VN TST 1992

Let $n,m$ be natural numbers, with $n\gt 1$. Find the least positive integer $k$ with the following property: among any $k$ integers $a_1,a_2,\ldots,a_k$ such that $a_i-a_j$ is not divisible by $n$ whenever $i\ne j$, there are two terms $a_p,a_s$, $p\ne s$, such that $m+a_p-a_s$ is divisible by $n$.

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

Dates 19–20 May 1992 are printed; the split into questions 1–3 and 4–6 is editorial because the source has no explicit day divider. All essay rubrics are editorial, and assessment is manual. Importer multipliers are correct 1, incorrect 0, blank 0.

Let $f(x)$ be a real polynomial of degree at least one. Prove that for every $c\gt 0$ there is a positive integer $n_0$ such that, for every monic real polynomial $P(x)$ of degree at least $n_0$, the number of integers $x$ satisfying $|f(P(x))|\le c$ is at most the degree of $P(x)$.

Let $ABC$ be a triangle with $BC=a,CA=b,AB=c$ and $a\ne b\ne c$. In its plane choose $A',B',C'$ so that:
1. The pairs $A,A'$; $B,B'$; $C,C'$ are either all on the same side, or all on opposite sides, of $BC,CA,AB$, respectively.
2. The triangles $A'BC,B'CA,C'AB$ are similar isosceles triangles.
Determine $\angle A'BC$ in terms of $a,b,c$ so that the lengths $AA',BB',CC'$ are not the side lengths of a triangle. A triangle is understood in the ordinary nondegenerate sense: its three vertices are not collinear.

In the plane there is a finite family of disks. Any two are externally disjoint or externally tangent, and no disk touches more than six other disks. Suppose a real number has been assigned to every disk that does not touch six other disks. Prove that there is at most one assignment of real numbers to the remaining disks such that the number on each remaining disk is the arithmetic mean of the six numbers on its six touching disks.

Find all pairs of positive integers $(x,y)$ satisfying $$x^2+y^2-5xy+5=0.$$

At a scientific conference the delegates collectively know $2n$ languages, where $n\ge2$. Each delegate knows exactly two languages, and any two delegates have at most one language in common. For each integer $k$ with $1\le k\le n-1$, there are at most $k-1$ languages each known by at most $k$ delegates. Prove that one can select a group of $2n$ delegates who collectively know all $2n$ languages, with exactly two members of the group knowing each language.