A triangulation of a polytope $P$ is a complex formed of simplices whose realization equals $P$. Show that every polytope admits a triangulation.

A complex $\mathcal{C}$ is a non-empty finite set of polytopes of $\mathbb{R}^n$ such that for all $P, Q \in \mathcal{C}$, $P \cap Q$ is either empty or simultaneously a face of both $P$ and $Q$. We denote $\chi(\mathcal{C}) = \sum_F (-1)^{\operatorname{dim} F}$ where $F$ ranges over the faces of $\mathcal{C}$.
Show that every complex $\mathcal{C}$ whose realization is convex satisfies $\chi(\mathcal{C}) = 1$.

Let $r$ and $s$ be two strictly positive natural integers such that $r > s$, and
$$J _ { r , s } = \frac { 1 } { r - s } \sum _ { k = s + 1 } ^ { r } \frac { 1 } { k }$$
Deduce that we can write
$$J _ { r , s } = \frac { p _ { r , s } } { q _ { r , s } }$$
with $p _ { r , s }$ and $q _ { r , s }$ natural integers and $q _ { r , s }$ dividing $d _ { r } ^ { 2 }$.

We define on $[ 0,1 ]$ the function $P _ { n }$ by:
$$\forall x \in [ 0,1 ] , \quad P _ { n } ( x ) = \frac { 1 } { n ! } \frac { \mathrm { d } ^ { n } \left( x ^ { n } ( 1 - x ) ^ { n } \right) } { \mathrm { d } x ^ { n } } .$$
Let $n \in \mathbb { N } ^ { * }$. Justify that $P _ { n }$ is a polynomial function on $[ 0,1 ]$ of degree $n$ with coefficients in $\mathbb { Z }$.

We define on $[ 0,1 ]$ the function $P _ { n }$ by:
$$\forall x \in [ 0,1 ] , \quad P _ { n } ( x ) = \frac { 1 } { n ! } \frac { \mathrm { d } ^ { n } \left( x ^ { n } ( 1 - x ) ^ { n } \right) } { \mathrm { d } x ^ { n } } .$$
We set
$$P _ { n } ( x ) = \sum _ { k = 0 } ^ { n } a _ { k } x ^ { k }, \quad ( 1 - y ) ^ { n } = \sum _ { k = 0 } ^ { n } b _ { k } y ^ { k }$$
with for all $k \in \llbracket 0 , n \rrbracket , a _ { k } \in \mathbb { Z }$ and $b _ { k } \in \mathbb { Z }$.
Let $n \in \mathbb { N } ^ { * }$. Justify the existence of
$$I _ { n } = \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } \frac { ( 1 - y ) ^ { n } P _ { n } ( x ) } { 1 - x y } \mathrm {~d} x \mathrm {~d} y$$
and show that
$$I _ { n } = \sum _ { \substack { r , s = 0 \\ r \neq s } } ^ { n } a _ { r } b _ { s } J _ { r , s } + \sum _ { r = 0 } ^ { n } a _ { r } b _ { r } J _ { r , r }$$

We define on $[ 0,1 ]$ the function $P _ { n }$ by:
$$\forall x \in [ 0,1 ] , \quad P _ { n } ( x ) = \frac { 1 } { n ! } \frac { \mathrm { d } ^ { n } \left( x ^ { n } ( 1 - x ) ^ { n } \right) } { \mathrm { d } x ^ { n } } .$$
We admit that $J _ { r , r } = \zeta ( 2 ) - \sum _ { k = 1 } ^ { r } \frac { 1 } { k ^ { 2 } }$.
Let $n \in \mathbb { N } ^ { * }$. Deduce that there exist two integers $p _ { n }$ and $q _ { n }$ such that
$$I _ { n } = \frac { p _ { n } + \zeta ( 2 ) q _ { n } } { d _ { n } ^ { 2 } }$$

We have
$$I _ { n } = ( - 1 ) ^ { n } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } \frac { x ^ { n } ( 1 - x ) ^ { n } y ^ { n } ( 1 - y ) ^ { n } } { ( 1 - x y ) ^ { n + 1 } } \mathrm {~d} x \mathrm {~d} y$$
Let $n \in \mathbb { N } ^ { * }$. Deduce that
$$\left| I _ { n } \right| \leqslant \zeta ( 2 ) \left( \frac { 5 \sqrt { 5 } - 11 } { 2 } \right) ^ { n }$$

We have $I _ { n } = \frac { p _ { n } + \zeta ( 2 ) q _ { n } } { d _ { n } ^ { 2 } }$ where $p _ { n }$ and $q _ { n }$ are non-zero integers for all $n \in \mathbb { N } ^ { * }$.
Show that there exists $N \in \mathbb { N } ^ { * }$ such that for all $n \geqslant N$,
$$0 < \left| p _ { n } + \zeta ( 2 ) q _ { n } \right| \leqslant \zeta ( 2 ) \left( \frac { 5 } { 6 } \right) ^ { n }$$
One may use, without proving it, the inequality $9 \frac { 5 \sqrt { 5 } - 11 } { 2 } \leqslant \frac { 5 } { 6 }$.

Show that $\zeta ( 2 )$ is an irrational number.

