### Real(ly) Numbers

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

### Overturning Fracsum

Solve the system of equations to find the values of x, y and z: xy/(x+y)=1/2, yz/(y+z)=1/3, zx/(z+x)=1/7

### Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

# LCM Sudoku II

### Rules of LCM Sudoku

Like the standard sudoku, this sudoku variant has the basic rule:
• Every row, every column and every $3\times 3$ box in the grids contains the digits $1$ through $9$.
The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times 9$ grid. At the bottom and right side of the $9\times 9$ grid are numbers, each of which is the Least Common Multiple (LCM) of a column or row of unknown digits marked by asterisks.

Altogether $18$ Least Common Multiples of $9$ columns and $9$ rows of unknown digits are given as clues for solving the puzzle.

As an example, the LCM of $3$, $9$, $6$, $4$, $2$ and $1$ is $36$. In the puzzle, the LCM of the four unknown digits in the second column beginning from the left of the $9\times 9$ grid is $36$, while the LCM of the four unknown digits in the seventh row beginning from the top of the $9\times 9$ grid is $120$.

After finding the values of all the unknown digits, the puzzle is solved by the usual sudoku technique and strategy.