Two mighty frogs jump once per unit time on the number line as described in the question.
The first frog is at $x = 2^i$ at time $t = i$. How many numbers of the form $7n+1$ (with $n$ an integer) does the frog visit from $t=0$ to $t=99$ (both endpoints included)? [3 points]

Which of the following statement(s) are true?
(A) Every prime ideal of a finite commutative ring with unity is maximal.
(B) A commutative ring with unity whose set of all ideals is countably infinite is necessarily a countable ring.
(C) Let $R$ be a unique factorisation domain and $K$ be its field of fractions. There exists an irreducible element $\alpha \in R$ and an element $\beta \in K$ such that $\beta ^ { 2 } = \alpha$.
(D) Every subring $R$ (with unity) of $\mathbb { Q }$ with $\mathbb { Z } \varsubsetneqq R \varsubsetneqq \mathbb { Q }$ has infinitely many prime ideals.

Two mighty frogs jump once per unit time on the number line as described in the question.
The second frog starts at $x=0$ and jumps $i+1$ steps to the right just after $t=i$, so that at times $t=0,1,2,3,\ldots$ this frog is at positions $x=0,1,3,6,\ldots$ respectively. How many numbers of the form $7n+1$ (with $n$ an integer) does the frog visit from $t=0$ to $t=99$ (both endpoints included)? [3 points]

An integer $d$ is called a factor of an integer $n$ if there is an integer $q$ such that $n = qd$. In particular the set of factors of $n$ contains $n$ and contains 1. You are given that $2024 = 8 \times 11 \times 23$.
Write the number of even positive integers that are factors of $2024^2$. [2 points]

An integer $d$ is called a factor of an integer $n$ if there is an integer $q$ such that $n = qd$. In particular the set of factors of $n$ contains $n$ and contains 1. You are given that $2024 = 8 \times 11 \times 23$.
Write the number of ordered pairs $(a,b)$ of positive integers such that $a^2 - b^2 = 2024^2$. If there are infinitely many such pairs, write the word infinite as your answer. [3 points]

As shown in the figure below, for a natural number $n$, $n$ terms $$\left[ \frac { n } { 1 } \right] \left[ \frac { n } { 2 } \right] \left[ \frac { n } { 3 } \right] , \cdots , \left[ \frac { n } { n } \right]$$ are arranged in the $n$-th row from column 1 to column $n$ in order. (Here, $[ x ]$ is the greatest integer not exceeding $x$.)
Select all correct statements from . [4 points]
ㄱ. In row $n$, the number of terms with value 1 is $\left[ \frac { n + 1 } { 2 } \right]$. ㄴ. In row 100, the number of terms with value 3 is 8. ㄷ. In column 3, the number of terms with value 5 is 5.
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄱ, ㄴ, ㄷ

As shown in the figure below, for a natural number $n$, the $n$ terms $$\left[ \frac { n } { 1 } \right] , \left[ \frac { n } { 2 } \right] , \left[ \frac { n } { 3 } \right] , \cdots , \left[ \frac { n } { n } \right]$$ are arranged in the $n$-th row from column 1 to column $n$ in order. (Here, $[ x ]$ is the greatest integer not exceeding $x$.)
Which of the following in $\langle$Remarks$\rangle$ are correct? [4 points]
$\langle$Remarks$\rangle$ ㄱ. In the $n$-th row, the number of terms with value 1 is $\left[ \frac { n + 1 } { 2 } \right]$. ㄴ. In the 100th row, the number of terms with value 3 is 8. ㄷ. In the 3rd column, the number of terms with value 5 is 5.
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄱ, ㄴ, ㄷ

Let $f ( n )$ denote the number of positive divisors of a natural number $n$, and let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \cdots , a _ { 9 }$ be all positive divisors of 36. What is the value of $\sum _ { k = 1 } ^ { 9 } \left\{ ( - 1 ) ^ { f \left( a _ { k } \right) } \times \log a _ { k } \right\}$? [4 points]
(1) $\log 2 + \log 3$
(2) $2 \log 2 + \log 3$
(3) $\log 2 + 2 \log 3$
(4) $2 \log 2 + 2 \log 3$
(5) $3 \log 2 + 2 \log 3$

Find all prime numbers $p$ such that $p ^ { 2 } + 2$ is also a prime number.

Throughout the rest of the problem, we fix an odd integer $\ell \geq 3$ and $q$ a primitive $\ell$-th root of unity. Show that $q^2$ is a primitive $\ell$-th root of unity.

