MathDB

1

Part of 2009 Junior Balkan Team Selection Tests - Romania

Problems(4)

a_n=2 33...3, n times digit 3, multiople of a_{2009}

Source: 2009 Romania JBMO TST 1.1

6/1/2020
For all positive integers nn define an=233...3⏟n timesa_n=2 \underbrace{33...3}_{n \, times}, where digit 33 occurs nn times. Show that the number a2009a_{2009} has infinitely many multiples in the set {an∣n∈N∗}\{a_n | n \in N*\}.
number theoryDigitsmultiple
7^a= 4^b + 5^c + 6^d diophantine

Source: 2009 Romania JBMO TST 2.1

6/1/2020
Find all non-negative integers a,b,c,da,b,c,d such that 7a=4b+5c+6d7^a= 4^b + 5^c + 6^d.
Diophantine equationdiophantinenumber theory
(AB -AC)^2(BC^2 + 4AB \cdot AC)^2 <= 2BC^6 in a right triangle (<A=90^o)

Source: 2009 Romania JBMO TST4 P1

5/26/2020
Show that in any triangle ABCABC with A=900A = 90^0 the following inequality holds: (AB−AC)2(BC2+4AB⋅AC)2≤2BC6(AB -AC)^2(BC^2 + 4AB \cdot AC)^2 \le 2BC^6
geometryright trianglegeometric inequalityinequalities
a/b+b/c+c/a>= 1/ab+1/bc+1/ca if a,b,c>0 and a+b+c>=1/a+1/b+1/c

Source: 2009 Romania JBMO TST 3.1

6/1/2020
Let a,b,ca, b, c be positive real number such that a+b+c≥1a+1b+1ca + b + c \ge \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c} . Prove that ab+bc+ca≥1ab+1bc+1ca \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\ge \frac{1}{ab}+ \frac{1}{bc}+ \frac{1}{ca} .
inequalitiesalgebra