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10.2

Part of VI Soros Olympiad 1999 - 2000 (Russia)

Problems(4)

D=6G, D=25R, D=120T, G=4R, G=21T, R=5T- VI Soros Olympiad 1999-00 Round 1 10.2

Source:

5/21/2024
The currency exchange trades dinars (D), guilders (G), reals (R) and thalers (T). Exchange players have the right to make a purchase and sale transaction with each pair of currencies no more than once a day. The exchange rates are as follows: D=6GD = 6G,; D=25RD=25R, D=120TD=120T,G=4RG = 4R; G=21TG=21T, R=5TR = 5T. For example, the entry D=6GD = 6G means that 11 dinar can be bought for 66 guilders (or 66 guilders can be sold for 11 dinar). In the morning the player had 3232 dinars. What is the maximum number that he can receive by evening a) in dinars? b) in thalers ?
number theory
(\pi-2)/2 + 2/(1+cos (2\sqrtx})+arcsin (VI Soros Olympiad 1990-00 R1 10.2)

Source:

5/28/2024
Solve the equation π−22+21+sin⁡(2x)+arccos(x3−8x−1)=tg2x−x4+x3−5x2−8x−24\frac{\pi-2}{2} + \frac{2}{1+\sin (2\sqrt{x})}+arccos(x^3-8x-1)=tg^2\sqrt{x}- \sqrt{x^4+x^3-5x^2-8x-24}
algebratrigonometry
37 points on plane (VI Soros Olympiad 1990-00 R2 10.2)

Source:

5/28/2024
3737 points are arbitrarily marked on the plane. Prove that among them there must be either two points at a distance greater than 66, or two points at a distance less than 1.51.5.
geometrycombinatorial geometrycombinatoricspoints
obtuse wanted, if < BAC >=60^o

Source: VI Soros Olympiad 1990-00 R3 10.2 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/28/2024
In the triangle ABCABC, the point XX is the projection of the touchpoint of the inscribed circle to the side BCBC on the middle line parallel to BCBC. It is known that ∠BAC≥60o\angle BAC \ge 60^o. Prove that the angle BXCBXC is obtuse.
anglesgeometry