Question bank VMO 1994

Given a parallelogram $ABCD$, let $E$ lie on side $BC$ and $F$ on side $CD$ so that triangles $ABE$ and $BCF$ have equal area. The diagonal $BD$ meets $AE$ at $M$ and $AF$ at $N$.
(a) Prove that a triangle exists with side lengths $BM,MN,ND$.
(b) As $E,F$ vary so that $MN$ decreases, prove that the circumradius of that triangle also decreases.

Editorial status: the literal statement allows endpoints and is false there. The source proofs establish the conditional version with $E,F$ strictly inside their sides, equivalently $0\lt BE/BC=CF/CD\lt1$. This additional interiority condition is a substantive repair, not implied by the two given area triangles. The original statement above is retained for review.

The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.

The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention.

The printed dates are 18–19 May 1994; grouping Q1–3 and Q4–6 into sessions is editorial, since no separate day boundary is printed. Import scoring multipliers are correct 1, incorrect 0, blank 0. All six essays require manual assessment; the rubrics are editorial, not official step marks.

For a given positive integer $N$, consider
$$x^2+y^2+z^2+t^2-Nxyzt-N=0.$$
(a) Prove that infinitely many values of $N$ admit a solution in positive integers $x,y,z,t$.
(b) If $N=4^k(8m+7)$, where $k,m$ are nonnegative integers, prove that there is no positive integer solution.

Let $P(x)$ be a polynomial of degree $4$ having four positive roots. Prove that
$$\frac{1-4x}{x^2}P(x)+\left(1-\frac{1-4x}{x^2}\right)P'(x)-P''(x)=0$$
also has four positive roots. If the four given roots are distinct, the four roots obtained below are distinct; the multiplicity version is also explained.

Let $ABC$ be an equilateral triangle and $M$ any point in its plane. Let $A',B',C'$ be the reflections of $A,B,C$, respectively, in the point $M$.
(a) Prove that there is a unique point $P$ equidistant from $A,B'$, from $B,C'$, and from $C,A'$.
(b) Let $D$ be the midpoint of $AB$. As $M$ varies, with $M\ne D$, let $N=DM\cap AP$. Prove that the circumcircle of triangle $MNP$ passes through a fixed point.
Construction domain for (b): a circumcircle is understood only when $M,N,P$ are three distinct noncollinear points. The printed condition $M\ne D$ alone does not ensure this; the exceptional undefined configurations are explicitly discussed in the solutions.

Determine all functions $f:\mathbb R\to\mathbb R$ satisfying
$$f(\sqrt2\,x)+f((4+3\sqrt2)x)=a f((2+\sqrt2)x)\qquad(x\in\mathbb R).$$
The parameter $a$ is retained exactly as printed. The classification below applies to every fixed real $a$; the supplied worked solution instead specializes to coefficient $2$. No continuity, measurability or other regularity is assumed.

Calculate the sum
$$T=\sum_{\substack{n_1,\ldots,n_{1994}\ge0\\n_1+2n_2+\cdots+1994n_{1994}=1994}}\frac{1}{n_1!n_2!\cdots n_{1994}!\,(n_2+2n_3+\cdots+1993n_{1994})!}.$$
The tuple entries are nonnegative integers, with $0!=1$. The source display omits the summation sign but immediately specifies that the expression is summed over all such tuples; the sign has been restored here.

Solve the system
$$\begin{cases}x^3+3x-3+\ln(x^2-x+1)=y,\\ y^3+3y-3+\ln(y^2-y+1)=z,\\ z^3+3z-3+\ln(z^2-z+1)=x.\end{cases}$$

Let $ABC$ be a triangle. Reflect $A,B,C$ in the lines containing their opposite sides to obtain $A',B',C'$, respectively. Determine a necessary and sufficient condition on $ABC$ for the triangles $ABC$ and $A'B'C'$ to have equal areas.

Define $x_0,x_1,x_2,\ldots$ by $x_0=a$, where $0\lt a\lt 1$, and
$$x_{n+1}=\frac4{\pi^2}\left(\arccos x_n+\frac\pi2\right)\arcsin x_n.$$
Prove that the sequence converges and determine its limit.

There are $n+1$ boxes arranged on a circle. One box contains $n$ stones; the others are empty. A move is defined as follows: choose two boxes $A$ and $B$, take two stones from $A$, put one into a box adjacent to $B$, and put the other into a box adjacent to $A$. Note that $A=B$ is allowed. For which $n$ can a sequence of such moves result in each stone being in a separate box?

Let $S$ be a sphere with center $O$, and let $G,G'$ be two great circles whose planes are perpendicular. Choose $A,B,C$ on $G$ and $D$ on $G'$ so that the four altitudes of the nondegenerate tetrahedron $ABCD$ are concurrent. Determine the locus of their common point.

Do there exist polynomials $p(x),q(x),r(x)$ whose coefficients are positive integers, satisfying
$$p(x)=(x^2-3x+3)q(x),\qquad q(x)=\left(\frac{x^2}{20}-\frac{x}{15}+\frac1{12}\right)r(x)?$$