Let $M=\bigoplus_{i \in \mathbb{Z}} \mathbb{C}e_i$ be an infinite dimensional $\mathbb{C}$-vector space, and let $\text{End}(M)$ denote the $\mathbb{C}$-algebra of $\mathbb{C}$-linear endomorphisms of $M$. Let $A$ and $B$ be two commuting elements in $\text{End}(M)$ satisfying the following condition: there exist integers $m \le n<0<p \le q$ satisfying $\text{gd}(-m,p)=\text{gcd}(-n,q)=1$, and such that for every $j \in \mathbb{Z}$, one has \[Ae_j=\sum_{i=j+m}^{j+n} a_{i,j}e_i, \quad \text{with } a_{i,j} \in \mathbb{C}, a_{j+m,j}a_{j+n,j} \ne 0,\] \[Be_j=\sum_{i=j+p}^{j+q} b_{i,j}e_i, \quad \text{with } b_{i,j} \in \mathbb{C}, b_{j+p,j}b_{j+q,j} \ne 0.\] Let $R \subset \text{End}(M)$ be the $\mathbb{C}$-subalgebra generated by $A$ and $B$. Note that $R$ is commutative and $M$ can be regarded as an $R$-module. (a) Show that $R$ is an integral domain and is isomorphic to $R \cong \mathbb{C}[x,y]/f(x,y)$, where $f(x,y)$ is a non-zero polynomial such that $f(A,B)=0$. (b) Let $K$ be the fractional field of $R$. Show that $M \otimes_R K$ is a $1$-dimensional vector space over $K$.

Let \( f(n) \) be a polynomial with integer coefficients. Prove that if \( f(-1) \), \( f(0) \), and \( f(1) \) are not divisible by 3, then \( f(n) \neq 0 \) for all integers \( n \).

What is the arithmetic mean of all values of the expression $ | a_1-a_2 | + | a_3-a_4 | $ Where $ a_1, a_2, a_3, a_4 $ is a permutation of the elements of the set {$ 1,2,3,4 $}?

Let $m \ge 2$ and $n \ge 1$ be integers and $A=(a_{ij})$ a square matrix of order $n$ with integer entries. Prove that for any permutation $\sigma \in S_n$ there is a function $\varepsilon : \{1,2,\ldots,n\} \to \{0,1\}$ such that replacing the entries $a_{\sigma(1)1},$ $a_{\sigma(2)2}, $ $\ldots,$ $a_{\sigma(n)n}$ of $A$ respectively by $$a_{\sigma(1)1}+\varepsilon(1), ~a_{\sigma(2)2}+\varepsilon(2), ~\ldots, ~a_{\sigma(n)n}+\varepsilon(n),$$ the determinant of the matrix $A_{\varepsilon}$ thus obtained is not divisible by $m.$

If we divide $2020$ by a prime $p$, the remainder is $6$. Determine the largest possible value of $p$.

For each $n$ find the number of ways one can put the numbers $\{1,2,3,...,n\}$ numbers on the circle, such that if for any $4$ numbers $a,b,c,d$ where $n|a+b-c-d$. The segments joining $a,b$ and $c,d$ do not meet inside the circle. (Two ways are said to be identical , if one can be obtained from rotaiting the other)

For each prime number $p$, determine the number of residue classes modulo $p$ which can be represented as $a^2+b^2$ modulo $p$, where $a$ and $b$ are arbitrary integers. [i](Daniel Holmes)[/i]

Determine the number of ordered quadruples of integers $(a,b,c,d)$ for which $0\leq a,b,c,d\leq 36$ and $37|a^2+b^2-c^3-d^3$.

Determine the last digit of $5^5+6^6+7^7+8^8+9^9$.

For a set with $n$ elements, how many subsets are there whose cardinality is respectively $\equiv 0$ (mod $3$), $\equiv 1$ (mod $3$), $ \equiv 2$ (mod $3$)? In other words, calculate $$s_{i,n}= \sum_{k\equiv i \;(\text{mod} \;3)} \binom{n}{k}$$ for $i=0,1,2$. Your result should be strong enough to permit direct evaluation of the numbers $s_{i,n}$ and to show clearly the relationship of $s_{0,n}, s_{1,n}$ and $s_{2,n}$ to each other for all positive integers $n$. In particular, show the relationships among these three sums for $n = 1000$.

