Question bank VMO 2014

Find all functions $f:\mathbb Z\to\mathbb Z$ such that
$$f(2m+f(m)+f(m)f(n))=nf(m)+m\qquad(m,n\in\mathbb Z).$$

Let $T=\{(x,y)\in\mathbb Z^2:-20\le x,y\le20,\ (x,y)\ne(0,0)\}$. Color exactly one of $(x,y)$ and $(-x,-y)$ for every point of $T$. For a coloring, let $N$ be the number of ordered pairs of colored points $(x_1,y_1),(x_2,y_2)$ satisfying $x_1\equiv2x_2$ and $y_1\equiv2y_2\pmod {41}$. Determine all possible values of $N$.

Let triangle $ABC$ be inscribed in circle $(O)$, and let $D$ lie on the arc $BC$ not containing $A$. Suppose $CD$ meets $AB$ at $E$, and $BD$ meets $AC$ at $F$. Let $(K)$ be the circle inside triangle $EBD$, tangent to $EB,ED$ and to $(O)$. Let $(L)$ be the circle inside triangle $FCD$, tangent to $FC,FD$ and to $(O)$.
(a) Let $M$ be the contact of $(K)$ with $BE$ and $N$ the contact of $(L)$ with $CF$. Prove that the circle with diameter $MN$ always passes through a fixed point as $D$ moves.
(b) The line through $M$ parallel to $CE$ meets $AC$ at $P$; the line through $N$ parallel to $BF$ meets $AB$ at $Q$. Prove that the circumcircles of $AMP$ and $ANQ$ are both tangent to a fixed circle as $D$ moves.
Source wording note: the source accidentally calls $(L)$ a “center”; its subsequent tangencies and the solution clearly use it as a circle. The full-arc formulation has a configuration issue, explicitly discussed in the solution.

Let $ABC$ be an acute, non-isosceles triangle with altitude $AD$, and let $P$ lie on $AD$. Lines $PB,PC$ meet $CA,AB$ at $E,F$, respectively.
(a) If $AEDF$ is cyclic, prove that $\dfrac{PA}{PD}=(\tan B+\tan C)\cot\dfrac A2$.
(b) Let $M$ be the intersection of $CP$ with the line through $B$ perpendicular to $AB$, and $N$ the intersection of $BP$ with the line through $C$ perpendicular to $AC$. Let $K$ be the perpendicular projection of $A$ onto $MN$. Prove that $\angle BKC+\angle MAN$ is constant as $P$ moves along $AD$.
All intersections and angles are understood in nondegenerate positions. Source-domain caveat: part (b), with ordinary angles on the whole altitude segment, is not valid as printed; the solution preserves the source’s configuration argument and gives an exact countercheck.

Find all integer-coefficient polynomials $P(x),Q(x)$ with the following property. The alternating sequence starts at $x_0=2014$ and satisfies
$$x_{2n+1}=P(x_{2n}),\qquad x_{2n+2}=Q(x_{2n+1})\qquad(n\ge0).$$
Every positive integer divides some nonzero term of the sequence.
Editorial indexing clarification: the source prints $x_{2n+1}=P(x_{2n}),\ x_{2n}=Q(x_{2n-1})$ with $n\ge1$, leaving $x_1$ undefined. The displayed recurrence makes explicit the intended initialization $x_1=P(2014)$, which the source solution uses.

Let $m,n,p$ be nonnegative integers, not all zero. Coordinate space is divided into unit cubes by equally spaced parallel coordinate planes. A filling of every unit cube with one of the numbers $1,2,\ldots,60$ is called a Dien Bien filling if the following holds: in every rectangular box formed by the grid planes whose three side lengths, in unit cubes and in any order, are $2m+1,2n+1,2p+1$, the number in its central unit cube is the arithmetic mean of the numbers in its eight corner unit cubes. How many Dien Bien fillings are there?
The source adds: two fillings are the same if, after choosing a cube as the center in each, cubes having the same position relative to the chosen centers carry equal numbers. The distinction between a fixed origin and equivalence under translation is discussed explicitly in the solution.

Let positive sequences $(x_n),(y_n)$ be determined by $x_1=1$, $y_1=\sqrt3$ and
$$\begin{cases}x_{n+1}y_{n+1}-x_n=0,\\x_{n+1}^2+y_n=2\end{cases}\qquad(n=1,2,\ldots).$$
Prove that both sequences converge and find their limits.

Let $P(x)=(x^2-7x+6)^{2n}+13$, where $n$ is a positive integer. Prove that $P(x)$ cannot be expressed as the product of $n+1$ nonconstant polynomials with integer coefficients.

A regular polygon has $103$ vertices. Color $79$ vertices red and all remaining vertices blue. Let $A$ be the number of adjacent red-red pairs and $B$ the number of adjacent blue-blue pairs.
(a) Find all possible pairs $(A,B)$.
(b) Find the number of colorings for which $B=14$, where two colorings are identified if one is obtained from the other by a rotation about the circumcenter.

Let an acute triangle $ABC$ be inscribed in a circle $(O)$, with $AB\lt AC$. Let $I$ be the midpoint of the arc $BC$ not containing $A$. Choose $K\ne C$ on segment $AC$ such that $IK=IC$. Line $BK$ meets $(O)$ again at $D\ne B$ and meets line $AI$ at $E$. Line $DI$ meets line $AC$ at $F$.
(a) Prove $EF=BC/2$.
(b) Choose $M$ on line $DI$ so that $CM\parallel AD$. Line $KM$ meets line $BC$ at $N$. The circumcircle of $BKN$ meets $(O)$ again at $P\ne B$. Prove that line $PK$ passes through the midpoint of $AD$.

An acute triangle $ABC$ is inscribed in a fixed circle $(O)$, with $B,C$ fixed and $A$ moving on $(O)$. On rays $AB,AC$, respectively, choose $M,N$ such that $MA=MC$ and $NA=NB$. The circumcircles of $AMN$ and $ABC$ meet again at $P\ne A$. Line $MN$ meets line $BC$ at $Q$.
(a) Prove that $A,P,Q$ are collinear.
(b) Let $D$ be the midpoint of $BC$. The circles centered at $M,N$ and passing through $A$ meet again at $K\ne A$. The line through $A$ perpendicular to $AK$ meets $BC$ at $E$. The circumcircle of $ADE$ meets $(O)$ again at $F\ne A$. Prove that $AF$ passes through a fixed point.

For positive real numbers $x,y,z$, find the greatest value of
$$T=\frac{x^3y^4z^3}{(x^4+y^4)(xy+z^2)^3}+\frac{y^3z^4x^3}{(y^4+z^4)(yz+x^2)^3}+\frac{z^3x^4y^3}{(z^4+x^4)(zx+y^2)^3}.$$

Find all collections of $2014$ rational numbers, not necessarily distinct, with the following property: after any one number is removed, the remaining $2013$ numbers can be partitioned into three disjoint groups of $671$ numbers each, with equal products in all three groups.