Subcontests
(4)How many numbers are there on the board?
The number 1 is written on the blackboard. After that a sequence of numbers is created as follows: at each step each number a on the blackboard is replaced by the numbers a−1 and a+1; if the number 0 occurs, it is erased immediately; if a number occurs more than once, all its occurrences are left on the blackboard. Thus the blackboard will show 1 after 0 steps; 2 after 1 step; 1,3 after 2 steps; 2,2,4 after 3 steps, and so on. How many numbers will there be on the blackboard after n steps? Smallest set that behaves like {1,2,...,n}
Find the smallest positive integer n, such that there exist n integers x1,x2,…,xn (not necessarily different), with 1≤xk≤n, 1≤k≤n, and such that
x_1 + x_2 + \cdots + x_n =\frac{n(n + 1)}{2}, \text{ and }x_1x_2 \cdots x_n = n!,
but {x1,x2,…,xn}={1,2,…,n}. This expression is an integer
The real numbers a,b,c are such that a2+b2=2c2, and also such that a=b,c=−a,c=−b. Show that
(a−b)(a+c)(b+c)(a+b+2c)(2a2−b2−c2)
is an integer.