Question bank VMO 2016

Find all positive integers $a,n$, with $a\gt 2$, such that every prime divisor of $a^n-1$ also divides $a^{3^{2016}}-1$.

Let $A$ be a set of $2000$ distinct integers and $B$ a set of $2016$ distinct integers. Let $K$ count the ordered pairs $(m,n)\in A\times B$ satisfying $|m-n|\le 1000$. Find the largest possible value of $K$.

Let triangle \(ABC\) be inscribed in a fixed circle \((O)\), with \(B,C\) fixed and \(A\) moving on one fixed arc \(BC\). The internal angle bisectors \(AD,BE,CF\) meet at \(I\). The circle through \(D\), tangent to line \(OA\) at \(A\), meets \((O)\) again at \(G\). Lines \(GE,GF\) meet \((O)\) again at \(M,N\), respectively, and \(H=BM\cap CN\).
(a) Prove that \(AH\) passes through a fixed point.
(b) Let \(K,L\) be the second intersections of \(BE,CF\) with \((O)\), and let \(P=AH\cap KL\). Choose \(Q\in EF\) with \(QP=QI\). Let \(J\ne I\) be the second intersection of the line through \(I\) perpendicular to \(IQ\) with \((BIC)\). Prove that the midpoint of \(IJ\) lies on a fixed circle.
As is necessary for the construction, consider positions in which all named circles, lines, and intersections exist and are distinct as used.

Let \(ABC\) be an acute triangle satisfying
\[\angle ACB\lt \angle ABC\lt \angle ACB+\frac12\angle BAC.\]
Choose \(D\) on segment \(BC\) so that \(\angle ADC=\angle ACB+\tfrac12\angle BAC\). The tangent at \(A\) to the circumcircle of \(ABC\) meets line \(BC\) at \(E\). The internal bisector of \(\angle AEB\) meets \(AD\) at \(G\) and the circle \((ADE)\) again at \(F\ne E\). Put \(H=DF\cap AE\).
(a) Prove that the circles with diameters \(AE,DF,GH\) have a common point.
(b) Choose \(K\) on the external angle-bisector line of \(\angle BAC\) and \(M\) on ray \(AC\) such that \(KB=KD=KM\). Similarly choose \(L\) on that external bisector and \(N\) on ray \(AB\) such that \(LC=LD=LN\). Let \(I\) be the midpoint of \(BC\), and let the circle \((MNI)\) meet \(BC\) again at \(P\ne I\). Prove that \(BM,CN,AP\) are concurrent.

Let $n\ge3$ and $a_1,\ldots,a_n\in\{0,1\}$. Let $S_i=(a_i,a_{i+1},\ldots,a_n,a_1,\ldots,a_{i-1})$ be the $n$ cyclic shifts. For a binary tuple $r=(b_1,\ldots,b_n)$ define $\omega(r)=\sum_{j=1}^n b_j2^{n-j}$. Suppose $\omega(S_1),\ldots,\omega(S_n)$ take exactly $k$ distinct values.
1. Prove $k\mid n$ and $\frac{2^n-1}{2^k-1}\mid\omega(S_i)$ for every $i$.
2. If $M$ and $m$ are their maximum and minimum, prove
$$M-m\ge\frac{(2^n-1)(2^{k-1}-1)}{2^k-1}.$$

Let $a_1,\ldots,a_{16}$ be distinct real numbers. For each real polynomial $P$, put $V(P)=\sum_{i=1}^{16}P(a_i)$. Prove there exists exactly one monic polynomial $Q$ of degree $8$ satisfying both:
1. $V(QP)=0$ for every polynomial $P$ of degree less than $8$;
2. $Q$ has $8$ real roots, counted with multiplicity.

Solve over the real numbers:
$$\begin{cases}6x-y+z^2=3,\\x^2-y^2-2z=-1,\\6x^2-3y^2-y-2z^2=0.\end{cases}$$

a) Let $a_n=\ln(2n^2+1)-\ln(n^2+n+1)$ for $n=1,2,\ldots$. Prove that only finitely many $n$ satisfy $\{a_n\}\lt 1/2$.
b) Let $b_n=\ln(2n^2+1)+\ln(n^2+n+1)$ for $n=1,2,\ldots$. Prove that infinitely many $n$ satisfy $\{b_n\}\lt 1/2016$.
Here $\{x\}=x-\lfloor x\rfloor$ denotes the fractional part of $x$.

Let $ABC$ be an acute triangle with fixed $B,C$ and variable $A$. Let $D$ be the midpoint of $BC$, and let $E,F$ be the perpendicular projections of $D$ onto $AB,AC$, respectively.
a) Let $O$ be the circumcenter of $ABC$. The line $EF$ meets $AO$ and $BC$ at $M$ and $N$, respectively. Prove that the circumcircle of $AMN$ passes through a fixed point.
b) The tangents at $E,F$ to the circumcircle of $AEF$ meet at $T$. Prove that $T$ belongs to a fixed line.

Two types of trees are planted on an $m\times n$ rectangular grid, one tree per cell. A planting is called impressive if:
i) The total numbers of the two types are equal.
ii) In every row the difference between the numbers of the two types is at least half the number of cells in that row, and in every column the analogous difference is at least half the number of cells in that column.
a) Exhibit an impressive planting for $m=n=2016$.
b) Prove that the existence of an impressive planting implies that both $m$ and $n$ are multiples of $4$.

Find all real numbers $a$ for which a function $f:\mathbb R\to\mathbb R$ exists satisfying
$$f(1)=2016,\qquad f(x+y+f(y))=f(x)+ay\quad(x,y\in\mathbb R).$$

Let $ABC$ be inscribed in a circle $(O)$ with center $O$, with the angles at $B,C$ acute. Choose $M$ on the arc $BC$ not containing $A$ such that $AM$ is not perpendicular to $BC$. Let $T$ be the intersection of $AM$ and the perpendicular bisector of $BC$. The circumcircle of $AOT$ meets $(O)$ again at $N\ne A$.
a) Prove that $\angle BAM=\angle CAN$.
b) Let $I$ be the incenter and $G$ the foot of the internal bisector at $A$. The lines $AI,MI,NI$ meet $(O)$ at $D,E,F$, respectively. Set $P=DF\cap AM$ and $Q=DE\cap AN$. The circle through $P$ tangent to $AD$ at $I$ meets $DF$ at $H$ ($H\ne D$); the circle through $Q$ tangent to $AD$ at $I$ meets $DE$ at $K$ ($K\ne D$). Prove that the circumcircle of $GHK$ is tangent to $BC$.

A positive integer is called perfect if it equals the sum of its positive divisors other than itself.
a) Prove that an odd perfect number has the form $n=p^s m^2$, where $p$ is a prime of the form $4k+1$, $s$ is a positive integer of the form $4h+1$, and $m$ is a positive integer not divisible by $p$.
b) Find all integers $n\gt 1$ for which both $n-1$ and $n(n+1)/2$ are perfect.