MathDB

11.3

Part of II Soros Olympiad 1995 - 96 (Russia)

Problems(3)

[2 sinx] =2cos (3x+\pi/4) (II Soros Olympiad 1995-96 R1 11.3)

Source:

6/3/2024
Solve the equation [2sin⁡x]=2cos⁡(3x+π4)[2 \sin x] =2\cos \left(3x+\frac{\pi}{4} \right) ([x][x] is the integer part of xx, [x][x] is equal to the largest integer not exceeding xx. For example, [3,33]=3[3,33] = 3, [2]=2[2] = 2, [−3.01]=−4[- 3.01] = -4).
trigonometryfloor function
cards with numbers, 300 problems (II Soros Olympiad 1995-96 R2 11.3)

Source:

6/3/2024
The math problem book contains 300300 problems. The teacher has cards with numbers. She pins these cards to a special stand and indicates the numbers of four problems that need to be solved during the lesson. What is the smallest number of cards that a teacher can use in order to be able to indicate the numbers of any four problems from the problem book?
combinatorics
3x3 system sin \pi}/ 2 xy =z (II Soros Olympiad 1997-98 R3 11.3)

Source:

6/6/2024
Solve the system of equations {sin⁡π2xy=zsin⁡π2yz=xsin⁡π2zx=y   ?\begin{cases} \sin \frac{\pi}{2}xy =z \\ \sin \frac{\pi}{2}yz =x \\ \sin \frac{\pi}{2}zx =y \end{cases} \,\,\, ?
algebrasystem of equationstrigonometry