MathDB

4

Part of 1997 Romania Team Selection Test

Problems(4)

Least n such that any n-subset of A contains x,y with x|y

Source: Romanian TST 1997

9/17/2011
Let p,q,rp,q,r be distinct prime numbers and let A={paqbrc∣0≤a,b,c≤5}A=\{p^aq^br^c\mid 0\le a,b,c\le 5\} Find the least n∈Nn\in\mathbb{N} such that for any B⊂AB\subset A where ∣B∣=n|B|=n, has elements xx and yy such that xx divides yy.
Ioan Tomescu
floor functionceiling functionnumber theoryprime numberscombinatorics proposedcombinatoricsposet
Circles tangent iff angle BAD is the same as angle CAD

Source: Romanian TST 1997

5/27/2009
Let ABCABC be a triangle, DD be a point on side BCBC, and let O\mathcal{O} be the circumcircle of triangle ABCABC. Show that the circles tangent to O,AD,BD\mathcal{O},AD,BD and to O,AD,DC\mathcal{O},AD,DC are tangent to each other if and only if ∠BAD=∠CAD\angle BAD=\angle CAD.
Dan Branzei
geometrycircumcircleincentergeometric transformationhomothetyratiopower of a point
Infinitely many (x,y) such that x|P(y) and y|P(x)

Source: Romanian TST 1997

9/17/2011
Let n≥2n\ge 2 be an integer and let P(X)=Xn+an−1Xn−1+…+a1X+1P(X)=X^n+a_{n-1}X^{n-1}+\ldots +a_1X+1 be a polynomial with positive integer coefficients. Suppose that ak=an−ka_k=a_{n-k} for all k∈1,2,…,n−1k\in 1,2,\ldots,n-1. Prove that there exist infinitely many pairs of positive integers x,yx,y such that x∣P(y)x|P(y) and y∣P(x)y|P(x).
Remus Nicoara
algebrapolynomialnumber theorygreatest common divisormodular arithmeticalgebra proposed
Inversion, Romania 1997

Source:

1/12/2008
Let ww be a circle and ABAB a line not intersecting ww. Given a point P0P_{0} on ww, define the sequence P0,P1,…P_{0},P_{1},\ldots as follows: P_{n\plus{}1} is the second intersection with ww of the line passing through BB and the second intersection of the line APnAP_{n} with ww. Prove that for a positive integer kk, if P_{0}\equal{}P_{k} for some choice of P0P_{0}, then P_{0}\equal{}P_{k} for any choice of P0P_{0}.
Gheorge Eckstein
geometrycircumcircleincentergeometry unsolved