MathDB

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Part of 2004 Federal Math Competition of S&M

Problems(3)

Diophantine Equation

Source: Federal Mathematical Competition of Serbia and Montenegro 2004

5/14/2018
Find all pairs of positive integers (a,b)(a,b) such that 5ab−b=20045a^b - b = 2004.
number theoryDiophantine equation
Number theory problem

Source: Federal Mathematical Competition of Serbia and Montenegro 2004

5/14/2018
Suppose that a,b,ca,b,c are positive numbers such that ab+bc+ca\frac{a}{b}+\frac{b}{c}+\frac{c}{a} is an integer. Show that abcabc is a perfect cube.
number theory
Geometry Problem

Source: Federal Mathematical Competition of Serbia and Montenegro 2004

5/14/2018
In a triangle ABCABC of the area SS, point HH is the orthocenter, D,E,FD,E,F are the feet of the altitudes from A,B,CA,B,C, and P,Q,RP,Q,R are the reflections of A,B,CA,B,C in BC,CA,ABBC,CA,AB, respectively. The triangles DEFDEF and PQRPQR have the same area TT. Given that T>35ST > \frac{3}{5}S, prove that T=ST = S.
geometry