Throughout this subsection II.B, we fix two natural integers $m$ and $n$. The Chebyshev polynomials of the second kind $U_n$ have roots $\cos\left(\frac{k\pi}{n+2}\right)$ for $k = 1, \ldots, n+1$.
Let $h$ be the $\gcd$ in $\mathbb{N}$ of $m+1$ and $n+1$. By examining the common roots of $U_n$ and $U_m$, show that $U_{h-1}$ is a $\gcd$ in $\mathbb{R}[X]$ of $U_n$ and $U_m$.

Throughout this subsection II.B, we fix two natural integers $m$ and $n$. Let $g > 0$ be the $\gcd$ of $m$ and $n$. We set $m_1 = m/g$ and $n_1 = n/g$.
a) Show that if $m_1$ and $n_1$ are odd, then $T_g$ is a $\gcd$ of $T_n$ and $T_m$.
b) Show that if one of the two integers $m_1$ or $n_1$ is even, then $T_n$ and $T_m$ are coprime.
c) What can be said about the gcds of $T_n$ and $T_m$ when $m$ and $n$ are odd? When $n$ and $m$ are two distinct powers of 2?

Let $p$ be an odd prime number. For every integer $r \in \mathbb{Z}$, we denote by $\varphi(r)$ the remainder of the Euclidean division of $r^2$ by $p$. We thus have $0 \leqslant \varphi(r) \leqslant p - 1$ and $r^2 - \varphi(r) \in p\mathbb{Z}$. a) Show that the restriction of $\varphi$ to $\left\{0, \ldots, \frac{p-1}{2}\right\}$ is injective. b) We consider the sets $X = \left\{p - \varphi(r) \left\lvert\, 0 \leqslant r \leqslant \frac{p-1}{2}\right.\right\}$ and $Y = \left\{\varphi(s) + 1 \left\lvert\, 0 \leqslant s \leqslant \frac{p-1}{2}\right.\right\}$.
Show that $X$ and $Y$ are contained in $\{1, \ldots, p\}$ and that their intersection is non-empty. Deduce that there exist $u, v \in \left\{0, \ldots, \frac{p-1}{2}\right\}$ and $m \in \{1, \ldots, p-1\}$ such that $u^2 + v^2 + 1 = mp$.

We denote $\mathbb{G} = \{xe + yI + zJ + tK \mid x, y, z, t \in \mathbb{Z}\}$ the set of ``integer'' quaternions. For $q = xe + yI + zJ + tK \in \mathbb{H}$, $q^* = xe - yI - zJ - tK$ and $N(q) = x^2 + y^2 + z^2 + t^2$. We still assume that $p$ is an odd prime number. Justify that there exist $m \in \{1, \ldots, p-1\}$ and $\mu = xe + yI + zJ + tK \in \mathbb{G} \backslash \{0\}$ such that $N(\mu) = mp$. We choose $m$ minimal and assume that $m > 1$. a) Show that if $m$ were even, an even number of the integers $x, y, z, t$ would be odd and lead to a contradiction.
You may write $\left(\frac{x-y}{2}\right)^2 + \left(\frac{x+y}{2}\right)^2 = \frac{x^2 + y^2}{2}$. b) We assume $m$ is odd. Show that there exists $\nu \in \mathbb{G}$ such that $N(\mu - m\nu) < m^2$. c) Prove that $\mu' = \frac{1}{m}\mu(\mu - m\nu)^*$ is in $\mathbb{G} \backslash \{0\}$ and that $N(\mu')$ is a multiple of $p$ strictly less than $mp$. Conclude.

Show that every natural integer is a sum of four squares of integers.

Let $n \geqslant 2$ and $a_1, \ldots, a_n$ be integers not all zero. The purpose of this question is to show that there exists a matrix in $M_n(\mathbb{Z})$ whose first column is $(a_1, \ldots, a_n)$ and whose determinant is $\gcd(a_1, \ldots, a_n)$. For this we proceed by induction on $n$.
Let $N \in M_{n-1}(\mathbb{Z})$ be a matrix whose first column is $(a_2, \ldots, a_n)$. Given $u, v \in \mathbb{Q}$, we consider the matrix $$M = \left(\begin{array}{cccc|c} a_1 & 0 & \cdots & 0 & u \\ \hline & & & & va_2 \\ & N & & & va_3 \\ & & & & \vdots \\ & & & & va_n \end{array}\right).$$
4a. Express $\det M$ as a function of $\det N$, $u$ and $v$.
4b. Suppose that $a_2, \ldots, a_n$ are not all zero and that $\det N = \gcd(a_2, \ldots, a_n)$. Show that we can choose $u, v$ so that $M$ answers the question.
4c. Conclude the induction.

