[expert { circle problem }] ........

Geometry Level 2

For each natural number k , let Ck denote the circle with radius k centimetres and the centre at origin . On the circle Ck , a particle moves k centimetres in the counter clockwise direction . After completing its motion on Ck , the particle moves to C(k+1) in the radial direction . the motion of the particle continues in this manner . The particle starts at (1,0) if the particle crosses the positive direction of x -axis for the first time on the circle Cn, then n =

7 8 1 11 5 6

1 solution

Ronald Overwater
Jan 23, 2015

Moving on a circle with radius $k$ , the angle covered by moving the length $k$ on the circle equals $\dfrac {k}{2k \pi} = \dfrac {1}{2 \pi}$ .

So you have to move to the circle with radius $N$ to cover $\dfrac{N} {2 \pi}$ .

For $N> 2 \pi \approx 6.28$ this is larger than $1$ , so minimal $N$ is $7$ .

[expert { circle problem }] ........

1 solution

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