PAMO 2001

Find all positive integers $n$ such that: $$\dfrac{n^3+3}{n^2+7}$$ is a positive integer.

Let $n$ be a positive integer. A child builds a wall along a line with $n$ identical cubes. He lays the first cube on the line and at each subsequent step, he lays the next cube either on the ground or on the top of another cube, so that it has a common face with the previous one. How many such distinct walls exist?

Let $ABC$ be an equilateral triangle and let $P_0$ be a point outside this triangle, such that $\triangle{AP_0C}$ is an isoscele triangle with a right angle at $P_0$ . A grasshopper starts from $P_0$ and turns around the triangle as follows. From $P_0$ the grasshopper jumps to $P_1$ , which is the symmetric point of $P_0$ with respect to $A$ . From $P_1$ , the grasshopper jumps to $P_2$ , which is the symmetric point of $P_1$ with respect to $B$ . Then the grasshopper jumps to $P_3$ which is the symmetric point of $P_2$ with respect to $C$ , and so on. Compare the distance $P_0P_1$ and $P_0P_n$ . $n \in N$ .

Let $n$ be a positive integer, and let $a>0$ be a real number. Consider the equation: $$\sum_{i=1}^{n}(x_i^2+(a-x_i)^2)= na^2$$ How many solutions ( $x_1, x_2 \cdots , x_n$ ) does this equation have, such that: $$0 \leq x_i \leq a, i \in N^+$$

Find the value of the sum: $$\sum_{i=1}^{2001} [\sqrt{i}]$$ where $[ {x} ]$ denotes the greatest integer which does not exceed $x$ .

Let $S_1$ be a semicircle with centre $O$ and diameter $AB$ .A circle $C_1$ with centre $P$ is drawn, tangent to $S_1$ , and tangent to $AB$ at $O$ . A semicircle $S_2$ is drawn, with centre $Q$ on $AB$ , tangent to $S_1$ and to $C_1$ . A circle $C_2$ with centre $R$ is drawn, internally tangent to $S_1$ and externally tangent to $S_2$ and $C_1$ . Prove that $OPRQ$ is a rectangle.