Let $C$ be a non-empty, convex and closed subset of $\mathbb{R}^d$. Determine explicitly $\operatorname{proj}_C$ in the following cases: $$\text{i) } C = \mathbb{R}_+^d, \quad \text{ii) } C = \left\{y \in \mathbb{R}^d : \|y\| \leqslant 1\right\}$$ $$\text{iii) } C = \left\{y \in \mathbb{R}^d : \sum_{i=1}^d y_i \leqslant 1\right\}, \quad \text{iv) } C = [-1,1]^d$$

To each function $f \in E$, we associate the function $U ( f )$ defined for all $x > 0$ by $$U ( f ) ( x ) = \left\langle k _ { x } \mid f \right\rangle = \int _ { 0 } ^ { + \infty } \left( \mathrm { e } ^ { \min ( x , t ) } - 1 \right) f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$$ Using the Cauchy-Schwarz inequality, show that for all functions $f \in E$, $$\lim _ { \substack { x \rightarrow 0 \\ x > 0 } } U ( f ) ( x ) = 0$$

We fix a choice of $\lambda$ such that $P_a(x) = x - \lambda x(x-a)(x-1)$ satisfies $P([0,1])=[0,1]$ and $P$ is increasing on $[0,1]$. Let $\left(P_a^{\circ n}\right)_{n\geq 0}$ be the sequence of polynomials defined recursively by $P_a^{\circ 0}(x) = x$ and $P_a^{\circ n+1}(x) = P_a\left(P_a^{\circ n}(x)\right)$.
Show that $P_a^{\circ n}$ converges uniformly to 1 on every compact subset of $]a,1]$ and uniformly to 0 on every compact subset of $[0,a[$.

Let $F:[0,2\pi]\times\mathbb{R}_+\rightarrow\mathcal{H}$ be the map defined by $$F(t,\theta) = \begin{pmatrix} \frac{1}{\sqrt{3}}\operatorname{sh}(t)\cos(\theta) \\ \frac{1}{\sqrt{3}}\operatorname{sh}(t)\sin(\theta) \\ \operatorname{ch}(t) \end{pmatrix}.$$ For all $(\theta,\alpha)\in[0,2\pi]\times\mathbb{R}_+$ show that $$d\left(F(t,\theta), F\left(t,\theta+\alpha e^{-t}\right)\right) \underset{t\rightarrow+\infty}{\longrightarrow} \operatorname{arcch}\left(1+\frac{\alpha^2}{8}\right)$$ and that the convergence is uniform on every compact subset of $[0,2\pi]\times\mathbb{R}_+$.

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let the application $f_A$ defined on $\mathbf{R}$ by $$f_A(t) = \operatorname{det}(A + tM).$$ Show that $f_A(t) \underset{t \rightarrow 0}{=} \operatorname{det}(A) + \operatorname{det}(A) \operatorname{Tr}(A^{-1}M) t + o(t)$.
Hint: You may begin by treating the case where $A = I_n$.

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$. Let the application $f _ { A }$ defined on $\mathbf { R }$ by
$$f _ { A } ( t ) = \operatorname { det } ( A + t M )$$
Show that $f _ { A } ( t ) \underset { t \rightarrow 0 } { = } \operatorname { det } ( A ) + \operatorname { det } ( A ) \operatorname { Tr } \left( A ^ { - 1 } M \right) t + o ( t )$.
Hint: You may begin by treating the case where $A = I _ { n }$.

We fix $\boldsymbol{x}, \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R})$. Show that if $V_{n}(\boldsymbol{x}) = \frac{1}{n} \sum_{i=1}^{n} |\boldsymbol{x}_{i} - \overline{\boldsymbol{x}}|^{2}$ and $V_{n}(\boldsymbol{y}) = \frac{1}{n} \sum_{i=1}^{n} |\boldsymbol{y}_{i} - \overline{\boldsymbol{y}}|^{2}$ then $$\delta(\boldsymbol{x}, \boldsymbol{y})^{2} = nV_{n}(\boldsymbol{x}) + nV_{n}(\boldsymbol{y}) - 2\sup_{R \in \mathrm{SO}_{d}(\mathbb{R})} \langle Z(\boldsymbol{x}, \boldsymbol{y}), R \rangle$$ where $Z(\boldsymbol{x}, \boldsymbol{y})$ is a matrix that we will specify.

