Question bank VMO 2008

In the plane, an angle $xOy$ is given. Let $M,N$ lie on the rays $Ox,Oy$, respectively. Let $d$ be the external angle-bisector line of $xOy$, and let $I$ be the intersection of $d$ with the perpendicular bisector of $MN$. Let $P,Q$ be distinct points on $d$ such that $IM=IN=IP=IQ$, and suppose $K=MQ\cap NP$.
1. Prove that $K$ lies on a fixed line.
2. Let $d_1$ be the line perpendicular to $IM$ at $M$, and $d_2$ the line perpendicular to $IN$ at $N$. Suppose $d_1,d_2$ intersect $d$ at $E,F$, respectively. Prove that $EN,FM,OK$ are concurrent.

Determine all positive integers $m$ for which there exist real-coefficient polynomials $P(x),Q(x),R(x,y)$ such that for all real numbers $a,b$ satisfying $a^m-b^2=0$, one has $P(R(a,b))=a$ and $Q(R(a,b))=b$.

Let $n>3$ be an integer and let $T=\{1,2,\ldots,n\}$. A subset $S$ of $T$ is called deficient in $T$ if there is a positive integer $c\le n/2$ such that $|s_1-s_2|\ne c$ for any two elements $s_1,s_2$ of $S$. What is the largest possible number of elements of a deficient subset?

Let $m,n$ be positive integers. Prove that $(2m+3)^n+1$ is divisible by $6m$ if and only if $3^n+1$ is divisible by $4m$.

Let $ABC$ be an acute, non-isosceles triangle with circumcenter $O$. Let $AD,BE,CF$ be its internal angle bisectors. Choose points $L,M,N$ on the lines $AD,BE,CF$, respectively, such that
$$\frac{AL}{AD}=\frac{BM}{BE}=\frac{CN}{CF}=k,$$
where $k$ is a positive constant. Let $(O_1),(O_2),(O_3)$ be the circles which respectively pass through $L$ and are tangent to $OA$ at $A$; pass through $M$ and are tangent to $OB$ at $B$; and pass through $N$ and are tangent to $OC$ at $C$.
1. Prove that for $k=\frac12$ the three circles have exactly two common points, and the line joining these points passes through the centroid of $ABC$.
2. Find all values of $k$ for which the three circles have exactly two common points.

Let $M=\{1,2,\ldots,2008\}$. Color every element blue, yellow, or red, with exactly one color per element and each color used at least once. Define
$$S_1=\{(x,y,z)\in M^3:x,y,z\text{ have the same color},\ x+y+z\equiv0\pmod{2008}\},$$
$$S_2=\{(x,y,z)\in M^3:x,y,z\text{ have pairwise different colors},\ x+y+z\equiv0\pmod{2008}\}.$$
Prove that $2|S_1|>|S_2|$. Here $M^3=M\times M\times M$.

Determine the number of real solutions $(x,y)$ of the system
$$\begin{cases}x^2+y^3=29,\\[3pt]\log_3x\cdot\log_2y=1.\end{cases}$$

In triangle $ABC$, let $E$ be the midpoint of $AB$ and suppose $\angle BEC$ is acute. On ray $EC$, take $M$ such that $\angle BME=\angle ECA$. If $\alpha=\angle BEC$, express $MC/AB$ in terms of $\alpha$.

Let $m=2007^{2008}$. How many natural numbers $n$ satisfy $n\lt m$ and $m\mid n(2n+1)(5n+2)$?

Let the real sequence $(x_n)$ be defined by $x_1=0$, $x_2=2$ and
$$x_{n+2}=2^{-x_n}+\frac12\qquad(n=1,2,3,\ldots).$$
Prove that it converges to a finite limit as $n\to+\infty$, and find the limit.

How many natural numbers are divisible by $9$, have at most $2008$ decimal digits, and contain at least two digits equal to $9$?

Let $x,y,z$ be pairwise distinct nonnegative real numbers. Prove
$$(xy+yz+zx)\left(\frac1{(x-y)^2}+\frac1{(y-z)^2}+\frac1{(z-x)^2}\right)\ge4.$$
When does equality occur?

Let $ABC$ be a triangle with median $AD$. Let a line $d$ be perpendicular to $AD$, and let $M$ vary on $d$. Let $E,F$ be the midpoints of $MB,MC$, respectively. The line through $E$ perpendicular to $d$ meets $AB$ at $P$, and the line through $F$ perpendicular to $d$ meets $AC$ at $Q$. Prove that the line through $M$ perpendicular to $PQ$ always passes through a fixed point as $M$ varies on $d$.