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9.6

Part of V Soros Olympiad 1998 - 99 (Russia)

Problems(3)

{x [x] } = 0.5 (V Soros Olympiad 1998-99 Round 1 9.6)

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5/25/2024
How many solutions satisfying the condition 1<x<51 < x < 5 does the equation {x[x]}=0.5\{x[x]\} = 0.5 have?
(Here [x][x] is the integer part of the number xx, {x}=x−[x]\{x\} = x - [x] is the fractional part of the number xx.)
algebrafractional part
ratio that (BMF) cuts segment BE (V Soros Olympiad 1998-99 Round 2 9.6)

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5/25/2024
On side ABAB of triangle ABCABC, points MM and KK are taken (MM on segment AKAK). It is known that AM:MK:MB=a:b:cAM: MK: MB = a: b: c. Straight lines CMCM and CKCK intersect for the second time the circumscribed circle of the triangle ABCABC at points EE and FF, respectively. In what ratio does the circumscribed circle of the triangle BMFBMF divide the segment BEBE?
ratiogeometry
|x- y| + |1 -x| + |у| = 1 (V Soros Olympiad 1998-99 Round 3 9.6)

Source:

5/25/2024
On the coordinate plane, draw all points M(x,y)M(x, y), whose coordinates satisfy the equation: ∣x−y∣+∣1−x∣+∣y∣=1 |x-y| + |1-x| + |y|=1
analytic geometrygeometry