We admit, only in this question, that $\zeta ( 2 ) = \frac { \pi ^ { 2 } } { 6 }$. Show that $\pi$ is an irrational number.

a) Show that if $a$ and $b$ are irrational numbers that are roots of a quadratic with rational coefficients, then $(a-b)^2$ is not a perfect square of any rational number.
b) i) If $a = r \pm \sqrt{s}$ is a quadratic surd, find a rational $x$ such that $a + x$ is irrational but $a_n = (r + (r^2 - s)) \pm \sqrt{s} \notin \mathbb{Q}$. If $a$ is not a surd, take $x = -a$.
ii) Find $y$ such that the required condition holds.

In a triangle, $E$ is the midpoint of $AC$. Let $\angle BCE = \angle ABE$. Prove that $AB + BD = CD$ (where $D$ is the midpoint of $BC$), i.e., $AB + BD = l_1 + l_2$.

Let $n_1, n_2, \ldots, n_k$ be positive integers with $\gcd(n_1, n_2, \ldots, n_k) = 1$. Show that every sufficiently large positive integer $n$ can be represented as $n = \sum_i c_i n_i$ with $c_i \geq 0$ integers.

The number of triplets $(a, b, c)$ of integers such that $a < b < c$ and $a, b, c$ are sides of a triangle with perimeter 21 is
(A) 7
(B) 8
(C) 11
(D) 12

Consider the squares of an $8 \times 8$ chessboard filled with the numbers 1 to 64 as in the figure below. If we choose 8 squares with the property that there is exactly one from each row and exactly one from each column, and add up the numbers in the chosen squares, show that the sum obtained is always 260.

1	2	3	4	5	6	7	8
9	10	11	12	13	14	15	16
17	18	19	20	21	22	23	24
25	26	27	28	29	30	31	32
33	34	35	36	37	38	39	40
41	42	43	44	45	46	47	48
49	50	51	52	53	54	55	56
57	58	59	60	61	62	63	64

Consider the squares of an $8 \times 8$ chessboard filled with the numbers 1 to 64 as in the figure below. If we choose 8 squares with the property that there is exactly one from each row and exactly one from each column, and add up the numbers in the chosen squares, show that the sum obtained is always 260.

1	2	3	4	5	6	7	8
9	10	11	12	13	14	15	16
17	18	19	20	21	22	23	24
25	26	27	28	29	30	31	32
33	34	35	36	37	38	39	40
41	42	43	44	45	46	47	48
49	50	51	52	53	54	55	56
57	58	59	60	61	62	63	64

For real numbers $x , y$ and $z$, show that $$| x | + | y | + | z | \leq | x + y - z | + | y + z - x | + | z + x - y |$$

If $n$ is a positive integer such that $8 n + 1$ is a perfect square, then
(a) $n$ must be odd
(b) $n$ cannot be a perfect square
(c) $2 n$ cannot be a perfect square
(d) none of the above.

The equation $x ^ { 3 } y + x y ^ { 3 } + x y = 0$ represents
(a) a circle
(b) a circle and a pair of straight lines
(c) a rectangular hyperbola
(d) a pair of straight lines.

A subset $W$ of the set of real numbers is called a ring if it contains 1 and if for all $a , b \in W$, the numbers $a - b$ and $a b$ are also in $W$. Let $S = \left\{ \left. \frac { m } { 2 ^ { n } } \right\rvert\, m , n \text{ integers} \right\}$ and $T = \left\{ \left. \frac { p } { q } \right\rvert\, p , q \text{ integers}, q \text{ odd} \right\}$. Then:
(a) neither $S$ nor $T$ is a ring
(b) $S$ is a ring, $T$ is not a ring.
(c) $T$ is a ring, $S$ is not a ring.
(d) both $S$ and $T$ are rings.

Suppose $a, b$ and $n$ are positive integers, all greater than one. If $a ^ { n } + b ^ { n }$ is prime, what can you say about $n$?
(A) The integer $n$ must be 2
(B) The integer $n$ need not be 2, but must be a power of 2
(C) The integer $n$ need not be a power of 2, but must be even
(D) None of the above is necessarily true

The number of triplets $(a, b, c)$ of integers such that $a < b < c$ and $a, b, c$ are sides of a triangle with perimeter 21 is
(A) 7
(B) 8
(C) 11
(D) 12

The number of triplets $(a, b, c)$ of integers such that $a < b < c$ and $a, b, c$ are sides of a triangle with perimeter 21 is
(A) 7
(B) 8
(C) 11
(D) 12

A subset $W$ of the set of real numbers is called a ring if it contains 1 and if for all $a , b \in W$, the numbers $a - b$ and $a b$ are also in $W$. Let $S = \left\{ \left. \frac { m } { 2 ^ { n } } \right\rvert\, m , n \text{ integers} \right\}$ and $T = \left\{ \left. \frac { p } { q } \right\rvert\, p , q \text{ integers}, q \text{ odd} \right\}$. Then
(A) neither $S$ nor $T$ is a ring
(B) $S$ is a ring, $T$ is not a ring
(C) $T$ is a ring, $S$ is not a ring
(D) both $S$ and $T$ are rings

Let $g : \mathbb { N } \rightarrow \mathbb { N }$ with $g ( n )$ being the product of the digits of $n$.
(a) Prove that $g ( n ) \leq n$ for all $n \in \mathbb { N }$.
(b) Find all $n \in \mathbb { N }$, for which $n ^ { 2 } - 12 n + 36 = g ( n )$.