MathDB

9.2

Part of V Soros Olympiad 1998 - 99 (Russia)

Problems(3)

x^4 + 4x^3 - 8x + 4 = 0 (V Soros Olympiad 1998-99 Round 1 9.2)

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5/25/2024
Solve the equation x4+4x3−8x+4=0x^4 + 4x^3 - 8x + 4 = 0.
algebra
3 equal non intersecting circles (V Soros Olympiad 1998-99 Round 2 9.2)

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5/25/2024
There are two equal non-intersecting circles on a plane. Two lines were drawn. Each of the lines intersects the circles at four points so that the three segments formed on each of the lines are equal (segments with ends at adjacent points of intersection are considered). For one line these segments have length aa, for the other they have length bb (a<ba < b). Find the radius of the circles.
geometry
simplify be reducing gives correct (V Soros Olympiad 1998-99 Round 3 9.2)

Source:

5/25/2024
As evidence that the correct answer does not mean the correctness of the proof, the teacher cited next example. Let's take the fraction 1995\frac{19}{95}. After crossing out 99 in the numerator and denominator (“reduction” by 99), we get 15\frac{1}{5} which is the correct answer. In the same way, a fraction 19999995\frac{1999}{9995} can be “reduced” by three nines (cross out 999999 in the numerator and denominator). Is it possible that as a result of such a “reduction” we also get the correct answer, equal to 13\frac13 ? (We consider fractions of the form 1aa3\frac{1a}{a3}. Here, with the letter aa we denote several numbers that follow in the same order in the numerator after 11, and in the denominator before 33. “Reduce” by aa.)
algebranumber theory