A function $f$ is continuous on the closed interval $[ 2,5 ]$ with $f ( 2 ) = 17$ and $f ( 5 ) = 17$. Which of the following additional conditions guarantees that there is a number $c$ in the open interval $( 2,5 )$ such that $f ^ { \prime } ( c ) = 0$ ?
(A) No additional conditions are necessary.
(B) $f$ has a relative extremum on the open interval $( 2,5 )$.
(C) $f$ is differentiable on the open interval $( 2,5 )$.
(D) $\int _ { 2 } ^ { 5 } f ( x ) d x$ exists.

Let $x$, $y$ and $z$ be three real numbers. We consider the following implications $\left( P _ { 1 } \right)$ and $\left( P _ { 2 } \right)$:
$$\begin{array} { l l } \left( P _ { 1 } \right) & ( x + y + z = 1 ) \Rightarrow \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \geqslant \frac { 1 } { 3 } \right) \\ \left( P _ { 2 } \right) & \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \geqslant \frac { 1 } { 3 } \right) \Rightarrow ( x + y + z = 1 ) \end{array}$$

Part A

Is the implication $\left( P _ { 2 } \right)$ true?

Part B

In space, we consider the cube $A B C D E F G H$ and we define the orthonormal coordinate system $( A ; \overrightarrow { A B } , \overrightarrow { A D } , \overrightarrow { A E } )$.

1. a. Verify that the plane with equation $x + y + z = 1$ is the plane $( B D E )$. b. Show that the line $( A G )$ is orthogonal to the plane $( B D E )$. c. Show that the intersection of the line $( A G )$ with the plane $( B D E )$ is the point $K$ with coordinates $\left( \frac { 1 } { 3 } ; \frac { 1 } { 3 } ; \frac { 1 } { 3 } \right)$.
2. Is the triangle $B D E$ equilateral?
3. Let $M$ be a point in space. a. Prove that if $M$ belongs to the plane $( B D E )$, then $A M ^ { 2 } = A K ^ { 2 } + M K ^ { 2 }$. b. Deduce that if $M$ belongs to the plane $( B D E )$, then $A M ^ { 2 } \geqslant A K ^ { 2 }$. c. Let $x$, $y$ and $z$ be arbitrary real numbers. By applying the result of the previous question to the point $M$ with coordinates $( x ; y ; z )$, show that the implication $\left( P _ { 1 } \right)$ is true.

It is intended to build a mosaic with the shape of a right triangle, having three pieces available, two of which are congruent right triangles and the third is an isosceles triangle. The figure presents five mosaics formed by three pieces.
In the figure, the mosaic that has the characteristics of the one intended to be built is
(A) 1.
(B) 2.
(C) 3.
(D) 4.
(E) 5.

There is a sequence of open intervals $I _ { n } \subset \mathbb { R }$ such that $\bigcap _ { n = 1 } ^ { \infty } I _ { n } = [ 0,1 ]$.

The set $S$ of real numbers of the form $\frac { m } { 10 ^ { n } }$ with $m , n \in \mathbb { Z }$ and $n \geq 0$ is a dense subset of $\mathbb { R }$.

There is a continuous bijection from $\mathbb { R } ^ { 2 } \rightarrow \mathbb { R }$.

There is a bijection between $\mathbb { Q }$ and $\mathbb { Q } \times \mathbb { Q }$.

If $\left\{ a _ { n } \right\} _ { n = 1 } ^ { \infty } , \left\{ b _ { n } \right\} _ { n = 1 } ^ { \infty }$ are two sequences of positive real numbers with the first converging to zero, and the second diverging to $\infty$, then the sequence of complex numbers $c _ { n } = a _ { n } e ^ { i b _ { n } }$ also converges to zero.

If $f : [ 0,1 ] \rightarrow [ - \pi , \pi ]$ is a continuous bijection then it is a homeomorphism.

A vector space of dimension $\geq 2$ can be expressed as a union of two proper subspaces.

There is a bijective analytic function from the complex plane to the upper half-plane.

Let $R$ be a commutative ring with 1 and $I$ and $J$ ideals of $R$. Choose the correct statement(s) from below:
(A) If $I$ or $J$ is maximal then $IJ = I \cap J$;
(B) If $IJ = I \cap J$, then $I$ or $J$ is maximal;
(C) If $IJ = I \cap J$, then $1 \in I + J$;
(D) If $1 \in I + J$ then $IJ = I \cap J$.

Let $(X, d)$ and $(Y, \rho)$ be metric spaces and $f : X \longrightarrow Y$ a homeomorphism. Choose the correct statement(s) from below:
(A) If $B \subseteq Y$ is compact, then $f^{-1}(B)$ is compact;
(B) If $B \subseteq Y$ is bounded, then $f^{-1}(B)$ is bounded;
(C) If $B \subseteq Y$ is connected, then $f^{-1}(B)$ is connected;
(D) If $\{y_n\}$ is Cauchy in $Y$, then $\{f^{-1}(y_n)\}$ is Cauchy in $X$.

