MathDB

9.4

Part of I Soros Olympiad 1994-95 (Rus + Ukr)

Problems(3)

triangle construction, given G,H,A (I Soros Olympiad 1994-95 R1 9.4)

Source:

7/30/2021
Use a compass and a ruler to construct a triangle, given the intersection point of its median, the orthocenter, and one from the vertices.
geometryconstruction
2 body time and speed problem (I Soros Olympiad 1994-95 R2 9.4)

Source:

5/25/2024
Two bodies simultaneously emerge from points AA and BB towards each other. Every time each body finds itself at point AA or point BB, it changes the direction of its movement to the opposite (the absolute values of the velocities of both bodies are constant). A body that initially leaves point AA travels the path ABAB in 101101 seconds. A body that initially left point BB travels the same path ABAB in 201201 seconds. After 2⋅101⋅2012 \cdot 101 \cdot 201 seconds, each body will be at its starting point. How many times during this time will the first body overtake the second?
algebra
permuting 2 digits (I Soros Olympiad 1994-95 Ukraine R2 9.4)

Source:

6/6/2024
The natural numbers XX and YY are obtained from each other by permuting the digits. Prove that the sums of the digits of the numbers 5X5X and 5Y5Y coincide.
number theorysum of roots