MathDB

10.4

Part of II Soros Olympiad 1995 - 96 (Russia)

Problems(3)

x^2+ [y]=10, y^2+[x]=13 (II Soros Olympiad 1995-96 R1 10.4)

Source:

6/3/2024
Solve the system of equations {x2+[y]=10y2+[x]=13\begin{cases} x^2+ [y]=10 \\ y^2+[x]=13 \end{cases} ([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).
system of equationsalgebrafloor functionnice
common tangent circle, parabola (II Soros Olympiad 1995-96 R2 10.4)

Source:

6/3/2024
Find the equation of the line tangent to the parabola y=1/3(x2−2x+4)y = 1/3(x^2-2x+4) and a circle of unit radius centered at the origin. (List all solutions.)
geometryanalytic geometryparabolaconics
2x2 system x+(x-y)/(x^2+y^2)=2 (II Soros Olympiad 1997-98 R3 10.4)

Source:

6/6/2024
Solve system of equations {x+x+yx2+y2=1x+x−yx2+y2=2\begin{cases} x+\dfrac{x+y}{x^2+y^2}=1 \\ x+\dfrac{x-y}{x^2+y^2}=2 \end{cases}
algebrasystem of equations