Let $a_1, a_2, ... , a_8$ be positive integers. It was discovered that sum of any 7 numbers from this list can only yield $56, 58, 61, 63, 64, 65, $ or $66$. What is the largest number on this list?
From a hunch, I think the given isn't enough, since the sum of any 7 numbers from 8 integers would be as many as $\dbinom{8}{7} = 8$ sums, and only $7$ sums given.
I gave the benefit of the doubt and assume that one of the sums is not unique. However, even if I tried the largest sum given, $66$, as the sum that is not unique, then the sum of $a_1, a_2, ... , a_8$ would be $$\dfrac{56+58+61+63+64+65+66+66}{7} = \dfrac{499}{7} = 71.2857143$$ (See EDIT for further clarification)
and applying to the largest 7-number sum to get the smallest integer would be $$71.2857143 - 66 = 5.2857143$$
... which contradicts the word "integer".
However, it actually has an answer, as the answer sheet said that it's $15$.
Did I do wrong on some point, or the problem is utterly bad?
EDIT : To further clarify, I try to expand my work a bit
$$a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7 = 56$$ $$a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_8 = 58$$ $$a_1 + a_2 + a_3 + a_4 + a_5 + a_7 + a_8 = 61$$ $$a_1 + a_2 + a_3 + a_4 + a_6 + a_7 + a_8 = 63$$ $$a_1 + a_2 + a_3 + a_5 + a_6 + a_7 + a_8 = 64$$ $$a_1 + a_2 + a_4 + a_5 + a_6 + a_7 + a_8 = 65$$ $$a_1 + a_3 + a_4 + a_5 + a_6 + a_7 + a_8 = 66$$ $$a_2 + a_3 + a_4 + a_5 + a_6 + a_7 + a_8 = 66$$
Adding all of those would be $$7 a_1 + 7 a_2 + 7 a_3 + 7 a_4 + 7 a_5 + 7 a_6 + 7 a_7 + 7 a_8= 499$$
Dividing by 7 would be $$a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7 + a_8= 71.2857143$$
I'm not sure why you're dividing by $8$, but I think you may have been at least on the right track. If one sums $7$ of the $8$ numbers, one is left out. If the duplicate sum is $x$, then $x+433$ must equal $7$ times the sum of all $8$ numbers. Using modular arithmetic, we have $433\equiv6\pmod7$, so the duplicate sum must be equivalent to $1\pmod7$. The only one that works is $64$. So the sum of all $8$ numbers is $\frac{433+64}7=71$. So our largest number must be $71-56=15$.