3

Part of 2003 Junior Balkan Team Selection Tests - Romania

Problems(3)

\sqrt{n}+\sqrt{n+1} < \sqrt{x}+\sqrt{y} <\sqrt{4n+2}, diophantine inequality

Source: 2003 Romania JBMO TST 1.3

n

be a positive integer. Prove that there are no positive integers

x

y

\sqrt{n}+\sqrt{n+1} < \sqrt{x}+\sqrt{y} <\sqrt{4n+2}

diophantinenumber theoryradicalinequalities

sums a + b,b + c and a + b + c are nonnegative, numbers around a circle

Source: 2003 Romania JBMO TST 2.3

Five real numbers of absolute values not greater than

1

and having the sum equal to

1

are written on the circumference of a circle. Prove that one can choose three consecutively disposed numbers

a, b, c

, such that all the sums

a + b,b + c

a + b + c

are nonnegative.

2 out of 2003 such their sum is not a divisor of the sum of other elements.

Source: 2003 Romania JBMO TST 3.3

2003

positive integers is given. Show that one can find two elements such that their sum is not a divisor of the sum of the other elements.

number theorydivisorSumdividescombinatorics