MathDB

11.2

Part of V Soros Olympiad 1998 - 99 (Russia)

Problems(3)

max C: (x + y +z + u)^2 >= Cyz (V Soros Olympiad 1998-99 Round 1 11.2)

Source:

5/25/2024
Find the greatest value of CC for which, for any x,y,z,ux, y, z,u, and such that for 0≤x≤y≤z≤u0\le x\le y \le z\le u, holds the inequality (x+y+z+u)2≥Cyz.(x + y +z + u)^2 \ge Cyz .
algebrainequalities
2 cyclists and 1 fly (V Soros Olympiad 1998-99 Round 2 11.2)

Source:

5/25/2024
From the two cities of Dobruisk and Bodruisk, the distance between which is 4040 km, two cyclists Dobi and Bodi simultaneously rode towards each other. Dobie was traveling at 2323 km/h and Bodie was traveling at 1717 km/h. Before departure, a fly landed on Dobie’s nose, which, at the moment of his departure from the city, flew towards Bodruisk at a speed of 4040 km/h. The fly met Bodie, immediately turned back and flew towards Dobruisk at a speed of 3030 km/h. (The fact is that the wind was blowing from Dobruisk to Bodruisk.) Having met Doby, the fly turned back again, etc. Determine the total path that the fly flew until the moment the cyclists met. (The speed of the fly in each direction did not change.)
algebra
triangular pyramid (V Soros Olympiad 1998-99 Round 3 11.2)

Source:

5/26/2024
Five edges of a triangular pyramid are equal to 11. Find the sixth edge if it is known that the radius of the ball circumscribed about this pyramid is equal to 11.
geometry3D geometrypyramid