1

Part of 2006 Junior Balkan Team Selection Tests - Moldova

Problems(2)

Poblem 1 (jbmo tst), moldova

Source: JBMO TST, Moldova

Five segments have lengths such that any three of them can be sides of a - possibly degenerate - triangle. Also, the lengths of these segments are nonzero and pairwisely different. Prove that there exists at least one acute-angled triangle among these triangles.

geometry proposedgeometry

Problem 1 (2nd jbmo tst)

Source: 2nd JBMO TST Moldova

x

. Using multiply and division operations of any 2 given or already given numbers we can obtain powers with natural exponent of the number

x

x\cdot x=x^{2}

x^{2}\cdot x^{2}=x^{4}

x^{4}: x=x^{3}

, etc). Determine the minimal number of operations needed for calculating

x^{2006}

algebra proposedalgebra