Fix a non-negative integer $d$. Let $$\mathcal{A}_d := \{A \subseteq \mathbb{C} : A \text{ is the zero-set of a polynomial of degree } \leq d \text{ in } \mathbb{C}[X]\}.$$ Let $\mathcal{T}$ be the coarsest topology on $\mathbb{C}$ in which $A$ is closed for every $A \in \mathcal{A}_d$.
(A) Determine whether $\mathcal{T}$ is Hausdorff.
(B) Show that for every polynomial $f(X) \in \mathbb{C}[X]$, the function $\mathbb{C} \longrightarrow \mathbb{C}$ defined by $z \mapsto f(z)$ is continuous, where $\mathbb{C}$ (on both the sides) is given the topology $\mathcal{T}$.

Let $U = \left\{(x, y) \in \mathbb{R}^{2} \mid x < y^{2} < 4\right\}$ and $V = \left\{(x, y) \in \mathbb{R}^{2} \mid 0 < xy < 4\right\}$, both taken with the subspace topology from $\mathbb{R}^{2}$. Which of the following statement(s) is/are true?
(A) There exists a non-constant continuous map $V \longrightarrow \mathbb{R}$ whose image is not an interval.
(B) Image of $U$ under any continuous map $U \longrightarrow \mathbb{R}$ is bounded.
(C) There exists an $\epsilon > 0$ such that given any $p \in V$ the open ball $B_{\epsilon}(p)$ with centre $p$ and radius $\epsilon$ is contained in $V$.
(D) If $C$ is a closed subset of $\mathbb{R}^{2}$ which is contained in $U$, then $C$ is compact.

Consider the function $f : \mathbb{R}^{2} \longrightarrow \mathbb{R}$ given by
$$f(x, y) = \left(1 - \cos \frac{x^{2}}{y}\right) \sqrt{x^{2} + y^{2}}$$
for $y \neq 0$ and $f(x, 0) = 0$. (The square root is chosen to be non-negative). Pick the correct statement(s) from below:
(A) $f$ is continuous at $(0,0)$.
(B) $f$ is an open map.
(C) $f$ is differentiable at $(0,0)$.
(D) $f$ is a bounded function.

Which of the following functions are uniformly continuous on $\mathbb{R}$?
(A) $f(x) = x$;
(B) $f(x) = x^{2}$;
(C) $f(x) = (\sin x)^{2}$;
(D) $f(x) = e^{-|x|}$.

Let $U$ and $V$ be non-empty open connected subsets of $\mathbb{C}$ and $f : U \longrightarrow V$ an analytic function. Which of the following statement(s) is/are true?
(A) $f^{\prime}(z) \neq 0$ for every $z \in U$.
(B) If $f$ is bijective, then $f^{\prime}(z) \neq 0$ for every $z \in U$.
(C) If $f^{\prime}(z) \neq 0$ for every $z \in U$, then $f$ is bijective.
(D) If $f^{\prime}(z) \neq 0$ for every $z \in U$, then $f$ is injective.

Let $U$ denote the unit open disc centred at 0. Let $f : U \backslash \{0\} \longrightarrow \mathbb{C}$ be an analytic function. Assume that $\lim_{z \longrightarrow 0} z f(z) = 0$.
(A) $\lim_{z \longrightarrow 0} |f(z)|$ exists and is in $\mathbb{R}$.
(B) $f$ has a pole of order 1 at 0.
(C) $zf(z)$ has a zero of order 1 at 0.
(D) There exists an analytic function $g : U \longrightarrow \mathbb{C}$ such that $g(z) = f(z)$ for every $z \in U \backslash \{0\}$.

Consider the following four propositions:
$p _ { 1 }$ : Three lines that are pairwise intersecting and do not pass through the same point must lie in the same plane.
$p _ { 2 }$ : Through any three points in space, there is exactly one plane.
$p _ { 3 }$ : If two lines in space do not intersect, then these two lines are parallel.
$p _ { 4 }$ : If line $l \subset$ plane $\alpha$ and line $m \perp$ plane $\alpha$ , then $m \perp l$ .
The sequence numbers of all true propositions among the following statements are $\_\_\_\_$.
(1) $p _ { 1 } \wedge p _ { 4 }$
(2) $p _ { 1 } \wedge p _ { 2 }$
(3) $\neg p _ { 2 } \vee p _ { 3 }$
(4) $\neg p _ { 3 } \vee \neg p _ { 4 }$

Given proposition $p : \forall x \in \mathbf { R } , | x + 1 | > 1$; proposition $q : \exists x > 0 , x ^ { 3 } = x$, then
A. Both $p$ and $q$ are true propositions
B. Both $\neg p$ and $q$ are true propositions
C. Both $p$ and $\neg q$ are true propositions
D. Both $\neg p$ and $\neg q$ are true propositions

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
With $q(x,y) = x^2 - y^2$ and $q'(x,y) = 2xy$ on $\mathbb{R}^2$ as defined in question II.B.1, does there exist a basis of $\mathbb{R}^2$ orthogonal for both $q$ and $q'$?

Does this result hold if the set $\Omega$ has infinitely many elements but is not assumed to be open?

By means of the function $\psi ( x ) = \sqrt { 1 + x ^ { 2 } }$, show that in the previous question hypothesis (1) cannot be replaced by $$\forall x \in \mathbb { R } , \left| \phi ^ { \prime } ( x ) \right| < 1$$

Let $F$ be a vector subspace of a symplectic space $(E , \omega)$, and let $F^{\omega} = \{ x \in E \mid \forall y \in F , \omega ( x , y ) = 0 \}$. Is the subspace $F ^ { \omega }$ necessarily in direct sum with $F$?