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11.4

Part of II Soros Olympiad 1995 - 96 (Russia)

Problems(3)

points M(a, b), with x^4+ax+b=0, 0<=x<=1 (II Soros Olympiad 1995-96 R1 11.4)

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6/3/2024
Draw on the coordinate plane a set of points M(a,b)M(a, b) such that the equation x4+ax+b=0x^4+ax+b=0 has a unique root satisfying the condition 0≤x≤10 \le x \le 1.
analytic geometryalgebra
x^6 - 100x+1 = 0 (II Soros Olympiad 1995-96 R2 11.4)

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6/3/2024
Prove that the equation x6−100x+1=0x^6 - 100x+1 = 0 has two roots, and both of these roots are positive. a) Find the first non-zero digit in the decimal notation of the lesser root of this equation. b) Find the first two non-zero digits in the decimal notation of the lesser root of this equation.
algebrapolynomial
>=6 tangents for a point to y = (1 -x^2)^3 (II Soros Olympiad 1997-98 R3 11.4)

Source:

6/6/2024
Consider the graph of the function y=(1−x2)3y = (1 -x^2)^3. Find the set of points M(x,y)M(x,y) through which you can draw at least 66 lines touching this graph.
algebraanalysiscalculus