In this subsection, we still assume that $J_n = J_n^{(\mathrm{C})}$. Moreover, we assume that $\beta = 1$ and $h = 0$.
We denote $Z_\infty = \int_{-\infty}^{+\infty} \exp\left(-\frac{x^4}{12}\right) \mathrm{d}x$ and we consider the function $$\varphi_\infty : x \longmapsto \frac{1}{Z_\infty} \exp\left(-\frac{x^4}{12}\right).$$
Justify that, for any continuous and bounded function $f$ on $\mathbb{R}$, the quantity $$E_{n,f} = \int_{-\infty}^{+\infty} \mathbb{E}\left(f\left(\frac{t}{n^{1/4}} + n^{1/4} M_n\right)\right) \exp\left(-\frac{t^2}{2}\right) \frac{\mathrm{d}t}{\sqrt{2\pi}}$$ is well defined.

In this subsection, we still assume that $J_n = J_n^{(\mathrm{C})}$. Moreover, we assume that $\beta = 1$ and $h = 0$.
We denote $Z_\infty = \int_{-\infty}^{+\infty} \exp\left(-\frac{x^4}{12}\right) \mathrm{d}x$ and we consider the function $$\varphi_\infty : x \longmapsto \frac{1}{Z_\infty} \exp\left(-\frac{x^4}{12}\right).$$
An analogous proof to that of the previous subsection allows us to show that, for any continuous and bounded function $f$ on $\mathbb{R}$, $$E_{n,f} \xrightarrow[n \rightarrow +\infty]{} \int_{-\infty}^{+\infty} f(u) \varphi_\infty(u) \mathrm{d}u$$
Let $K \in \mathbb{R}_+^*$, and let $f$ be a $K$-Lipschitz function and bounded on $\mathbb{R}$. Show that $$\left|E_{n,f} - \mathbb{E}\left(f\left(n^{1/4} M_n\right)\right)\right| \leqslant \frac{2K}{n^{1/4}\sqrt{2\pi}}$$ and deduce that $$\mathbb{E}\left(f\left(n^{1/4} M_n\right)\right) \xrightarrow[n \rightarrow +\infty]{} \int_{-\infty}^{+\infty} f(u) \varphi_\infty(u) \mathrm{d}u$$

In this subsection, we still assume that $J_n = J_n^{(\mathrm{C})}$. Moreover, we assume that $\beta = 1$ and $h = 0$.
We are given $x \in \mathbb{R}$ and $\varepsilon > 0$. Let $k$ be a non-zero natural integer such that $k \geqslant \frac{2}{\varepsilon Z_\infty}$. We define the function $$f_k : u \in \mathbb{R} \longmapsto \begin{cases} 1 & \text{if } u \leqslant x \\ 1 - k(u-x) & \text{if } x < u \leqslant x + \frac{1}{k} \\ 0 & \text{otherwise} \end{cases}$$
Show that $f_k$ is $k$-Lipschitz on $\mathbb{R}$.

In this subsection, we still assume that $J_n = J_n^{(\mathrm{C})}$. Moreover, we assume that $\beta = 1$ and $h = 0$.
We denote $Z_\infty = \int_{-\infty}^{+\infty} \exp\left(-\frac{x^4}{12}\right) \mathrm{d}x$ and we consider the function $$\varphi_\infty : x \longmapsto \frac{1}{Z_\infty} \exp\left(-\frac{x^4}{12}\right).$$
We are given $x \in \mathbb{R}$ and $\varepsilon > 0$. Let $k$ be a non-zero natural integer such that $k \geqslant \frac{2}{\varepsilon Z_\infty}$. We define the function $$f_k : u \in \mathbb{R} \longmapsto \begin{cases} 1 & \text{if } u \leqslant x \\ 1 - k(u-x) & \text{if } x < u \leqslant x + \frac{1}{k} \\ 0 & \text{otherwise} \end{cases}$$
Deduce that there exists $n_0 \in \mathbb{N}$ such that, for all $n \geqslant n_0$, $$\mathbb{P}\left(n^{1/4} M_n \leqslant x\right) \leqslant \frac{\varepsilon}{2} + \int_{-\infty}^{x + \frac{1}{k}} \varphi_\infty(u) \mathrm{d}u$$

