For $(n,N) \in \mathbf{N} \times \mathbf{N}^*$, we denote by $P_{n,N}$ the set of lists $(a_1, \ldots, a_N) \in \mathbf{N}^N$ such that $\sum_{k=1}^{N} k a_k = n$. If this set is finite, we denote by $p_{n,N}$ its cardinality.
Let $n \in \mathbf{N}$. Show that $P_{n,N}$ is included in $[0,n]^N$ and non-empty for all $N \in \mathbf{N}^*$, that the sequence $(p_{n,N})_{N \geq 1}$ is increasing and that it is constant from rank $\max(n,1)$ onwards.

For $p \in \mathbb { N } ^ { * }$, let $h(x) = \mathrm{e}^{-x} P(x)$ where $P$ is a polynomial solution of $(E_p)$. We denote by $\left( b _ { n } \right) _ { n \in \mathbb { N } }$ the sequence of coefficients of the power series expansion of $h$, so that for all $x \in \mathbb { R }$, $h ( x ) = \sum _ { n = 0 } ^ { + \infty } b _ { n } x ^ { n }$. These coefficients satisfy $$\left\{ \begin{array} { l } b _ { 0 } = 0 \\ n ( n + 1 ) b _ { n + 1 } = - ( n + p ) b _ { n } , \quad \forall n \in \mathbb { N } ^ { * } . \end{array} \right.$$ Establish that, for all $n \in \mathbb { N } ^ { * } , b _ { n } = \frac { ( - 1 ) ^ { n - 1 } ( n + p - 1 ) ! } { p ! n ! ( n - 1 ) ! } b _ { 1 }$.

We apply the results of question 40 to the endomorphism $L$ defined by $Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$. We denote by $(\ell_n)_{n \in \mathbb{N}}$ its associated sequence of polynomials.
Verify that, for $n \in \mathbb{N}^*$, $$\ell_n' = \ell_{n-1}' - \ell_{n-1}$$ and $$X\ell_n'' - X\ell_n' + n\ell_n = 0$$ and $$\ell_n(X) = \sum_{k=1}^n (-1)^k \binom{n-1}{k-1} \frac{X^k}{k!}$$

Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$. An application $h$ from a non-trivial interval $J$ of $\mathbf{R}$ to $\mathbf{R}$ is said to be log-convex if, and only if, it takes values in $\mathbf{R}_+^*$ and $\ln \circ h$ is convex on $J$. The functional equation referred to as (1) is: $$\forall x \in I, (x+1)f(x) = (x+2)f(x+2)$$
Conclude that $f$ is the unique application from $I$ to $\mathbf{R}$, which is log-convex, which satisfies (1) and such that $$f(0) = \frac{\pi}{2}$$

Show that $(B_{n})_{n \in \mathbb{N}}$ is the unique sequence of polynomials satisfying $$\begin{cases} B_{0} = 1 \\ \forall n \in \mathbb{N}^{*},\, B_{n}' = n B_{n-1} \\ \forall n \in \mathbb{N}^{*},\, \displaystyle\int_{0}^{1} B_{n}(t)\,\mathrm{d}t = 0 \end{cases}$$

We denote by $\lambda_1, \cdots, \lambda_\ell$ the eigenvalues of $A$, with $\lambda_i \neq \lambda_j$ if $i \neq j$. We denote by $m_1 \geqslant 1, \cdots, m_\ell \geqslant 1$ the multiplicities of $\lambda_1, \cdots, \lambda_\ell$ respectively as roots of $\varphi_A$. We set $u(A) = Q(A)$ where $Q$ is the unique polynomial in $\mathbb{C}_{m-1}[X]$ satisfying $\forall i \in \llbracket 1; \ell \rrbracket, \forall k \in \llbracket 0; m_i - 1 \rrbracket, Q^{(k)}(\lambda_i) = U^{(k)}(\lambda_i)$.
Let $P \in \mathbb{C}[X]$. Show that $u(A) = P(A)$ if and only if $$\forall i \in \llbracket 1; \ell \rrbracket, \forall k \in \llbracket 0; m_i - 1 \rrbracket, P^{(k)}(\lambda_i) = U^{(k)}(\lambda_i)$$

Define two sequences $(w_{n,k})_{n,k \geq 0}$ and $(w_n(k))_{n,k \geq 0}$ by the formulas $$w_{n,k} = n! \sum_{i=0}^{n-k} \frac{u_i}{i!} \quad \text{and} \quad \sum_{n=0}^{\infty} w_n(k) x^n = \left(1 - s_1 x - \cdots - s_r x^r\right)^k \sum_{n=0}^{\infty} w_{n,k}\, x^n.$$ Show the equality $w_n(k) = v_n(k)$ for all $n$ and $k$ such that $n \geq kr$.

Deduce that there exists an integer $k_0$ such that $$v_n(k) = 0 \text{ for all } k \geq k_0 \text{ and } kr \leq n \leq 10kr.$$

Conclude that $v_n(k_0) = 0$ for all $n \geq k_0 r$.

