Determine whether there exists a positive integer $n$ such that the sum of the digits of $n^2$ is $2002$.

Three chords $AB, CD$ and $EF$ of a circle intersect at the midpoint $M$ of $AB$. Show that if $CE$ produced and $DF$ produced meet the line $AB$ at the points $P$ and $Q$ respectively, then $M$ is also the midpoint of $PQ$.

In how many ways can $n^2$ distinct real numbers be arranged into an $n\times n$ array $(a_{ij })$ such that max$_{j}$ min $_i \,\, a_{ij} $= min$_i$ max$_j \,\, a_{ij}$?

Let $A = \{3 + 10k, 6 + 26k, 5 + 29k, k = 1, 2, 3, 4, ...\}$. Determine the smallest positive integer $r$ such that there exists an integer $b$ with the property that the set $B = \{b + rk, k = 1, 2, 3, 4, ...\}$ is disjoint from $A$.

Let $M$ be a point on the diameter $AB$ of a semicircle $\Gamma$. The perpendicular at $M$ meets the semicircle $\Gamma$ at $P$. A circle inside $\Gamma$. touches $\Gamma$. and is tangent to $PM$ at $Q$ and $AM$ at $R$. Prove that $P B = RB$.

Determine all functions $f : Z\to Z$, where $Z$ is the set of integers, such that $$f(m + f(f(n))) = -f(f(m + 1)) - n$$for all integers $m$ and $n$.