OBM 2000

A rectangular piece of paper has top edge $AD$ . A line $L$ from $A$ to the bottom edge makes an angle $x$ with the line $AD$ . We want to trisect $x$ . We take $B$ and $C$ on the vertical ege through $A$ such that $AB = BC$ . We then fold the paper so that $C$ goes to a point $C'$ on the line $L$ and $A$ goes to a point $A'$ on the horizontal line through $B$ . The fold takes $B$ to $B'$ . Show that $AA'$ and $AB'$ are the required trisectors.

Let $s(n)$ be the sum of all positive divisors of $n$ , so $s(6) = 12$ . We say $n$ is almost perfect if $s(n) = 2n - 1$ . Let $\mod(n, k)$ denote the residue of $n$ modulo $k$ (in other words, the remainder of dividing $n$ by $k$ ). Put $t(n) = \mod(n, 1) + \mod(n, 2) + \cdots + \mod(n, n)$ .

Show that $n$ is almost perfect if and only if $t(n) = t(n-1)$ .

Define $f$ on the positive integers by $f(n) = k^2 + k + 1$ , where $n=2^k(2l+1)$ for some $k,l$ nonnegative integers. Find the smallest $n$ such that $f(1) + f(2) + ... + f(n) \geq 123456$ .

An infinite road has traffic lights at intervals of 1500m. The lights are all synchronised and are alternately green for $\frac 32$ minutes and red for 1 minute. For which $v$ can a car travel at a constant speed of $v$ m/s without ever going through a red light?

Let $X$ the set of all sequences $\{a_1, a_2,\ldots , a_{2000}\}$ , such that each of the first 1000 terms is 0, 1 or 2, and each of the remaining terms is 0 or 1. The distance between two members $a$ and $b$ of $X$ is defined as the number of $i$ for which $a_i$ and $b_i$ are different.

Find the number of functions $f : X \to X$ which preserve the distance.

Let it be is a wooden unit cube. We cut along every plane which is perpendicular to the segment joining two distinct vertices and bisects it. How many pieces do we get?