Let $\left( a _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ and $\left( c _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ be two sequences of strictly positive real numbers. We define the sequence $\left( b _ { n } \right) _ { n \in \mathbb { N } }$ by
$$\left\{ \begin{array} { l } b _ { 0 } = - 1 \\ \forall n \in \mathbb { N } ^ { * } , \quad b _ { n } = - \frac { 1 } { n } \sum _ { k = 1 } ^ { n } \frac { 1 } { k + 1 } b _ { n - k } \end{array} \right.$$
By considering $c _ { n } = \frac { ( n + 1 ) ^ { n } } { n ^ { n - 1 } }$, deduce the Carleman-Yang inequality:
$$\sum _ { n = 1 } ^ { + \infty } \left( \prod _ { k = 1 } ^ { n } a _ { k } \right) ^ { 1 / n } \leqslant \mathrm { e } \sum _ { n = 1 } ^ { + \infty } \left( 1 - \sum _ { k = 1 } ^ { + \infty } \frac { b _ { k } } { ( n + 1 ) ^ { k } } \right) a _ { n }$$

Let $k \in \mathbb{N}^*$ and $(a_1, \ldots, a_k) \in (\mathbb{N}^*)^k$ a $k$-tuple of strictly positive integers. When $k \geq 2$, we assume they are coprime as a set. We define a function $P : \mathbb{N} \rightarrow \mathbb{C}$ by setting for all $n \in \mathbb{N}$: $$P(n) = \operatorname{Card}\left\{(n_1, \ldots, n_k) \in \mathbb{N}^k : n_1 a_1 + \cdots + n_k a_k = n\right\}.$$ We assume $k = 2$. We assume in this question that $(a_1, a_2) = (2, 3)$. Construct a function $\phi : \mathbb{Z} \rightarrow \mathbb{Z}$ of period 6 such that $P(n) = \frac{n + \phi(n)}{6}$ for all $n \in \mathbb{N}$.

Find the general term $T_r$ of the sequence $2, 7, 14, 23, 34, \ldots$

Let $a _ { n }$ be the number of subsets of $\{ 1,2 , \ldots , n \}$ that do not contain any two consecutive numbers. Then
(A) $a _ { n } = a _ { n - 1 } + a _ { n - 2 }$
(B) $a _ { n } = 2 a _ { n - 1 }$
(C) $a _ { n } = a _ { n - 1 } - a _ { n - 2 }$
(D) $a _ { n } = a _ { n - 1 } + 2 a _ { n - 2 }$.

Consider the sequence $1,2,2,3,3,3,4,4,4,4,5,5,5,5,5 , \ldots$ obtained by writing one 1 , two 2's, three 3's and so on. What is the $2020 ^ { \text {th} }$ term in the sequence?
(A) 62
(B) 63
(C) 64
(D) 65

Consider a sequence $P_1, P_2, \ldots$ of points in the plane such that $P_1, P_2, P_3$ are non-collinear and for every $n \geq 4$, $P_n$ is the midpoint of the line segment joining $P_{n-2}$ and $P_{n-3}$. Let $L$ denote the line segment joining $P_1$ and $P_5$. Prove the following:
(a) The area of the triangle formed by the points $P_n, P_{n-1}, P_{n-2}$ converges to zero as $n$ goes to infinity.
(b) The point $P_9$ lies on $L$.

There is a rectangular plot of size $1 \times n$. This has to be covered by three types of tiles - red, blue and black. The red tiles are of size $1 \times 1$, the blue tiles are of size $1 \times 1$ and the black tiles are of size $1 \times 2$. Let $t _ { n }$ denote the number of ways this can be done. For example, clearly $t _ { 1 } = 2$ because we can have either a red or a blue tile. Also, $t _ { 2 } = 5$ since we could have tiled the plot as: two red tiles, two blue tiles, a red tile on the left and a blue tile on the right, a blue tile on the left and a red tile on the right, or a single black tile.
(a) Prove that $t _ { 2 n + 1 } = t _ { n } \left( t _ { n - 1 } + t _ { n + 1 } \right)$ for all $n > 1$.
(b) Prove that $t _ { n } = \sum _ { d \geq 0 } \binom { n - d } { d } 2 ^ { n - 2 d }$ for all $n > 0$.
Here,
$$\binom { m } { r } = \begin{cases} \frac { m ! } { r ! ( m - r ) ! } , & \text { if } 0 \leq r \leq m , \\ 0 , & \text { otherwise } , \end{cases}$$
for integers $m , r$.