In this subsection, we still assume that $J_n = J_n^{(\mathrm{C})}$. Moreover, we assume that $\beta = 1$ and $h = 0$.
We denote $Z_\infty = \int_{-\infty}^{+\infty} \exp\left(-\frac{x^4}{12}\right) \mathrm{d}x$ and we consider the function $$\varphi_\infty : x \longmapsto \frac{1}{Z_\infty} \exp\left(-\frac{x^4}{12}\right).$$
Show finally that $$\mathbb{P}\left(n^{1/4} M_n \leqslant x\right) \underset{n \rightarrow +\infty}{\longrightarrow} \int_{-\infty}^{x} \varphi_\infty(u) \mathrm{d}u$$

Show that $n^4 + 4^n$ is composite for all integers $n > 1$.

Prove that $2^n < \dbinom{2n}{n} < \dfrac{2^n}{\prod_{j=0}^{n-1}\left(1 - \frac{j}{n}\right)}$ for all positive integers $n$.

Consider the function $f ( x ) = x ^ { n } ( 1 - x ) ^ { n } / n !$, where $n \geq 1$ is a fixed integer. Let $f ^ { ( k ) }$ denote the $k$-th derivative of $f$. Which of the following is true for all $k \geq 1$?
(a) $f ^ { ( k ) } ( 0 )$ and $f ^ { ( k ) } ( 1 )$ are integers.
(b) $f ^ { ( k ) } ( 0 )$ is an integer, but not $f ^ { ( k ) } ( 1 )$
(c) $f ^ { ( k ) } ( 1 )$ is an integer, but not $f ^ { ( k ) } ( 0 )$
(d) Neither $f ^ { ( k ) } ( 1 )$ nor $f ^ { ( k ) } ( 0 )$ is an integer.

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

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that for any two real numbers $x$ and $y$, $$|f(x) - f(y)| \leq 7|x - y|^{201}$$ Then,
(A) $f(101) = f(202) + 8$
(B) $f(101) = f(201) + 1$
(C) $f(101) = f(200) + 2$
(D) None of the above.

Let $k , n$ and $r$ be positive integers.
(a) Let $Q ( x ) = x ^ { k } + a _ { 1 } x ^ { k + 1 } + \cdots + a _ { n } x ^ { k + n }$ be a polynomial with real coefficients. Show that the function $\frac { Q ( x ) } { x ^ { k } }$ is strictly positive for all real $x$ satisfying
$$0 < | x | < \frac { 1 } { 1 + \sum _ { i = 1 } ^ { n } \left| a _ { i } \right| }$$
(b) Let $P ( x ) = b _ { 0 } + b _ { 1 } x + \cdots + b _ { r } x ^ { r }$ be a non-zero polynomial with real coefficients. Let $m$ be the smallest number such that $b _ { m } \neq 0$. Prove that the graph of $y = P ( x )$ cuts the $x$-axis at the origin (i.e. $P$ changes sign at $x = 0$) if and only if $m$ is an odd integer.

Let $a \geq b \geq c > 0$ be real numbers such that for all $n \in \mathbb { N }$, there exist triangles of side lengths $a ^ { n } , b ^ { n } , c ^ { n }$. Prove that the triangles are isosceles.

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be two functions. Consider the following two statements: $\mathbf { P ( 1 ) }$: If $\lim _ { x \rightarrow 0 } f ( x )$ exists and $\lim _ { x \rightarrow 0 } f ( x ) g ( x )$ exists, then $\lim _ { x \rightarrow 0 } g ( x )$ must exist. $\mathbf { P ( 2 ) }$: If $f , g$ are differentiable with $f ( x ) < g ( x )$ for every real number $x$, then $f ^ { \prime } ( x ) < g ^ { \prime } ( x )$ for all $x$. Then, which one of the following is a correct statement?
(A) Both $\mathrm { P } ( 1 )$ and $\mathrm { P } ( 2 )$ are true.
(B) Both $P ( 1 )$ and $P ( 2 )$ are false.
(C) $\mathrm { P } ( 1 )$ is true and $\mathrm { P } ( 2 )$ is false.
(D) $\mathrm { P } ( 1 )$ is false and $\mathrm { P } ( 2 )$ is true.