Let $M \in M_n(\mathbb{Z})$ with nonzero determinant. We wish to show that there exists a matrix $A$ in $\mathrm{GL}_n(\mathbb{Z})$ such that $MA$ is upper triangular and, denoting $MA = (c_{ij})$, we have the inequalities $0 < c_{11}$ and $0 \leqslant c_{ij} < c_{ii}$ for all $i, j \in \{1, \ldots, n\}$ such that $i < j$.
5a. We denote $M = (x_1 | \cdots | x_n)$. Let $x_1', \ldots, x_n'$ be the elements of $\mathbb{Z}^{n-1}$ obtained by taking the last $(n-1)$ coordinates of $x_1, \ldots, x_n$. Show that there exist $a_1, \ldots, a_n$ in $\mathbb{Q}$, not all zero, such that $\sum_{i=1}^n a_i x_i' = 0$. Show that we can choose the $a_i$ to be integers that are coprime as a set.
5b. Show that there exists a matrix $A_1$ in $\mathrm{GL}_n(\mathbb{Z})$ such that the first column of $\tilde{C} = MA_1$ has all its coefficients $\tilde{c}_{i1}$ zero except the first $\tilde{c}_{11}$ which we can take to be strictly positive.
5c. By considering for all $j = 2, \ldots, n$ the Euclidean division $\tilde{c}_{1j} = q_j \tilde{c}_{11} + r_j$, $0 \leqslant r_j < \tilde{c}_{11}$, show that we can assume $\tilde{c}_{11} > \tilde{c}_{1j}$, if necessary by changing $A_1$.
5d. Conclude by induction.

Let $M \in M_n(\mathbb{Z})$ with nonzero determinant. Show that there exists a matrix $A$ in $\mathrm{GL}_n(\mathbb{Z})$ such that $AM$ is lower triangular and, denoting $AM = (c_{ij})$, we have the inequality $0 \leqslant c_{ij} < c_{jj}$ for all $i, j \in \{1, \ldots, n\}$ such that $j < i$.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$. Show that the set $\Lambda_X$ defined in question 8a is closed under addition and that $r\left(\Lambda_X\right) = 0$.

Let $d \in \mathbb { N } ^ { * }$. Construct a map $$\left\lvert \, \begin{aligned} r : \quad \mathbb { R } _ { d } [ X ] & \rightarrow \mathbb { R } \\ P & \mapsto r ( P ) \end{aligned} \right.$$ polynomial in the coefficients of $P$, such that, if $r ( P )$ is nonzero, then the roots of $P$ in $\mathbb { C }$ are simple.
Hint: You may use the previous question.

Let $d \in \mathbb { N } ^ { * }$ and $f$ a polynomial function on $\mathbb { R } ^ { d }$. Suppose that the function $f$ is nonzero. Show that $f ^ { - 1 } ( \mathbb { R } \backslash \{ 0 \} )$ is dense in $\mathbb { R } ^ { d }$.
Hint: You may use the fact that a nonzero polynomial in one variable has only finitely many roots.

Let $X$ be a random variable that follows the zeta distribution with parameter $x > 1$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ Show that, for all $a \in \mathbb{N}^{*}$, $$\mathbb{P}\left(X \in a\mathbb{N}^{*}\right) = \frac{1}{a^{x}}$$

Show that $I ( \alpha )$ is an ideal of $\mathbb { Q } [ X ]$, different from $\{ 0 \}$, where $\alpha$ is a fixed algebraic number and $I ( \alpha ) = \{ P \in \mathbb { Q } [ X ] \mid P ( \alpha ) = 0 \}$.

Show that $\alpha$ has degree 1 if and only if $\alpha \in \mathbb { Q }$, where the degree of $\alpha$ is the degree of its minimal polynomial $\Pi_{\alpha}$.

(a) Show that $\Pi _ { \alpha }$ is irreducible in $\mathbb { Q } [ X ]$.
(b) Let $P \in \mathbb { Q } [ X ]$ be a monic polynomial, irreducible in $\mathbb { Q } [ X ]$. Show that if $z$ is a complex root of $P$, then $P$ is the minimal polynomial of $z$.