Question 93
Consider five positive real numbers sorted by $0\lt a \leq b \leq c \leq d \leq e$. It is known that whenever we take one of these numbers, we can separate the other four into two groups such that the sum of the numbers of one group is equal to the sum of the numbers of the other group. If a $(a, b, c, d, e)$ sequence satisfies this condition, we say that it is quasi-balanced. There are sequences that meet a more restricted condition: if we remove a number we can separate the four remaining numbers into two groups with the same number of numbers such that the sum of the numbers of one group is equal to the sum of the numbers of the other group. If a sequence of positive real numbers $(a, b, c, d, e)$ satisfies this more restricted condition, we say that this sequence is balanced.
a) Show an example of five positive real numbers ordered by $0\lt a \leq b \leq c \leq d \leq e$, with $a\lt e$, which form an almost-balanced sequence. See that we can make some of them the same if this is convenient.
b) If a balanced sequence has three equal terms, show that the five numbers are necessarily equal.
c) Consider a balanced sequence $(a, b, c, d, e)$ with $0\lt a \leq b \leq c \leq d \leq e$. It is also known that $(e+c)=(b+d)$ and that $(e+a)=(c+d)$. Prove that the five numbers are the same.
a) Show an example of five positive real numbers ordered by $0\lt a \leq b \leq c \leq d \leq e$, with $a\lt e$, which form an almost-balanced sequence. See that we can make some of them the same if this is convenient.
b) If a balanced sequence has three equal terms, show that the five numbers are necessarily equal.
c) Consider a balanced sequence $(a, b, c, d, e)$ with $0\lt a \leq b \leq c \leq d \leq e$. It is also known that $(e+c)=(b+d)$ and that $(e+a)=(c+d)$. Prove that the five numbers are the same.