Question 1
Solve over the real numbers:
$$\begin{cases}6x-y+z^2=3,\\x^2-y^2-2z=-1,\\6x^2-3y^2-y-2z^2=0.\end{cases}$$
$$\begin{cases}6x-y+z^2=3,\\x^2-y^2-2z=-1,\\6x^2-3y^2-y-2z^2=0.\end{cases}$$
VMO 2016
Question 2
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 $x$.
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 $x$.
Question 3
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.
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.
Question 4
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 $4$.
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 $4$.
Question 5
Find all real numbers $a$ for which a function $f:\mathbb R\to\mathbb R$ exists satisfying
$$f(1)=2016,\qquad f(x+y+f(y))=f(x)+ay\quad(x,y\in\mathbb R).$$
$$f(1)=2016,\qquad f(x+y+f(y))=f(x)+ay\quad(x,y\in\mathbb R).$$
Question 6
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) 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 $BC$.
Question 7
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.
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.