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Prove that the number $\sum_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k}$ is not divisible by 5 for any integer $n \geq 0$.

Let $1 \leq r \leq n$ and consider all subsets of $r$ elements of the set $\{1, 2, \ldots, n\}$. Each of these subsets has a smallest member. Let $F(n, r)$ denote the arithmetic mean of these smallest numbers; prove that $$F(n, r) = \frac{n + 1}{r + 1}.$$

Given any set $A = \{a_1, a_2, a_3, a_4\}$ of four distinct positive integers, we denote the sum $a_1 + a_2 + a_3 + a_4$ by $s_A$. Let $n_A$ denote the number of pairs $(i,j)$ with $1 \leq i < j \leq 4$ for which $a_i + a_j$ divides $s_A$. Find all sets $A$ of four distinct positive integers which achieve the largest possible value of $n_A$.

Let $a$ and $b$ be positive integers. Prove that there exist positive integers $x$ and $y$ such that $$\binom{x + y}{2} = ax + by.$$

(a) Prove that for every positive integer $m$ there exists an integer $n \geq m$ such that

$$\left\lfloor \frac {n}{1} \right\rfloor \cdot \left\lfloor \frac {n}{2} \right\rfloor \cdots \left\lfloor \frac {n}{m} \right\rfloor = \binom {n} {m}. \tag{*}$$

(b) Denote by $p(m)$ the smallest integer $n \geq m$ such that the equation (*) holds. Prove that $p(2018) = p(2019)$.

Remark: For a real number $x$, we denote by $\lfloor x \rfloor$ the largest integer not larger than $x$.

Dokažite da je $$\binom{n}{p} - \left\lfloor \frac{n}{p} \right\rfloor$$ djeljivo s $p$, za svaki prost broj $p$ i svaki prirodan broj $n \geq p$.

Izračunaj $$\sum_{k=1}^{1011} \binom{2022}{2k-1}2^{k-1}$$