MathDB

2

Part of 1993 Romania Team Selection Test

Problems(4)

Romanian TST 1993

Source: Romanian TST 1993 - Day 1 - Problem 2

4/9/2012
Let ABCABC be a triangle inscribed in the circle C(O,R)\mathcal{C}(O,R) and circumscribed to the circle C(L,r)\mathcal{C}(L,r). Denote d=RrR+rd=\dfrac{Rr}{R+r}. Show that there exists a triangle DEFDEF such that for any interior point MM in ABCABC there exists a point XX on the sides of DEFDEF such that MX≤dMX\le d.
Dan Brânzei
geometry proposedgeometry
f(x) = x^{m+n} -x^{m+1} -x+1, g(x) = x^{m+n} +x^{n+1} -x+1

Source: Romania IMO TST 1993 2.2

2/17/2020
For coprime integers m>n>1m > n > 1 consider the polynomials f(x)=xm+n−xm+1−x+1f(x) = x^{m+n} -x^{m+1} -x+1 and g(x)=xm+n+xn+1−x+1g(x) = x^{m+n} +x^{n+1} -x+1. If ff and gg have a common divisor of degree greater than 11, find this divisor.
algebrapolynomialdivisor
x²+y²+z² = 1993 then x+y+z can't be a perfect square

Source:

4/29/2008
x^2 \plus{} y^2 \plus{} z^2 \equal{} 1993 then prove x \plus{} y \plus{} z can't be a perfect square:
number theory unsolvednumber theory
Romania TST 1993,problem 2,day 3

Source:

8/14/2009
Suppose that D,E,F D,E,F are points on sides BC,CA,AB BC,CA,AB of a triangle ABC ABC respectively such that BD\equal{}CE\equal{}AF and \angle BAD\equal{}\angle CBE\equal{}\angle ACF.Prove that the triangle ABC ABC is equilateral.
geometry unsolvedgeometry