MathDB

1

Part of 2005 ITAMO

Problems(2)

find all possible values of AH/CH

Source: ItaMO 2005, P1

3/9/2012
Let ABCABC be a right angled triangle with hypotenuse ACAC, and let HH be the foot of the altitude from BB to ACAC. Knowing that there is a right-angled triangle with side-lengths AB,BC,BHAB, BC, BH, determine all the possible values ​​of AHCH\frac{AH}{CH}
geometry proposedgeometry
find the greatest n

Source: ItaMO 2005, P4

3/9/2012
Determine all n≥3n \geq 3 for which there are nn positive integers a1,⋯ ,ana_1, \cdots , a_n any two of which have a common divisor greater than 11, but any three of which are coprime. Assuming that, moreover, the numbers aia_i are less than 50005000, find the greatest possible nn.
number theory proposednumber theory