MathDB

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Part of 2002 Junior Balkan Team Selection Tests - Romania

Problems(4)

A square of side 1 ...

Source:

6/17/2009
A square of side 1 is decomposed into 9 equal squares of sides 1/3 and the one in the center is painted in black. The remaining eight squares are analogously divided into nine squares each and the square in the center is painted in black. Prove that after 1000 steps the total area of black region exceeds 0.999
geometrycombinatorics proposedcombinatorics
a + b is a divisor of a^n + b^n , n even

Source: 2002 Romania JBMO TST 3.1

5/31/2020
Let nn be an even positive integer and let a,ba, b be two relatively prime positive integers. Find aa and bb such that a+ba + b is a divisor of an+bna^n + b^n.
number theorydividesdivisibledivisorSum of powers
\sqrt{3 -x^2} and \sqrt{a - x^3} cannot be rational, if x, x^3 < 3

Source: 2002 Romania JBMO TST 4.1

5/31/2020
Let aa be an integer. Prove that for any real number x,x3<3x, x^3 < 3, both the numbers 3−x2\sqrt{3 -x^2} and a−x3\sqrt{a - x^3} cannot be rational.
rationalalgebraradicalirrational
x^n+y^n = 2^m diophantine

Source: 2002 Romania JBMO TST 5.1

5/31/2020
Let m,n>1m,n > 1 be integer numbers. Solve in positive integers xn+yn=2mx^n+y^n = 2^m.
number theoryDiophantine equationdiophantine