f : A_n \to A_n$ satisfying the following conditions:

(1) $f(k) \le f(k+1)$ for $k = 1, 2, \dots, n-1$,

(2) $f(k) = f(f(k+1))$ for $k = 1, 2, \dots, n-1$.

Find the number of functions in $F$.

Suppose that $S$ is a family of spheres (i.e., surfaces of balls of positive radius) in $\mathbb{R}^n$, $n \ge 2$, such that the intersection of any two contains at most one point. Prove that the set $M$ of those points that belong to at least two different spheres from $S$ is countable.

Let $f : (0,1) \to [0,\infty)$ be a function that is zero except at the distinct points $a_1, a_2, \dots$. Let $b_n = f(a_n)$.

(a) Prove that if $\sum\limits_{n=1}^{\infty} b_n < \infty$, then $f$ is differentiable at at least one point $x \in (0,1)$.

(b) Prove that for any sequence of non-negative real numbers $(b_n)_{n=1}^{\infty}$, with $\sum\limits_{n=1}^{\infty} b_n = \infty$, there exists a sequence $(a_n)_{n=1}^{\infty}$ such that the function $f$ defined as above is nowhere differentiable.