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11.1

Part of VI Soros Olympiad 1999 - 2000 (Russia)

Problems(3)

x^3 + ax^2 + bx + c.=0, 3 real roots - VI Soros Olympiad 1999-00 Round 1 11.1

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5/21/2024
The game involves two players AA and BB. Player AA sets the value of one of the coefficients a,ba, b or cc of the polynomial x3+ax2+bx+c.x^3 + ax^2 + bx + c. Player BB indicates the value of any of the two remaining coefficients . Player AA then sets the value of the last coefficients. Is there a strategy for player A such that no matter how player BB plays, the equation x3+ax2+bx+c=0x^3 + ax^2 + bx + c = 0 to have three different (real) solutions?
algebrapolynomial
2x2 system with arcsin and radical (VI Soros Olympiad 1990-00 R2 11.1)

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5/28/2024
Solve the system of equations {x2+arcsiny=y2+arcsinxx2+y2−3x=2yx2−2x−y+1\begin{cases} x^2+arc siny =y^2+arcsin x \\ x^2+y^2-3x=2y\sqrt{x^2-2x-y}+1 \end{cases}
algebrasystem of equationsRadicalstrigonometry
2 coprime among 16 naturals (VI Soros Olympiad 1990-00 R3 11.1)

Source:

5/28/2024
1616 different natural numbers are written on the board, none of which exceeds 3030. Prove that there must be two coprime numbers among the written numbers.
number theorycombinatorics