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9.2

Part of III Soros Olympiad 1996 - 97 (Russia)

Problems(3)

y = x^2 + px + q, y = 2x +p are tangent (III Soros Olympiad 1996-97 R1 9.2)

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5/29/2024
It is known that the graph of a quadratic trinomial y=x2+px+qy = x^2 + px + q touches the graph of a straight line y=2x+py = 2x + p. Prove that all such quadratic trinomials have the same minimum value. Find this smallest value.
analytic geometrygeometryconics
4 bells ring in different times (III Soros Olympiad 1996-97 R2 9.2)

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5/31/2024
Three bells begin to ring simultaneously. The intervals between strikes for these bells are, respectively, 43\frac43 seconds, 53\frac53 second and 22 seconds. Impacts that coincide in time are perceived as one. How many beats will be heard in 11 minute? (Include first and last.)
algebra
\sqrt{x^2-4}+\sqrt{2x^2-7x+5}=a (III Soros Olympiad 1996-97 R3 9.2)

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5/31/2024
How many solutions, depending on the value of the parameter aa, has the equation x2−4+2x2−7x+5=a?\sqrt{x^2-4}+\sqrt{2x^2-7x+5}=a ?
algebraRadicalsparameter