MathDB

4

Part of 1995 Romania Team Selection Test

Problems(4)

Romanian TST 1995

Source:

8/28/2018
Let m,nm,n be positive integers, greater than 2.Find the number of polynomials of degree 2n−12n-1 with distinct coefficients from the set {1,2,…,m}\left\{ 1,2,\ldots,m\right\} which are divisible by xn−1+xn−2+…+1.x^{n-1}+x^{n-2}+\ldots+1.
algebra
convex set S on a plane, not lying on a line, is painted in p colors

Source: Romania TST 1995 2.4

2/17/2020
A convex set SS on a plane, not lying on a line, is painted in pp colors. Prove that for every n≥3n \ge 3 there exist infinitely many congruent nn-gons whose vertices are of the same color.
combinatorial geometrycombinatoricsColoringconvex
A sequence of integers

Source: Romania TST 1995 Test 3 P4

2/22/2014
Find a sequence of positive integers f(n)f(n) (n∈Nn \in \mathbb{N}) such that: (i) f(n)≤n8f(n) \leq n^8 for any n≥2n \geq 2; (ii) for any distinct a1,⋯ ,ak,na_1, \cdots, a_k, n, f(n)≠f(a1)+⋯+f(ak)f(n) \neq f(a_1) + \cdots+ f(a_k).
number theory
similar isosceles on sides of convex ABCD, square condition, rhombus wanted

Source: Romania TST 1995 4.4

2/17/2020
Let ABCDABCD be a convex quadrilateral. Suppose that similar isosceles triangles APB,BQC,CRD,DSAAPB, BQC, CRD, DSA with the bases on the sides of ABCDABCD are constructed in the exterior of the quadrilateral such that PQRSPQRS is a rectangle but not a square. Show that ABCDABCD is a rhombus.
geometryrhombussquareisosceles