MathDB

4

Part of 2003 Moldova Team Selection Test

Problems(3)

Infinitely many solutions (a,b,c)

Source: Moldova IMO-BMO TST 2003, day 1, problem 4

8/15/2008
Prove that the equation \frac {1}{a} \plus{} \frac {1}{b} \plus{} \frac {1}{c}\plus{}\frac{1}{abc} \equal{} \frac {12}{a \plus{} b \plus{} c} has infinitely many solutions (a,b,c) (a,b,c) in natural numbers.
Minimum number of coins on a chesstable

Source: Moldova IMO-BMO TST 2003, day 2, problem 4

8/16/2008
On the fields of a chesstable of dimensions n×n n\times n, where n≥4 n\geq 4 is a natural number, are being put coins. We shall consider a diagonal of table each diagonal formed by at least 2 2 fields. What is the minimum number of coins put on the table, s.t. on each column, row and diagonal there is at least one coin? Explain your answer.
Balanced table containing the numbers 1,2,...,n^2

Source: Moldova IMO-BMO TST 2003, day 3, problem 4

8/16/2008
A square-table of dimensions n×n n\times n, where n∈N∗ n\in N^*, is filled arbitrarly with the numbers 1,2,...,n2 1,2,...,n^2 such that every number appears on the table exactly one time. From each row of the table is chosen the least number and then denote by x x the biggest number from the numbers chosen. From each column of the table is chosen the least number and then denote by y y the biggest number from the numbers chosen. The table is called balanced iff x \equal{} y. How many balanced tables we can obtain?