Question bank APMO 2000

Compute the sum
$$S=\sum_{i=0}^{101}\frac{x_i^3}{1-3x_i+3x_i^2},\qquad x_i=\frac{i}{101}.$$

Given the following triangular arrangement of circles:



Each of the numbers $1,2,\ldots,9$ is to be written into one circle, so that each circle contains exactly one number and
(i) the sums of the four numbers on each side of the triangle are equal;
(ii) the sums of the squares of the four numbers on each side are equal.
Find all ways in which this can be done.

Let $ABC$ be a triangle. Let $M,N$ be the points where the median and angle bisector at $A$, respectively, meet $BC$. Let $Q,P$ be the points where the perpendicular at $N$ to $NA$ meets $MA,BA$, respectively, and $O$ the point where the perpendicular at $P$ to $BA$ meets $AN$ produced. Prove that $QO\perp BC$.

Let $n,k$ be positive integers with $n\gt k$. Prove that
$$\frac1{n+1}\frac{n^n}{k^k(n-k)^{n-k}}\lt \frac{n!}{k!(n-k)!}\lt \frac{n^n}{k^k(n-k)^{n-k}}.$$

Given a permutation $(a_0,a_1,\ldots,a_n)$ of $0,1,\ldots,n$, a transposition of $a_i$ with $a_j$ is called legal if $a_i=0$ for $i\gt 0$ and $a_{i-1}+1=a_j$. The permutation is called regular if after a number of legal transpositions it becomes $(1,2,\ldots,n,0)$. For which $n$ is $(1,n,n-1,\ldots,3,2,0)$ regular?