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$. We now assume that $h > 0$.
We set $\gamma_h = G_h''(u_h)$ and we denote $f_h : x \longmapsto \frac{\widehat{G}_h(x)}{x^2}$.
Show then that $$\int_{-\infty}^{+\infty} \mathrm{e}^{-n\widehat{G}_h\left(\frac{t}{\sqrt{n}}\right)} \mathrm{d}t \xrightarrow[n \rightarrow +\infty]{} \sqrt{\frac{2\pi}{\gamma_h}}$$ then conclude that $\psi(h) = -G_h(u_h)$.

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 $$\mathbb{P}\left(n^{1/4} M_n \leqslant x\right) \underset{n \rightarrow +\infty}{\longrightarrow} \int_{-\infty}^{x} \varphi_\infty(u) \mathrm{d}u$$

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 $P ( x )$ be a polynomial with real coefficients. Let $\alpha _ { 1 } , \ldots , \alpha _ { k }$ be the distinct real roots of $P ( x ) = 0$. If $P ^ { \prime }$ is the derivative of $P$, show that for each $i = 1,2 , \ldots , k$,
$$\lim _ { x \rightarrow \alpha _ { i } } \frac { \left( x - \alpha _ { i } \right) P ^ { \prime } ( x ) } { P ( x ) } = r _ { i } ,$$
for some positive integer $r _ { i }$.

If $R$ is the smallest equivalence relation on the set $\{ 1,2,3,4 \}$ such that $\{ ( 1,2 ) , ( 1,3 ) \} \subset R$, then the number of elements in $R$ is
(1) 10
(2) 12
(3) 8
(4) 15

Let the relations $R _ { 1 }$ and $R _ { 2 }$ on the set $X = \{ 1,2,3 , \ldots , 20 \}$ be given by $R _ { 1 } = \{ ( x , y ) : 2 x - 3 y = 2 \}$ and $R _ { 2 } = \{ ( x , y ) : - 5 x + 4 y = 0 \}$. If $M$ and $N$ be the minimum number of elements required to be added in $R _ { 1 }$ and $R _ { 2 }$, respectively, in order to make the relations symmetric, then $M + N$ equals
(1) 12
(2) 16
(3) 8
(4) 10

Let $A = \{ 1,2,3 \}$. The number of relations on $A$, containing $( 1,2 )$ and $( 2,3 )$, which are reflexive and transitive but not symmetric, is $\_\_\_\_$

Consider the integral expression
$$P = ( x - 1 ) ^ { 2 } ( y + 5 ) + ( 2 x - 3 ) ( y + 4 ) - ( x - 1 ) ^ { 2 } .$$
(1) $P$ can be transformed into
$$P = \left( x ^ { 2 } - \mathbf { K } \right) ( y + \mathbf { L } ) .$$
(2) The pairs $( x , y )$ of integers $x$ and $y$ which give $P = 7$ are
$$( \pm \mathbf { M } , \mathbf { N O P } ) , \quad ( \pm \mathbf { Q } , \mathbf { R S } ) .$$
(3) Let $a$ be a rational number. If $x = \sqrt { 2 } + 2 \sqrt { 3 }$ and $y = a + \sqrt { 6 }$, then the value of $a$ such that the value of $P$ is a rational number is $\mathbf { T U }$.

Let $n$ be a natural number and $a$ be a real number, where $a \neq 0$. Suppose that the integral expression $x ^ { n } + y ^ { n } + z ^ { n } + a ( x y + y z + z x )$ can be expressed as the product of $x + y + z$ and an integral expression $P$ in $x$, $y$ and $z$, i.e.
$$x ^ { n } + y ^ { n } + z ^ { n } + a ( x y + y z + z x ) = ( x + y + z ) P .$$
We are to find the values of $n$ and $a$.
The equality (1) holds for all $x$, $y$ and $z$. So, consider for example, two triples of $x$, $y$ and $z$ that satisfy $x + y + z = 0$:
$$x = y = 1 , \quad z = - \mathbf { A }$$
and
$$x = y = - \frac { \mathbf { B } } { \mathbf { C } } , \quad z = 1 .$$
By substituting each triple in (1), we obtain the following two equations:
$$\left( - \mathbf { A } \right) ^ { n } = \mathbf { D } \text{ a} - \frac{\mathbf{E}}{\mathbf{F}}$$
$$\left( - \frac { \mathbf { B } } { \mathbf { C } } \right) ^ { n } = \frac { \mathbf{E} }{ \mathbf{F} } \text{ a} - \frac { \mathbf { H } } { \mathbf { I } } .$$
From these two equations, we get an equation in $a$. Solving this, we obtain $a = \mathbf { K }$ and hence by the first equation that $n = \mathbf { L }$.
Conversely, when $a = \mathbf{K}$ and $n = \mathbf{L}$, there exists a $P$ such that (1) holds, and hence these values of $a$ and $n$ are the solution.