A subset $S$ of the plane is called convex if given any two points $x$ and $y$ in $S$, the line segment joining $x$ and $y$ is contained in $S$. A quadrilateral is called convex if the region enclosed by the edges of the quadrilateral is a convex set. Show that given a convex quadrilateral $Q$ of area 1, there is a rectangle $R$ of area 2 such that $Q$ can be drawn inside $R$.

There are three cities each of which has exactly the same number of citizens, say $n$. Every citizen in each city has exactly a total of $n + 1$ friends in the other two cities. Show that there exist three people, one from each city, such that they are friends. We assume that friendship is mutual (that is, a symmetric relation).

Let $f : \mathbb { Z } \rightarrow \mathbb { Z }$ be a function satisfying $f ( 0 ) \neq 0 = f ( 1 )$. Assume also that $f$ satisfies equations $( \mathbf { A } )$ and $( \mathbf { B } )$ below.
$$\begin{aligned} f ( x y ) & = f ( x ) + f ( y ) - f ( x ) f ( y ) \\ f ( x - y ) f ( x ) f ( y ) & = f ( 0 ) f ( x ) f ( y ) \end{aligned}$$
for all integers $x , y$.
(i) Determine explicitly the set $\{ f ( a ) : a \in \mathbb { Z } \}$.
(ii) Assuming that there is a non-zero integer $a$ such that $f ( a ) \neq 0$, prove that the set $\{ b : f ( b ) \neq 0 \}$ is infinite.

If a given equilateral triangle $\Delta$ of side length $a$ lies in the union of five equilateral triangles of side length $b$, show that there exist four equilateral triangles of side length $b$ whose union contains $\Delta$.

Let $a , b , c$ be three real numbers which are roots of a cubic polynomial, and satisfy $a + b + c = 6$ and $a b + b c + a c = 9$. Suppose $a < b < c$. Show that
$$0 < a < 1 < b < 3 < c < 4$$

Let $a , b , c$ and $d$ be four non-negative real numbers where $a + b + c + d = 1$. The number of different ways one can choose these numbers such that $a ^ { 2 } + b ^ { 2 } + c ^ { 2 } + d ^ { 2 } = \max \{ a , b , c , d \}$ is
(A) 1 .
(B) 5 .
(C) 11 .
(D) 15 .

Amongst all polynomials $p ( x ) = c _ { 0 } + c _ { 1 } x + \cdots + c _ { 10 } x ^ { 10 }$ with real coefficients satisfying $| p ( x ) | \leq | x |$ for all $x \in [ - 1,1 ]$, what is the maximum possible value of $\left( 2 c _ { 0 } + c _ { 1 } \right) ^ { 10 }$ ?
(A) $4 ^ { 10 }$
(B) $3 ^ { 10 }$
(C) $2 ^ { 10 }$
(D) 1

Let $f : [ 0,1 ] \rightarrow \mathbb { R }$ be a continuous function which is differentiable on $( 0,1 )$. Prove that either $f$ is a linear function $f ( x ) = a x + b$ or there exists $t \in ( 0,1 )$ such that $| f ( 1 ) - f ( 0 ) | < \left| f ^ { \prime } ( t ) \right|$.

Suppose $n \geq 2$. Consider the polynomial
$$Q _ { n } ( x ) = 1 - x ^ { n } - ( 1 - x ) ^ { n } .$$
Show that the equation $Q _ { n } ( x ) = 0$ has only two real roots, namely 0 and 1.

Let $ABCD$ be a quadrilateral with all internal angles $< \pi$. Squares are drawn on each side as shown in the picture below. Let $\Delta _ { 1 } , \Delta _ { 2 } , \Delta _ { 3 }$ and $\Delta _ { 4 }$ denote the areas of the shaded triangles shown. Prove that
$$\Delta _ { 1 } - \Delta _ { 2 } + \Delta _ { 3 } - \Delta _ { 4 } = 0 .$$

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function which is differentiable at 0. Define another function $g : \mathbb { R } \rightarrow \mathbb { R }$ as follows:
$$g ( x ) = \begin{cases} f ( x ) \sin \left( \frac { 1 } { x } \right) & \text { if } x \neq 0 \\ 0 & \text { if } x = 0 \end{cases}$$
Suppose that $g$ is also differentiable at 0. Prove that
$$g ^ { \prime } ( 0 ) = f ^ { \prime } ( 0 ) = f ( 0 ) = g ( 0 ) = 0$$