# Recent questions tagged isi2017-pcb-a

1
Suppose all the roots of the equation $x^3 +bx-2017=0$ (where $b$ is a real number) are real. Prove that exactly one root is positive.
Let $a, b, c$ and $d$ be real numbers such that $a+b=c+d$ and $ab=cd$. Prove that $a^n+b^n=c^n+d^n$ for all positive integers $n$.
Let $B=\{1, 2, 3, 4\}$. A set $S \subseteq B \times B$ called a symmetric set of $B$ if for all $x, y \in B$, $(x, y) \in S \Rightarrow (y,x) \in S.$ Find the number of symmetric sets of $B$.
Let $\lceil x \rfloor$ denote the integer nearest to $x$. For example, $\lceil 1.1 \rfloor =1, \lceil 1.5 \rfloor =1$ and $\lceil 1.6 \rfloor$ =2. Draw the graph of the function $y= \mid x - \lceil x \rfloor \mid$ for $0 \leq x \leq 4$. Find all the points $x, \: 0 \leq x \leq 4$, where the function is not differentiable. Justify your answer.