MathDB

3

Part of 1995 Romania Team Selection Test

Problems(4)

Coloring of polygon

Source: Romania TST 1995 Test 1 P3

2/21/2014
Let n≥6n \geq 6 and 3≤p<n−p3 \leq p < n - p be two integers. The vertices of a regular nn-gon are colored so that pp vertices are red and the others are black. Prove that there exist two congruent polygons with at least [p/2]+1[p/2] + 1 vertices, one with all the vertices red and the other with all the vertices black.
geometrygeometric transformationrotationcombinatorics proposedcombinatorics
if MNPQ is square, then ABCD is also square

Source: Romania TST 1995 2.3

2/17/2020
Let M,N,P,QM, N, P, Q be points on sides AB,BC,CD,DAAB, BC, CD, DA of a convex quadrilateral ABCDABCD such that AQ=DP=CN=BMAQ = DP = CN = BM. Prove that if MNPQMNPQ is a square, then ABCDABCD is also a square.
geometrysquare
f(x^3) is irreducible when f is irreducible integer monic polynomial

Source: Romania TST 1995 3.3

2/17/2020
Let ff be an irreducible (in Z[x]Z[x]) monic polynomial with integer coefficients and of odd degree greater than 11. Suppose that the modules of the roots of ff are greater than 11 and that f(0)f(0) is a square-free number. Prove that the polynomial g(x)=f(x3)g(x) = f(x^3) is also irreducible
algebrapolynomialInteger PolynomialIrreducible
altitudes are integers, inradius is prime, sidelengths ?

Source: Romania TST 1995 4.3

2/17/2020
The altitudes of a triangle have integer length and its inradius is a prime number. Find all possible values of the sides of the triangle.
geometryinradiusaltitudessidelenghts