1. Let $f(x) = ax^6 - bx^4 + 3x - \sqrt{2}$, where $a, b$ are non-zero real numbers. Then the value of $f(5) - f(-5)$ is
(1) $-30$
(2) $0$
(3) $2\sqrt{2}$
(4) $30$
(5) Cannot be determined (depends on $a, b$)

As shown in the figure, consider a rectangular stone block with a vertex $A$ and a face containing point $A$. Let the midpoints of the edges of this face be $B , E , F , D$ respectively. Another face of the rectangular block containing point $B$ has its edge midpoints as $B , C , H , G$ respectively. Given that $\overline { B C } = 8$ and $\overline { B D } = \overline { D C } = 9$. The stone block is now cut to remove eight corners, such that the cutting plane for each corner passes through the midpoints of the three adjacent edges of that corner.
How many faces does the stone block have after cutting the corners? (Single choice question, 3 points)
(1) Octahedron
(2) Decahedron
(3) Dodecahedron
(4) Tetradecahedron
(5) Hexadecahedron

The diagram shows a square-based pyramid with base $P Q R S$ and vertex $O$. All the edges of the pyramid are of length 20 metres.
Find the shortest distance, in metres, along the outer surface of the pyramid from $P$ to the midpoint of $O R$.
A $10 \sqrt { 5 - 2 \sqrt { 3 } }$ B $10 \sqrt { 3 }$ C $10 \sqrt { 5 }$ D $10 \sqrt { 7 }$ E $10 \sqrt { 5 + 2 \sqrt { 3 } }$

Nine people are sitting in the squares of a 3 by 3 grid，one in each square，as shown． Two people are called neighbours if they are sitting in squares that share a side． （People in diagonally adjacent squares，which only have a point in common，are not called neighbours．）
Each of the nine people in the grid is either a truth－teller who always tells the truth， or a liar who always lies．
Every person in the grid says：＇My neighbours are all liars＇． Given only this information，what are the smallest number and the largest number of people who could be telling the truth？

The function f is such that
$$\mathrm { f } ( m n ) = \begin{cases} \mathrm { f } ( m ) \mathrm { f } ( n ) & \text { if } m n \text { is a multiple of } 3 \\ m n & \text { if } m n \text { is not a multiple of } 3 \end{cases}$$
for all positive integers $m$ and $n$. Given that $\mathrm { f } ( 9 ) + \mathrm { f } ( 16 ) - \mathrm { f } ( 24 ) = 0$, what is the value of $\mathrm { f } ( 3 )$ ? A $\frac { 8 } { 3 }$ B $2 \sqrt { 2 }$ C 3 D $\frac { 16 } { 5 }$ E $3 \sqrt { 2 }$ F 4

A rectangular piece of paper ABCD shown below is folded so that vertices B and D coincide. Let E be the folding point on side [AB] such that $|AE| = 1$ unit.
As a result of the folding, the overlapping parts of the paper form a dark-colored equilateral triangular region DEF.
Accordingly, what is the area of the paper in square units?
A) $6\sqrt{2}$ B) $2\sqrt{2}$ C) $4\sqrt{3}$ D) $3\sqrt{3}$ E) $4\sqrt{2}$

Teacher Cemal conducted the following activity step by step with his students in a geometry lesson and asked them a question at the end of the activity.

• Let us draw a line segment AB of length 8 cm.
• Let us open our compass to 5 cm.
• By placing the sharp point of the compass first at point A and then at point B, let us draw two circles.
• Let us name the intersection points of these two circles as C and D.
• Let us form the quadrilateral ACBD with vertices at points A, B, C, and D.

• What is the area of the quadrilateral region ACBD in $\mathrm { cm } ^ { 2 }$?

According to this, what is the answer to the question asked by Teacher Cemal?
A) 20
B) 24
C) 25
D) 26
E) 32

A wooden piece in the shape of a square right prism with a square base has a base edge length equal to 2 times its height. When a cube with an edge length equal to the height of the wooden piece is removed from inside this wooden piece, the surface area of the resulting shape in the final state is 8 square units more than the surface area of the wooden piece in the initial state.
Accordingly, what is the volume of the wooden piece in the initial state in cubic units?
A) 32
B) 80
C) 108
D) 144
E) 256

The volume of a rectangular prism is equal to the product of its base area and height.
A closed glass container in the shape of a rectangular prism contains 360 cubic units of water. When the container is placed on a flat surface with different faces completely touching the surface, the height of the water is 2 units, 4 units, and 5 units respectively.
Accordingly, what is the volume of the container in cubic units?
A) 540
B) 720
C) 840
D) 960
E) 4080