MathDB

10.3

Part of III Soros Olympiad 1996 - 97 (Russia)

Problems(3)

2 geometric progressions (III Soros Olympiad 1996-97 R1 10.3)

Source:

5/29/2024
An infinite sequence of numbers a,b,c,d,...a, b, c, d,... is obtained by term-by-term addition of two geometric progressions. Can this sequence begin with the following numbers:. a) 1,1,3,51,1,3,5 ? b) 1,2,3,51,2,3,5 ? c) 1,2,3,41,2,3, 4 ?
algebrageometric progression
cos<AOB by two tangents in trinomial (III Soros Olympiad 1996-97 R2 10.3)

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5/31/2024
Let's consider the graph of a square trinomial having roots 11 and 44. Let's draw two tangents to it from point OO (the origin of coordinates), touching it at points AA and BB. What values can the cosine of angle ∠AOB\angle AOB take?
analytic geometry
\sqrt{x(x+7)}+\sqrt{(x+7)(x+17)}+... (III Soros Olympiad 1996-97 R3 10.3)

Source:

5/31/2024
Solve the equation x(x+7)+(x+7)(x+17)+(x+17)(x+24)=12+172\sqrt{x(x+7)}+\sqrt{(x+7)(x+17)}+\sqrt{(x+17)(x+24)}=12+17\sqrt2
algebraRadicals