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9.1

Part of VI Soros Olympiad 1999 - 2000 (Russia)

Problems(4)

3 boxes TV show - VI Soros Olympiad 1999-00 Round 1 9.1

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5/21/2024
In the television program “Field of Miracles,” the presenter played the prize as follows. The player was shown three boxes, one of which contained a prize. The player pointed to one of the boxes, after which the leader opened one of the other two remaining boxes, which turned out to be empty. After this, the player could either insist on the original choice, or change it and choose the third box. In what case does his chance of winning increase? (There are three possible answers: both boxes are equal, it is better to keep the original choice, it is better to change it. Try to justify your answer.)
combinatorics
k^{1999} - k^{1998} = 2k + 2 (VI Soros Olympiad 1990-00 R1 9.1)

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5/27/2024
Prove that there is no natural number kk such that k1999−k1998=2k+2k^{1999} - k^{1998} = 2k + 2.
number theoryDiophantine equation
bus, cyclist, motorcyclist, pedestrian (VI Soros Olympiad 1990-00 R2 9.1)

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5/28/2024
A car and a motorcyclist left point AA in the direction of point BB at 1010 o'clock, and half an hour later a cyclist left point BB (in the direction of point A) and a pedestrian left (in the direction of point AA) The car met the cyclist at 1111 o'clock hour and half an hour later overtook the pedestrian, and the motorcyclist overtook the pedestrian at 12:3012:30 p.m. At what time did the motorcyclist and the cyclist meet? (Speeds and directions of movement of ALL participants)
algebra
compare radicals (VI Soros Olympiad 1990-00 R3 9.1)

Source:

5/28/2024
Which of the two numbers is bigger :
1997+21999+22001+2003\sqrt{1997}+2\sqrt{1999} + 2\sqrt{2001} + \sqrt{2003} or 21998+22000+220022\sqrt{1998} +2\sqrt{2000}+2\sqrt{2002} ?
algebraRadicals