Školjka

%V0 Let $n$ be a composite natural number and $p$ a proper divisor of $n.$ Find the binary representation of the smallest natural number $N$ such that $$\frac{(1 + 2^p + 2^{n-p})N - 1}{2^n}$$ is an integer.

%V0 Determine for which positive integers $k$ the set $$X = \{1990, 1990 + 1, 1990 + 2, \ldots, 1990 + k\}$$ can be partitioned into two disjoint subsets $A$ and $B$ such that the sum of the elements of $A$ is equal to the sum of the elements of $B.$

%V0 Assume that the set of all positive integers is decomposed into $r$ (disjoint) subsets $A_1 \cup A_2 \cup \ldots \cup A_r = \mathbb{N}.$ Prove that one of them, say $A_i,$ has the following property: There exists a positive $m$ such that for any $k$ one can find numbers $a_1, a_2, \ldots, a_k$ in $A_i$ with $0 < a_{j + 1} - a_j \leq m,$ $(1 \leq j \leq k - 1)$.

%V0 The integer $9$ can be written as a sum of two consecutive integers: $9 = 4+5.$ Moreover, it can be written as a sum of (more than one) consecutive positive integers in exactly two ways: $9 = 4+5 = 2+3+4.$ Is there an integer that can be written as a sum of $1990$ consecutive integers and that can be written as a sum of (more than one) consecutive positive integers in exactly $1990$ ways?

%V0 $(GBR 2)$ Let $a, b, x, y$ be positive integers such that $a$ and $b$ have no common divisor greater than $1$. Prove that the largest number not expressible in the form $ax + by$ is $ab - a - b$. If $N(k)$ is the largest number not expressible in the form $ax + by$ in only $k$ ways, find $N(k).$

%V0 $(GBR 1)$ The polynomial $P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$, where $a_0,\cdots, a_k$ are integers, is said to be divisible by an integer $m$ if $P(x)$ is a multiple of $m$ for every integral value of $x$. Show that if $P(x)$ is divisible by $m$, then $a_0 \cdot k!$ is a multiple of $m$. Also prove that if $a, k,m$ are positive integers such that $ak!$ is a multiple of $m$, then a polynomial $P(x)$ with leading term $ax^k$can be found that is divisible by $m.$