On the plane, there is drawn a parallelogram $P$ and a point $X$ outside of $P$. Using only an ungraded rule, determine the point $W$ that is symmetric to $X$ with respect to the center $O$ of $P$.
2015 Chile National Olympiad
Find all prime numbers that do not have a multiple ending in $2015$.
Consider a horizontal line $L$ with $n\ge 4$ different points $P_1, P_2, ..., P_n$. For each pair of points $P_i$ ,$P_j $a circle is drawn such that the segment $P_iP_j$ is a diameter. Determine the maximum number of intersections between circles that can occur, considering only those that occur strictly above $L$. original wordingConsidere una recta horizontal $L$ con $n\ge 4$ puntos $P_1, P_2, ..., P_n$ distintos en ella. Para cada par de puntos $P_i,P_j$ se traza un circulo de manera tal que el segmento $P_iP_j$ es un diametro. Determine la cantidad maxima de intersecciones entre circulos que pueden ocurrir, considerando solo aquellas que ocurren estrictamente arriba de $L$.
Find the number of different numbers of the form $\left\lfloor\frac{i^2}{2015} \right\rfloor$, with $i = 1,2, ..., 2015$.
A quadrilateral $ABCD$ is inscribed in a circle. Suppose that $|DA| =|BC|= 2$ and$ |AB| = 4$. Let $E $ be the intersection point of lines $BC$ and $DA$. Suppose that $\angle AEB = 60^o$ and that $|CD| <|AB|$. Calculate the radius of the circle.
On the plane, a closed curve with simple auto intersections is drawn continuously. In the plane a finite number is determined in this way from disjoint regions. Show that each of these regions can be completely painted either white or blue, so that every two regions that share a curve segment at its edges, they always have different colors. Clarification: a car intersection is simple if looking at a very small disk around from her, the curve looks like a junction $\times$.