MathDB

10.3

Part of V Soros Olympiad 1998 - 99 (Russia)

Problems(3)

5x^6 - 16x^4 - 33x^3 - 40x^2 +8 = 0 (V Soros Olympiad 1998-99 Round 1 10.3)

Source:

5/25/2024
Find two roots of the equation 5x6−16x4−33x3−40x2+8=0,5x^6 - 16x^4 - 33x^3 - 40x^2 +8 = 0, whose product is equal to 11.
algebra
P (cos x/ 1999t) = 1/2 (V Soros Olympiad 1998-99 Round 2 10.3)

Source:

5/25/2024
It is known that sin⁡3x=3sin⁡x−4sin⁡3x\sin 3x = 3 \sin x - 4 \sin^3x. It is also easy to prove that sin⁡nx\sin nx for odd nn can be represented as a polynomial of degree nn of sin⁡x\sin x. Let sin⁡1999x=P(sin⁡x)\sin 1999x = P(\sin x), where P(t)P(t) is a polynomial of the 19991999th degree of tt. Solve the equation P(cos⁡x1999)=12.P \left(\cos \frac{x}{1999}\right) = \frac12 .
algebratrigonometry
29^{200} 2^{151} ? 5^{279} 3^{300} (V Soros Olympiad 1998-99 Round 3 10.3)

Source:

5/26/2024
Without using a calculator, find out which number is greater: 29200⋅2151   or   5279⋅330029^{200}\cdot 2^{151} \,\,\, or \,\,\, 5^{279} \cdot 3^{300}
algebra