Let $f : R \rightarrow R$ be a function which satisfies $f ( x + y ) = f ( x ) + f ( y ) , \forall x , y \in R$. If $f ( 1 ) = 2$ and $g ( n ) = \sum _ { k = 1 } ^ { ( n - 1 ) } f ( k ) , n \in N$ then the value of $n$, for which $g ( n ) = 20$, is
(1) 5
(2) 20
(3) 4
(4) 9

Q82. Let the positive integers be written in the form : [Figure]
If the $k ^ { \text {th } }$ row contains exactly $k$ numbers for every natural number $k$, then the row in which the number 5310 will be, is $\_\_\_\_$

5. For ALL APPLICANTS.

Let $n$ be a positive integer. An $n$-brick is a rectangle of height 1 and width $n$. A 1 -tower is defined as a 1 -brick. An $n$-tower, for $n \geqslant 2$, is defined as an $n$-brick on top of which exactly two other towers are stacked: a $k _ { 1 }$-tower and a $k _ { 2 }$-tower such that $1 \leqslant k _ { 1 } \leqslant n - 1$ and $k _ { 1 } + k _ { 2 } = n$. The $k _ { 1 }$-tower is placed to the left of the $k _ { 2 }$-tower so that side-by-side they fit exactly on top of the $n$-brick. For example, here is a 4 -tower: [Figure]
(i) Draw the four other 4 -towers.
(ii) What is the maximum height of an $n$-tower? Justify your answer.
(iii) The area of a tower is defined as the sum of the widths of its bricks. For example, the 4 -tower drawn above has area $4 + 4 + 3 + 2 = 13$. Give an expression for the area of an $n$-tower of maximum height.
(iv) Show that there are infinitely many $n$ such that there is an $n$-tower of height exactly $1 + \log _ { 2 } n$.
(v) Write $t _ { n }$ for the number of $n$-towers. We have $t _ { 1 } = 1$. For $n \geqslant 2$ give a formula for $t _ { n }$ in terms of $t _ { k }$ for $k < n$. Use your formula to compute $t _ { 6 }$.
(vi) Show that $t _ { n }$ is odd if and only if $t _ { 2 n }$ is odd.

Let $f _ { 1 }$ be a positive constant function on $[ 0,1 ]$ with $f _ { 1 } ( x ) = c$, and let $p$ and $q$ be positive real numbers with $1 / p + 1 / q = 1$. Moreover, let $\left\{ f _ { n } \right\}$ be the sequence of functions on $[ 0,1 ]$ defined by
$$f _ { n + 1 } ( x ) = p \int _ { 0 } ^ { x } \left( f _ { n } ( t ) \right) ^ { 1 / q } \mathrm {~d} t \quad ( n = 1,2 , \ldots )$$
Answer the following questions.
(1) Let $\left\{ a _ { n } \right\}$ and $\left\{ c _ { n } \right\}$ be the sequences of real numbers defined by $a _ { 1 } = 0 , c _ { 1 } = c$ and
$$\begin{aligned} & a _ { n + 1 } = q ^ { - 1 } a _ { n } + 1 \quad ( n = 1,2 , \ldots ) \\ & c _ { n + 1 } = \frac { p \left( c _ { n } \right) ^ { 1 / q } } { a _ { n + 1 } } \quad ( n = 1,2 , \ldots ) \end{aligned}$$
Show that $f _ { n } ( x ) = c _ { n } x ^ { a _ { n } }$.
(2) Let $g _ { n }$ be the function on $[ 0,1 ]$ defined by $g _ { n } ( x ) = x ^ { a _ { n } } - x ^ { p }$ for $n \geq 2$. Noting that $a _ { n } \geq 1$ holds true for $n \geq 2$, show that $g _ { n }$ attains its maximum at a point $x = x _ { n }$, and find the value of $x _ { n }$.
(3) Show that $\lim _ { n \rightarrow \infty } g _ { n } ( x ) = 0$ for any $x \in [ 0,1 ]$.
(4) Let $d _ { n }$ be defined by $d _ { n } = \left( c _ { n } \right) ^ { q ^ { n } }$. Show that $d _ { n + 1 } / d _ { n }$ converges to a finite positive value as $n \rightarrow \infty$. You may use the fact that $\lim _ { t \rightarrow \infty } ( 1 - 1 / t ) ^ { t } = 1 / \mathrm { e }$.
(5) Find the value of $\lim _ { n \rightarrow \infty } c _ { n }$.
(6) Show that $\lim _ { n \rightarrow \infty } f _ { n } ( x ) = x ^ { p }$ for any $x \in [ 0,1 ]$.

The following steps are applied to the number 123 in sequence to change the positions of its digits, and a three-digit number is obtained at each step.

1. In step 1, a number is obtained by switching the positions of the digits in the tens and hundreds places.
2. In step 2, a number is obtained by switching the positions of the digits in the ones and tens places of the number obtained in the previous step.

Continuing in this way, if the step number is odd, numbers are obtained by switching the positions of the digits in the tens and hundreds places of the number obtained in the previous step, and if the step number is even, by switching the positions of the digits in the ones and tens places of the number obtained in the previous step. Accordingly, which of the following is the number obtained after step 75?
A) 321
B) 312
C) 231
D) 213
E) 132