MathDB

Problem 1

Part of 2005 Slovenia National Olympiad

Problems(4)

xyz=1, evaluate expression

Source: Slovenia National MO 2005 1st Grade P1

4/5/2021
If x,y,zx,y,z are real numbers such that xyz=1xyz=1, evaluate x+1xy+x+1+y+1yz+y+1+z+1zx+z+1.\frac{x+1}{xy+x+1}+\frac{y+1}{yz+y+1}+\frac{z+1}{zx+z+1}.
algebra
Sum of floor of log function

Source: Slovenia National MO 2005 3rd Grade P1

4/5/2021
Evaluate the sum ⌊log⁡21⌋+⌊log⁡22⌋+⌊log⁡23⌋+…+⌊log⁡2256⌋\left\lfloor\log_21\right\rfloor+\left\lfloor\log_22\right\rfloor+\left\lfloor\log_23\right\rfloor+\ldots+\left\lfloor\log_2256\right\rfloor.
floor functionfunction
x^3-y^3=7(x-y) and x^3+y^3=5(x+y)

Source: Slovenia National MO 2005 2nd Grade P1

4/5/2021
Find all real numbers x,yx,y such that x3−y3=7(x−y)x^3-y^3=7(x-y) and x3+y3=5(x+y)x^3+y^3=5(x+y).
algebrasystem of equations
floor equation

Source: Slovenia National MO 2005 4th Grade P1

4/5/2021
Find all positive numbers xx such that 20{x}+0.5⌊x⌋=200520\{x\}+0.5\lfloor x\rfloor = 2005.
algebrafloor function