MathDB

1

Part of 1995 Romania Team Selection Test

Problems(4)

Ab=ac or a=90

Source: Romanian TST 1995,problem1

3/9/2006
Let AD be the altitude of a triangle ABC and E , F be the incenters of the triangle ABD and ACD , respectively. line EF meets AB and AC at K and L. prove tht AK=AL if and only if AB=AC or A=90
geometryincentercircumcircleconicsellipseperpendicular bisectorgeometry unsolved
Sequence of integers

Source: Romania TST 1995

9/18/2009
The sequence (xn) (x_n) is defined by x_1\equal{}1,x_2\equal{}a and x_n\equal{}(2n\plus{}1)x_{n\minus{}1}\minus{}(n^2\minus{}1)x_{n\minus{}2} ∀n≥3 \forall n \geq 3, where a∈N∗ a \in N^*.For which value of a a does the sequence have the property that xi∣xj x_i|x_j whenever i<j i<j.
number theory proposednumber theory
(a_1^5 + ...+ a_n^5) + (a_1^7 + ...+ a_n^7) &gt;= 2(a_1^3 + ...+ a_n^3)^2

Source: Romania TST 1995 3.1

2/17/2020
Let a1,a2,....,ana_1, a_2,...., a_n be distinct positive integers. Prove that (a15+...+an5)+(a17+...+an7)≥2(a13+...+an3)2(a_1^5 + ...+ a_n^5) + (a_1^7 + ...+ a_n^7) \ge 2(a_1^3 + ...+ a_n^3)^2 and find the cases of equality.
SumSum of powersinequalitiesalgebra
colorings of an n-gon in p colors

Source: Romania TST 1995 4.1

2/17/2020
How many colorings of an nn-gon in p≥2p \ge 2 colors are there such that no two neighboring vertices have the same color?
Coloringpolygoncombinatoricscombinatorial geometry