2015 UMO Problems
Problem 1
Three trolls have divided 
pancakes among themselves such that:
• Each troll has a positive integer number of pancakes.
• The greatest common divisor of the number of pancakes held by any two trolls is bigger than 
.
• The three greatest common divisors obtained in this way are all distinct.
What is the smallest possible value of ?
Problem 2
In 
and 
, and the length of the altitude from 
to
is 
. If 
is the midpoint of , find all possible length(s) of 
, and demonstrate that these
length(s) are achievable.
Problem 3
Find, with proof, all positive integers with 
such that the greatest common divisor of the coefficients of 
is equal to exactly 
.
Problem 4
Anastasia and Balthazar need to go to the grocery store, which is 
km away. Anastasia
walks at 
km/hr, and Balthazar walks at 
km/hr. However, they also own a single bike, and
each of them bikes at 
km/hr. They are allowed to go forwards or backwards, and the bike
will not get stolen if they drop it off along the way for the other person to pick up. What is
the shortest amount of time necessary for both of them to get to the grocery store?
Problem 5
A 
grid is filled with integers (positive or negative) such that the product of the integers
in any row or column is equal to 
. For example, one possible grid is:
In how many ways can this be done?
Problem 6
A triangular pyramid with apex 
and base 
has the property that the perimeter of 
is 
. Additionally, one can place a cylinder of radius and height completely inside the pyramid such that one of its bases is in the same plane as .
What is the minimum possible height from to apex ? Show that this height is achievable.
