We classify cycles of length $k$ into three subsets:

• the set $\mathcal{A}_{k}$, consisting of cycles where at least one edge appears only once;
• the set $\mathcal{B}_{k}$, consisting of cycles where all edges appear exactly twice;
• the set $\mathcal{C}_{k}$, consisting of cycles where all edges appear at least twice and there exists at least one that appears at least three times.

Show that, for every cycle $\vec{\imath}$ belonging to $\mathcal{C}_{k}$, $|\vec{\imath}| \leqslant \frac{k+1}{2}$.

Show that the number of cycles of length $k$ in $\llbracket 1,n \rrbracket$ passing through $\ell$ distinct vertices is at most $n^{\ell} \ell^{k}$.

Cycles of length $k$ are classified into three subsets: the set $\mathcal{A}_{k}$, consisting of cycles where at least one edge appears only once; the set $\mathcal{B}_{k}$, consisting of cycles where all edges appear exactly twice; the set $\mathcal{C}_{k}$, consisting of cycles where all edges appear at least twice and there exists at least one that appears at least three times.
Show that, for any cycle $\vec{\imath}$ belonging to $\mathcal{C}_{k}$, $|\vec{\imath}| \leqslant \frac{k+1}{2}$.

Let $n \in \mathbb { N }$ and $\gamma = \left( \gamma _ { 1 } , \ldots , \gamma _ { 2 n + 2 } \right)$ be a Dyck path of length $2 n + 2$. Let $r = \max \left\{ i \in \llbracket 0 , n \rrbracket \mid s _ { \gamma } ( 2 i ) = 0 \right\}$. We assume $0 < r < n$ and we consider the paths $\alpha = \left( \gamma _ { 1 } , \ldots , \gamma _ { 2 r } \right)$ and $\beta = \left( \gamma _ { 2 r + 2 } , \ldots , \gamma _ { 2 n + 1 } \right)$. Justify using a figure that $\gamma _ { 2 r + 1 } = 1 , \gamma _ { 2 n + 2 } = - 1$ and that $\alpha$ and $\beta$ are Dyck paths.

Deduce $\forall n \in \mathbb { N } , C _ { n } = \frac { 1 } { n + 1 } \binom { 2 n } { n }$.

Show that for all $k\geq 1$, we have $$|S_{\mathrm{prim}}(p^{2k})| \geq \frac{1}{2}p^{2k}.$$

Let $(u,v)\in\mathbb{Z}^2\setminus(0,0)$ and $p$ a prime number congruent to 1 modulo 4. Show that if $p$ does not divide $u^2+v^2$, then $$|L_{\mathrm{prim}}(u,v,p)| \leq 2 \quad \text{and} \quad |L_{\mathrm{prim}}(u,v,p^2)| \leq 1.$$

Let $d$ be an odd integer. Show that $$|S_{\mathrm{prim}}(d^2)| \geq d^2\, 2^{-|\mathcal{P}(d)|}.$$ Deduce that, for all $\varepsilon>0$, we have $$d^{2-\varepsilon} = \underset{d\rightarrow+\infty}{o}\left(S_{\mathrm{prim}}(d^2)\right).$$

Let $(u,v)\neq(0,0)$. Show the existence of a constant $C$ (depending on $(u,v)$) such that for all $d>0$, $$|L_{\mathrm{prim}}(u,v,d)| \leq C\, 2^{|\mathcal{P}(d)|}.$$ Deduce that, for all $\varepsilon>0$, we have $$|L_{\mathrm{prim}}(u,v,d)| = \underset{d\rightarrow+\infty}{o}\left(d^\varepsilon\right).$$

Let $x$ be a positive real number greater than or equal to 1 and $q \in \mathbb{N}^{*}$. Justify that the quantity $$\operatorname{Card}\{n \in \mathbb{N} \cap [1,x] : n \equiv 0 \pmod{q}\} - \frac{x}{q}$$ is bounded in absolute value by a real number independent of $x$ and $q$.

For any non-zero natural integer $n$, we set $$\omega(n) = \operatorname{Card}\{p \text{ prime} : p \mid n\} = \sum_{\substack{p \mid n \\ p \text{ prime}}} 1.$$ We define $\mathscr{S} = \left\{n \geqslant 3 : \left|\frac{\omega(n) - \ln_{2}(n)}{\sqrt{\ln_{2}(n)}}\right| \geqslant \left(\ln_{2}(n)\right)^{1/4}\right\}$. Show that $$\lim_{x \rightarrow +\infty} \frac{1}{x} \operatorname{Card}\{n \leqslant x : n \in \mathscr{S}\} = 0$$ One may begin by writing $\operatorname{Card}(\mathscr{S} \cap [1,x]) \underset{x \rightarrow +\infty}{=} \operatorname{Card}(\mathscr{S} \cap [\sqrt{x}, x]) + O(\sqrt{x})$ and note that in the sum on the right-hand side, the difference $\left|\ln_{2}(n) - \ln_{2}(x)\right|$ remains bounded.

