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9.8

Part of II Soros Olympiad 1995 - 96 (Russia)

Problems(2)

R=? h_a, m_a, d_a given (II Soros Olympiad 1995-96 R1 9.8)

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6/2/2024
The altitude, angle bisector and median coming from one vertex of the triangle are equal to 3\sqrt3, 22 and 6\sqrt6, respectively. Find the radius of the circle circumscribed round this triangle.
geometrycircumradius
{5{4{3{2{х}}}}}=1 nested fractional parts (II Soros Olympiad 1995-96 R3 9.8)

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6/4/2024
Let a{a} be the fractional part of the number aa, that is, {a}=a−[a]\{a\} = a - [a], where[a] [a] is the integer part of a a. (For example, {1.7}=1.7−1=0.7\{1.7\} = 1.7 -1 = 0.7,{−2}=−2−(−3)=3−2\{-\sqrt2 \}= -\sqrt2 -(-3) = 3-\sqrt2.) a) How many solutions does the equation have {5{4{3{2{x}}}}}=1  ? \{5\{4\{3\{2\{x\}\}\}\}\}=1\,\, ? b) Find its greatest solution.
fractional partalgebra