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 $A = I _ { n }$.

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 $\mathbf { T U }$.

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

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

Regarding a two-digit natural number $AB$, the following propositions are given: p: The number $AB$ is even. q: The number $A^{AB}$ is prime. r: $A + B = 11$
If the proposition $(p \Rightarrow q) \wedge (q' \wedge r)$ is true, what is the product $A \cdot B$?
A) 18
B) 20
C) 24
D) 28
E) 30

The surface area of a rectangular prism with edge lengths $a, b$ and $c$ is calculated with the formula
$$A = 2(a \cdot b + a \cdot c + b \cdot c)$$
Two identical rectangular prisms are placed in three different ways such that they share one face each. The surface areas of the resulting Figure 1, Figure 2, and Figure 3 are calculated as 18, 20, and 22 square units respectively.
Accordingly, what is the surface area of one of the identical prisms in square units?
A) 12 B) 13 C) 14 D) 15 E) 16

Three faces of a square right prism-shaped board are painted white, and the other three faces are painted red. The sum of the areas of the white-painted faces is 76 square units, and the sum of the areas of the red-painted faces is 12 square units.
Accordingly, what is the volume of this board in cubic units?
A) 18 B) 24 C) 27 D) 32 E) 36

A toy bear, a toy horse, and a cactus plant are placed on 3 wall shelves, each at a different height from the ground, first as shown in Figure 1, and then as shown in Figure 2. The heights that are equal in Figure 1 and Figure 2 are shown with dashed lines. It is known that the sum of the heights of the toy bear, toy horse, and cactus plant is 15 units.
Given that the height of the leftmost shelf from the ground is 18 units, what is the sum of the heights of the other two shelves from the ground?
A) 45 B) 48 C) 51 D) 54 E) 57