Let $x$ be a positive real number greater than or equal to 1 and $q \in \mathbb{N}^*$. Justify that the quantity $$\operatorname{Card}\{n \in \mathbb{N} \cap [1,x] : n \equiv 0 \pmod{q}\} - \frac{x}{q}$$ is bounded in absolute value by a real number independent of $x$ and $q$.

For any non-zero natural integer $n$, we set $$\omega(n) = \operatorname{Card}\{p \text{ prime} : p \mid n\} = \sum_{\substack{p \mid n \\ p \text{ prime}}} 1.$$ We set $\mathscr{S} = \left\{n \geqslant 3 : \left|\frac{\omega(n) - \ln_2(n)}{\sqrt{\ln_2(n)}}\right| \geqslant \left(\ln_2(n)\right)^{1/4}\right\}$. Show that $$\lim_{x \rightarrow +\infty} \frac{1}{x} \operatorname{Card}\{n \leqslant x : n \in \mathscr{S}\} = 0$$ One may begin by writing $\operatorname{Card}(\mathscr{S} \cap [1,x]) \underset{x \rightarrow +\infty}{=} \operatorname{Card}(\mathscr{S} \cap [\sqrt{x}, x]) + O(\sqrt{x})$ and note that in the sum on the right-hand side, the difference $\left|\ln_2(n) - \ln_2(x)\right|$ remains bounded.

a) Show that among any 2-coloring (Red/Blue) of the vertices of a regular hexagon and its center, there must exist a monochromatic equilateral triangle.
b) Extend the argument to show the result holds for a larger configuration of points.

a) $n$ lines are drawn through a point $A$ inside a circle, creating chords. Show that the number of regions created inside the circle is $(n+1)^2$, assuming no three lines meet at any point other than $A$.
b) Using the result from part a), find the total number of regions when additional lines are drawn, and show the total is $(n+1)^2 + (2n+1)n$.

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

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 number is called a palindrome if it reads the same backward or forward. For example, 112211 is a palindrome. How many 6-digit palindromes are divisible by 495?
(A) 10
(B) 11
(C) 30
(D) 45

An examination has 20 questions. For each question the marks that can be obtained are either $-1$ or $0$ or $4$. Let $S$ be the set of possible total marks that a student can score in the examination. Then, the number of elements in $S$ is
(A) 93
(B) 94
(C) 95
(D) 96 .

There are 128 numbers $1,2 , \ldots , 128$ which are arranged in a circular pattern in clockwise order. We start deleting numbers from this set in a clockwise fashion as follows. First delete the number 2, then skip the next available number (which is 3 ) and delete 4 . Continue in this manner, that is, after deleting a number, skip the next available number clockwise and delete the number available after that, till only one number remains. What is the last number left ?
(A) 1
(B) 63
(C) 127
(D) None of the above.

Prove that every positive rational number can be expressed uniquely as a finite sum of the form
$$a _ { 1 } + \frac { a _ { 2 } } { 2 ! } + \frac { a _ { 3 } } { 3 ! } + \cdots + \frac { a _ { n } } { n ! } ,$$
where $a _ { n }$ are integers such that $0 \leq a _ { n } \leq n - 1$ for all $n > 1$.

A $3 \times 3$ magic square is a $3 \times 3$ rectangular array of positive integers such that the sum of the three numbers in any row, any column or any of the two major diagonals, is the same. For the following incomplete magic square

27	36
31

the column sum is
(A) 90
(B) 96
(C) 94
(D) 99

Suppose $f : \mathbb { Z } \rightarrow \mathbb { Z }$ is a non-decreasing function. Consider the following two cases: $$\begin{aligned} & \text { Case 1. } f ( 0 ) = 2 , f ( 10 ) = 8 \\ & \text { Case 2. } f ( 0 ) = - 2 , f ( 10 ) = 12 \end{aligned}$$ In which of the above cases it is necessarily true that there exists an $n$ with $f ( n ) = n$?
(A) In both cases.
(B) In neither case.
(C) In Case 1. but not necessarily in Case 2.
(D) In Case 2. but not necessarily in Case 1.

Let $A = \{ n \in [ 100,700 ] \cap \mathbb { N } : n$ is neither a multiple of 3 nor a multiple of $4 \}$. Then the number of elements in $A$ is
(1) 290
(2) 280
(3) 300
(4) 310