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10.7

Part of II Soros Olympiad 1995 - 96 (Russia)

Problems(2)

inradius , 3 lines divide plane in 7 parts, distances of 3 points from lines

Source: II Soros Olympiad 1995-96 R1 10.7 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/3/2024
Three straight lines ℓ1\ell_1, ℓ2\ell_2 and ℓ3\ell_3, forming a triangle, divide the plane into 77 parts. Each of the points M1M_1, M2M_2 and M3M_3 lies in one of the angles, vertical to some angle of the triangle. The distance from M1M_1 to straight lines ℓ1\ell_1, ℓ2\ell_2 and ℓ3\ell_3 are equal to 7,37,3 and 11 respectively The distance from M2M_2 to the same lines are 44, 11 and 33 respectively. For M3M_3 these distances are 33, 55 and 22. What is the radius of the circle inscribed in the triangle?
[hide=second sentence in Russian]Каждая из точек М_1, М_2 и М_з лежит в одном из углов, вертикальном по отношению к какому-то углу треугольника.
geometryinradiuslines
2x2 system, <19x>=, <96y>=x (II Soros Olympiad 1997-98 R3 10.7)

Source:

6/6/2024
Let us denote by <a><a> the distance from aa to the nearest integer. (For example, <1,2>=0.2<1,2> = 0.2, <3>=2−3<\sqrt3> = 2-\sqrt3) How many solutions does the system of equations have {<19x>=y<96y>=x   ?\begin{cases} <19x>=y \\ <96y>=x \end{cases} \,\,\, ?
algebrasystem of equationsnumber theory