Given a positive integer $n \ge 2$, we define $f(n)$ to be the sum of all remainders obtained by dividing $n$ by all positive integers less than $n$. For example dividing $5$ with $1, 2, 3$ and $4$ we have remainders equal to $0, 1, 2$ and $1$ respectively. Therefore $f(5) = 0 + 1 + 2 + 1 = 4$. Find all positive integers $n \ge 3$ such that $f(n) = f(n - 1) + (n - 2)$.

If $x,y,z \in\mathbb{N}$ and $2x^2+3y^3=4z^4$, Prove that $6|x,y,z$

Let $a,b,c$ and $m$ ($0 \le m \le 26$) be integers such that $a + b + c = (a - b)(b- c)(c - a) = m$ (mod $27$) then $m$ is (A): $0$, (B): $1$, (C): $25$, (D): $26$ (E): None of the above.

Find all positive integers $n \ge 2$ such that equality $i+j \equiv C_{n}^{i} + C_{n}^{j}$ (mod $2$) is true for arbitrary $0 \le i \le j \le n$.

Prove that there exist a natural number $a$, for which 999 divides $2^{5n}+a.5^n$ for $\forall$ odd $n\in \mathbb{N}$ and find the smallest such $a$.

Let $p > 3$ be a prime number and $ x$ an integer, denote by $r ( x )\in \{ 0 , 1 , ... , p - 1 \}$ to the rest of $x$ modulo $p$ . Let $x_1, x_2, ... , x_k$ ( $2 < k < p$) different integers modulo $p$ and not divisible by $p$. We say that a number $a \in \{ 1 , 2 ,..., p -1 \}$ is [i]good [/i] if $r ( a x_1) < r ( a x_2) <...< r ( a x_k)$. Show that there are at most $\frac{2 p}{k + 1}-{ 1}$ [i]good [/i] numbers.

Let $n\ge 2$ be a positive integer. For any integer $a$, let $P_a(x)$ denote the polynomial $x^n+ax$. Let $p$ be a prime number and define the set $S_a$ as the set of residues mod $p$ that $P_a(x)$ attains. That is, $$S_a=\{b\mid 0\le b\le p-1,\text{ and there is }c\text{ such that }P_a(c)\equiv b \pmod{p}\}.$$Show that the expression $\frac{1}{p-1}\sum\limits_{a=1}^{p-1}|S_a|$ is an integer. [i]Proposed by fattypiggy123[/i]

Let $p$ and $q=4p+1$ be prime numbers. Determine the least power $i$ of 2 for which $2^i\equiv 1\,(mod\, q)$.

A prime number $p> 3$ is given. Prove that integers less than $p$, it is possible to divide them into two non-empty sets such that the sum of the numbers in the first set will be congruent modulo p to the product of the numbers in the second set.

Let $p_1, p_2, p_3$, and $p$ be prime numbers. Prove that there exist $x,y\in \mathbb{Z}$ such that $y^2\equiv p_1 x^4-p_1 p_2^2 p_3^2\, (mod\, p)$.

Put $1,2,....,2018$ (2018 numbers) in a row randomly and call this number $A$. Find the remainder of $A$ divided by $3$.

Let $m, n,$ and $k$ be natural numbers, where $n$ is odd. Prove that $\frac{1}{m}+\frac{1}{m+n}+...+\frac{1}{m+kn}$ is not a natural number.

Let $n\ge 2$ be a positive integer. For any integer $a$, let $P_a(x)$ denote the polynomial $x^n+ax$. Let $p$ be a prime number and define the set $S_a$ as the set of residues mod $p$ that $P_a(x)$ attains. That is, $$S_a=\{b\mid 0\le b\le p-1,\text{ and there is }c\text{ such that }P_a(c)\equiv b \pmod{p}\}.$$Show that the expression $\frac{1}{p-1}\sum\limits_{a=1}^{p-1}|S_a|$ is an integer. [i]Proposed by fattypiggy123[/i]

Find all prime numbers $p$ for which $x^4\equiv -1\, (mod\, p)$ has a solution.

A set of $k$ integers is said to be a [i]complete residue system modulo[/i] $k$ if no two of its elements are congruent modulo $k$. Find all positive integers $m$ so that there are infinitely many positive integers $n$ wherein $\{ 1^n,2^n, \dots , m^n \}$ is a complete residue system modulo $m$.