Determine all integers $m \geq 2$ such that every $n$ with $\frac{m}{3} \leq n \leq \frac{m}{2}$ divides the binomial coefficient $\binom{n}{m-2n}$.

For a prime number $p$, let $GL_2(\mathbb{Z}/p\mathbb{Z})$ be the group of invertible $2 \times 2$ matrices of residues modulo $p$, and let $S_p$ be the symmetric group (the group of all permutations) on $p$ elements. Show that there is no injective group homomorphism $\phi : GL_2(\mathbb{Z}/p\mathbb{Z}) \rightarrow S_p$.

Find all pairs of prime numbers $p, q$ such that the number $pq + p - 6$ is also prime.

Determine with proof all triples $(a, b, c)$ of positive integers satisfying $\frac{1}{a}+ \frac{2}{b} +\frac{3}{c} = 1$, where $a$ is a prime number and $a \le b \le c$.

Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[ 2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots (\mbox{mod} \; n) \] is eventually constant. [The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$. Also $a_i \; (\mbox{mod} \; n)$ means the remainder which results from dividing $a_i$ by $n$.]

For a nonnegative integer $n$ define $\operatorname{rad}(n)=1$ if $n=0$ or $n=1$, and $\operatorname{rad}(n)=p_1p_2\cdots p_k$ where $p_1<p_2<\cdots <p_k$ are all prime factors of $n$. Find all polynomials $f(x)$ with nonnegative integer coefficients such that $\operatorname{rad}(f(n))$ divides $\operatorname{rad}(f(n^{\operatorname{rad}(n)}))$ for every nonnegative integer $n$.

Find the exponent of $37$ in the representation of the number $111...... 11$ with $3\cdot 37^{2000}$ digits equals to $1$, as product of prime powers

Find all prime numbers $p$ and $q$ such that $$(2p-q)^2=17p-10q$$

Archimedes planned to count all of the prime numbers between $2$ and $1000$ using the Sieve of Eratosthenes as follows: (a) List the integers from $2$ to $1000$. (b) Circle the smallest number in the list and call this $p$. (c) Cross out all multiples of $p$ in the list except for $p$ itself. (d) Let $p$ be the smallest number remaining that is neither circled nor crossed out. Circle $p$. (e) Repeat steps $(c)$ and $(d)$ until each number is either circled or crossed out. At the end of this process, the circled numbers are prime and the crossed out numbers are composite. Unfortunately, while crossing out the multiples of $2$, Archimedes accidentally crossed out two odd primes in addition to crossing out all the even numbers (besides $2$). Otherwise, he executed the algorithm correctly. If the number of circled numbers remaining when Archimedes finished equals the number of primes from $2$ to $1000$ (including $2$), then what is the largest possible prime that Archimedes accidentally crossed out?

Let $n \ge 50 $ be a natural number. Prove that $n$ is expressible as sum of two natural numbers $n=x+y$, so that for every prime number $p$ such that $ p\mid x$ or $p\mid y $ we have $ \sqrt{n} \ge p $. For example for $n=94$ we have $x=80, y=14$.

Set $S = \{1, 2, 3, ..., 2005\}$. If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$.

Let $p \ge 5$ be a prime number. For a positive integer $k$, let $R(k)$ be the remainder when $k$ is divided by $p$, with $0 \le R(k) \le p-1$. Determine all positive integers $a < p$ such that, for every $m = 1, 2, \cdots, p-1$, $$ m + R(ma) > a. $$

Find all primes $1 < p < 100$ such that the equation $x^2-6y^2=p$ has an integer solution $(x, y)$.

Let $p>3$ be a prime and $n$ a positive integer such that $p^n$ has $20$ digits. Prove that at least one digit appears more than twice in this number.

Find all natural numbers $N$ (in decimal system) with the following properties: (i) $N =\overline{aabb}$, where $\overline{aab}$ and $\overline{abb}$ are primes, (ii) $N = P_1P_2P_3$, where $P_k (k = 1,2,3)$ is a prime consisting of $k$ (decimal) digits.