Find all functions $f$ defined on the positive real numbers and taking positive real values that satisfy the following conditions:
(i) $f(xf(y))=yf(x)$ for all positive real $x,y$.
(ii) $f(x)\to0$ as $x\to+\infty$.
Let $K$ be one of the two intersection points of the circles $W_1$ and $W_2$. Let $O_1$ and $O_2$ be the centers of $W_1$ and $W_2$. The two common tangents to the circles meet $W_1$ and $W_2$ respectively in $P_1$ and $P_2$, the first tangent, and $Q_1$ and $Q_2$, the second tangent. Let $M_1$ and $M_2$ be the midpoints of $P_1Q_1$ and $P_2Q_2$, respectively. Prove that
$$\angle O_1KO_2=\angle M_1KM_2.$$
Let $a,b,c$ be positive integers satisfying $(a,b)=(b,c)=(c,a)=1$. Show that $2abc-ab-bc-ca$ is the largest integer not representable as
$$xbc+yca+zab$$
with nonnegative integers $x,y,z$.
Let $ABC$ be an equilateral triangle. Let $E$ be the set of all points from segments $AB,BC,$ and $CA$ (including $A,B,$ and $C$). Is it true that for any partition of the set $E$ into two disjoint subsets, there exists a right-angled triangle all of whose vertices belong to the same subset in the partition?
Prove or disprove the following statement: In the set $\{1,2,3,\ldots,10^5\}$ a subset of $1983$ elements can be found that does not contain any three consecutive terms of an arithmetic progression.