Let $p$ be an odd prime such that $2^h\not\equiv1\pmod p$ for every integer $1\le h\lt p-1$. Let $a$ be an even integer with $p/2\lt a\lt p$. Define $a_0=a$ and $a_{n+1}=p-b_n$ for $n\ge0$, where $b_n$ is the greatest odd divisor of $a_n$. Prove that $(a_n)$ is periodic and determine its least positive period.
Two real polynomials $f(x),g(x)$ are called proportional if there exists $q\in\mathbb R$ such that $f(x)=qg(x)$ for every real $x$.
1. Prove that there exists a real polynomial $P(x)$ of degree 1999 for which $P(x)^2-4$ and $(P'(x))^2(x^2-4)$ are proportional.
2. How many degree-1999 polynomials satisfy this condition?
Let $H$ be a convex polygon. Prove that for every real number $a\in(0,1)$ there exist six distinct points $A_1,A_2,\ldots,A_6$ on its sides, listed in clockwise order, such that
1. $\overrightarrow{A_1A_2}=\overrightarrow{A_5A_4}=a\overrightarrow{A_6A_3}$;
2. the lines $A_1A_2$ and $A_5A_4$ are at equal distances from the line $A_6A_3$.
Let $(u_n)$ be a sequence of positive real numbers. For each positive integer $n$, let $k_n$ be the least positive integer satisfying $\sum_{i=1}^{k_n}1/i\ge\sum_{i=1}^n u_i$. Prove that $(k_{n+1}/k_n)$ has a finite limit if and only if $(u_n)$ has a finite limit.
Triangle $A_1A_2A_3$ is inscribed in a circle $(O)$. A circle $(K_1)$ is tangent to the sides $A_1A_2,A_1A_3$ and internally tangent to $(O)$ at $M_1,N_1,P_1$, respectively. The points $M_1,N_2,P_2$ and $M_3,N_3,P_3$ are defined similarly. Prove that $M_1N_1,M_2N_2,M_3N_3$ intersect at the midpoint of each segment.
Let $p>2$ be prime. A monkey sits at every vertex of a $p$-gon. The source says “$p$ beans are given to each monkey”, under the following rule: give the first bean to an arbitrary monkey, the next to the monkey at the third vertex clockwise from that monkey, the third to the monkey at the fifth vertex clockwise from the previous recipient, and so on. In general the $(k+1)$st bean goes to the monkey at the $(2k+1)$st vertex clockwise from the recipient of the $k$th bean.
1. How many monkeys receive no beans?
2. How many edges have monkeys receiving beans at both endpoints?
Let $ABC$ be a triangle. Let $A'$ be the midpoint of the arc $BC$ of its circumcircle that does not contain $A$; define $B'$ and $C'$ similarly. The segments $A'B',B'C',C'A'$ meet the sides of $ABC$ at six points, two on each side, dividing each side into three parts. Prove that the three middle subsegments are equal if and only if $ABC$ is equilateral.
The sequence $a_1,a_2,a_3,\ldots$ is defined by $a_1=1$, $a_2=2$, and $a_{n+2}=3a_{n+1}-a_n$. The sequence $b_1,b_2,b_3,\ldots$ is defined by $b_1=1$, $b_2=4$, and $b_{n+2}=3b_{n+1}-b_n$. Prove that positive integers $a,b$ satisfy $5a^2-b^2=4$ if and only if $a=a_n$ and $b=b_n$ for some positive integer $n$.
Let $OA,OB,OC,OD$ be four rays in space such that the angle between any two is the same. Prove that, as a ray $OX$ varies, the sum of the cosines of $\angle XOA,\angle XOB,\angle XOC,\angle XOD$ is constant, and the sum of the squares of those cosines is also constant.
Determine all functions $f(n)$ defined on the nonnegative integers, with values in $\{0,1,2,\ldots,2000\}$, satisfying:
(a) $f(n)=n$ for $0\le n\le2000$;
(b) $f(f(m)+f(n))=f(m+n)$ for all positive integers $m,n$.