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The numbers $$1$$, $$5$$, $$8$$, $$9$$, $$10$$, $$12$$, and $$15$$ are divided into groups of one or more numbers. The sum of the numbers in each group is the same. What is the largest possible number of groups? (2016 Math Kangaroo Problem, Level 1-2, Question \#24) 
Overseas Competition->Knowledge Point->Combinatorics->Combinatorics Involving Extreme Values->Extreme Value with Fixed Sums
$1+5+8+9+10+12+15=60$  $2$ groups: sum of each group is $30$, $30=15+5+10=1+8+9+12$  $3$ groups: sum of each group is $20$, $20=15+5=12+8=10+9+1$  $4$ groups: sum of each group is $15$, $15-12=3$, but there is no $3$ in given numbers  Thus, the largest possible number of groups is $3$. 