Let $a_1,a_2,\cdots,a_n$ be a sequence of real numbers lying between $1$ and $-1$, i.e. $-1<a_i<1$, for $1\leq i \leq n$ and such that (i) $a_1+a_2+\cdots+a_n=0$ (ii) $a_1^2+a_2^2+\cdots+a_n^2=40$ Determine the smallest possible value of $n$

Find all integral ordered triples $(x,y,z)$ such that $\displaystyle\sqrt{\frac{2015}{x+y}}+\sqrt{\frac{2015}{y+z}}+\sqrt{\frac{2015}{x+z}}$ are positive integers

Let $ABC$ be a triangle. Let $D$ and $E$ be respectively points on the segments $AB$ and $AC$, and such that $DE||BC$. Let $M$ be the midpoint of $BC$. Let $P$ be a point such that $DB=DP$, $EC=EP$ and such that the open segments (segments excluding the endpoints) $AP$ and $BC$ intersect. Suppose $\angle BPD=\angle CME$. Show that $\angle CPE=\angle BMD$

Given an integer $n\geq 2$. There are $N$ distinct circle on the plane such that any two circles have two distinct intersections and no three circles have a common intersection. Initially there is a coin on each of the intersection points of the circles. Starting from $X$, players $X$ and $Y$ alternatively take away a coin, with the restriction that one cannot take away a coin lying on the same circle as the last coin just taken away by the opponent in the previous step. The one who cannot do so will lost. In particular, one loses where there is no coin left. For what values of $n$ does $Y$ have a winning strategy?