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.

A factory used a 3D printer to produce the prototype of a part. The prototype has the shape of a convex polyhedron, obtained by the juxtaposition of two distinct solids, one with the shape of a regular hexagonal prism and the other with the shape of a straight hexagonal pyramid frustum. The larger base of the pyramid frustum coincides with one of the bases of the prism.
After printing the prototype, it was sent to the customization sector for painting its surface. The criterion defined for painting considers that congruent faces must be painted with the same color, and non-congruent faces must have different colors. What is the quantity of colors used to paint the prototype?
(A) 9
(B) 8
(C) 6
(D) 4
(E) 3

A polynomial $f ( x )$ has integer coefficients such that $f ( 0 )$ and $f ( 1 )$ are both odd numbers. Prove that $f ( x ) = 0$ has no integer solutions.

Prove that $$\frac { 2 } { 0 ! + 1 ! + 2 ! } + \frac { 3 } { 1 ! + 2 ! + 3 ! } + \cdots + \frac { n } { ( n - 2 ) ! + ( n - 1 ) ! + n ! } = 1 - \frac { 1 } { n ! }$$

If 8 points in a plane are chosen to lie on or inside a circle of diameter 2 cm then show that the distance between some two points will be less than 1 cm.

If $f ( x ) = \frac { x ^ { n } } { n ! } + \frac { x ^ { n - 1 } } { ( n - 1 ) ! } + \cdots + x + 1$, then show that $f ( x ) = 0$ has no repeated roots.

Using the fact that $\sqrt { n }$ is an irrational number whenever $n$ is not a perfect square, show that $\sqrt { 3 } + \sqrt { 7 } + \sqrt { 21 }$ is irrational.

(a) Show that the area of a right-angled triangle with all side lengths integers is an integer divisible by 6.
(b) If all the sides and area of a triangle were rational numbers then show that the triangle is got by 'pasting' two right-angled triangles having the same property.

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.

For any polynomial $f ( x )$ with real coefficients and of degree 2011 , there is a real number $b$ such that $f ( b ) = f ^ { \prime } ( b )$.

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.

A differentiable function $f : \mathbb { R } \rightarrow \mathbb { R }$ satisfies $f ( 1 ) = 2 , f ( 2 ) = 3$ and $f ( 3 ) = 1$. Show that $f ^ { \prime } ( x ) = 0$ for some $x$.

Show that $\frac { \ln ( 12 ) } { \ln ( 18 ) }$ is irrational.

a) We want to choose subsets $A _ { 1 } , A _ { 2 } , \ldots , A _ { k }$ of $\{ 1,2 , \ldots , n \}$ such that any two of the chosen subsets have nonempty intersection. Show that the size $k$ of any such collection of subsets is at most $2 ^ { n - 1 }$. b) For $n > 2$ show that we can always find a collection of $2 ^ { n - 1 }$ subsets $A _ { 1 } , A _ { 2 } , \ldots$ of $\{ 1,2 , \ldots , n \}$ such that any two of the $A _ { i }$ intersect, but the intersection of all $A _ { i }$ is empty.

For $n > 1$, a configuration consists of $2n$ distinct points in a plane, $n$ of them red, the remaining $n$ blue, with no three points collinear. A pairing consists of $n$ line segments, each with one blue and one red endpoint, such that each of the given $2n$ points is an endpoint of exactly one segment. Prove the following. a) For any configuration, there is a pairing in which no two of the $n$ segments intersect. (Hint: consider total length of segments.) b) Given $n$ red points (no three collinear), we can place $n$ blue points such that any pairing in the resulting configuration will have two segments that do not intersect. (Hint: First consider the case $n = 2$.)

Let $N$ be the set of non-negative integers. Suppose $f : N \rightarrow N$ is a function such that $f ( f ( f ( n ) ) ) < f ( n + 1 )$ for every $n \in N$. Prove that $f ( n ) = n$ for all $n$ using the following steps or otherwise. a) If $f ( n ) = 0$, then $n = 0$. b) If $f ( x ) < n$, then $x < n$. (Start by considering $n = 1$.) c) $f ( n ) < f ( n + 1 )$ and $n < f ( n + 1 )$ for all $n$. d) $f ( n ) = n$ for all $n$.

Let $ABCD$ be a parallelogram. Let $O$ be a point in its interior such that $\angle AOB + \angle DOC = 180^{\circ}$. Show that $\angle ODC = \angle OBC$.