Let $A$ be a set with at least $3$ integers, and let $M$ be the maximum element in $A$ and $m$ the minimum element in $A$. it is known that there exist a polynomial $P$ such that: $m<P(a)<M$ for all $a$ in $A$. And also $p(m)<p(a)$ for all $a$ in $A-(m,M)$. Prove that $n<6$ and there exist integers $b$ and $c$ such that $p(x)+x^2+bx+c$ is cero in $A$.

Let $a_1,a_2,...,a_{15}$ be positive integers for which the number $a_k^{k+1} - a_k$ is not divisible by $17$ for any $k = 1,...,15$. Show that there are integers $b_1,b_2,...,b_{15}$ such that: (i) $b_m - b_n$ is not divisible by $17$ for $1 \le m < n \le 15$, and (ii) each $b_i$ is a product of one or more terms of $(a_i)$.

Let $n\ge 3$ be an integer and let $A_{1}, A_{2},\dots, A_{n}$ be $n$ distinct subsets of $S=\{1, 2,\dots, n\}$. Show that there exists $x\in S$ such that the n subsets $A_{i}-\{x\}, i=1,2,\dots n$ are also disjoint. what i have is thiswe may assume that the union of the $A_{i}$s is $S$.

Let $\Gamma$ be the circumcircle of a fixed triangle $ABC$, and let $M$ and $N$ be the midpoints of the arcs $BC$ and $CA$, respectively. For any point $X$ on the arc $AB$, let $O_1$ and $O_2$ be the incenters of $\vartriangle XAC$ and $\vartriangle XBC$, and let the circumcircle of $\vartriangle XO_1O_2$ intersect $\Gamma$ at $X$ and $Q$. Prove that triangles $QNO_1$ and $QMO_2$ are similar, and find all possible locations of point $Q$.

Let $f(n)=\sum_{k=0}^{n-1}x^ky^{n-1-k}$ with, $x$, $y$ real numbers. If $f(n)$, $f(n+1)$, $f(n+2)$, $f(n+3)$, are integers for some $n$, prove $f(n)$ is integer for all $n$.

Suppose that $n - 1$ items $A_1,A_2,...,A_{n-1}$ have already been arranged in the increasing order, and that another item $A_n$ is to be inserted to preserve the order. What is the expected number of comparisons necessary to insert $A_n$?