Given $51$ natural numbers whose sum is $100$.
Show that it is possible to split them to $2$ sets such that for each the sum is $50$.
Also, another interesting question is, what happens instead of natural numbers we have integers?
Note: Assumption here that $0$ doesn't count as a natural number.
Claim: If $a_1,\cdots,a_{m+1}$ are natural numbers that add up to $2m$ then we can find a subset that adds up to $m.$
Proof: By induction.
If $m=1$ then we have two natural numbers, $a_1,a_2$ that add up to $2.$ But that can only be $a_1=a_2=1$, and then $a_1=1$ is the subset.
Assume true for $m-1,$ with $m>1.$ Given $a_1,\cdots,a_{m+1}$ natural numbers that add up to $2m,$ we will reduce the question to a case of $m-1.$
We know some $a_i=1,$ since otherwise, $a_i\geq 2$ for all $i,$ and the sum is at least $2(m+1)>2m.$
We also know that some $a_j>1,$ since otherwise, all the values are $1$ and the sum is $m+1<2m,$ since $m>1.$
If $a_i=1$ and $a_j>1,$ then we can remove $a_i$ and replace $a_j$ with $a_j-1.$ This gives us $m=m-1+1$ natural numbers adding up to $2(m-1).$ So, by the induction proposition, it must have a subset adding up to $m-1.$ But then, replacing $a_j-1$ with $a_j$ if it is in the subset, we have a subset adding up to $m-1$ or $m.$ If $m-1,$ we add the $1.$
Your request is the case $m=50.$
We can find $a_1=a_2=\cdots=a_{m-1}=1$ and $a_m=m+1$ to get $m$ numbers that add up to $2m$ without a subset adding to $m.$ If $m$ is odd, you could also choose all $a_i=2$ and not get a sum of $m.$
We can slightly improve this algorithm for larger $a_j.$
Assume $a_j>1.$ Let $u$ be the number of values $i$ with $a_i=1.$ Then:
$$2m=\sum_{i=1}^{m+1} a_i \geq a_j + u + 2(m-u)=a_j+2m-u$$
So $u\geq a_j.$ That means we can remove $a_j-1$ values $a_i=1$ and replace $a_j$ with $1.$ Then you have $m-(a_j-1)+1$ natural numbers adding up to $2m-2(a_j-1).$ You can find a subset of these numbers that add up to $m-(a_j-1).$ If the subset does not contain the index $j,$ take the complement to get a subset containing $j.$ Then that subset works for $m,$ too.
We will show that the above holds if even we allow there are at least 51 positive integers that sum to 100.
We first observe the following:
Claim 1: Let $K$ be the largest number. Then of the 51 $+m$ natural numbers; $m$ a nonegative integer; that sum to 100, there must be at least $K$ 1s.
Indeed, let $J$ be the number of 1s in the remaining 50 integers. Then the remaining $50+m-J$ numbers must be at least 2 and sum to no more than $100-K-J$. Thus we observe the inequality $2(50+m-J) \le 100-K-J$ which gives $J \ge K$ for all such $m$. So Claim 1 follows. $\surd$
Then Claim 2 follows almost immediately from Claim 1
Claim 2: The largest number can be no larger than 50.
Indeed, if the largest number is $Y \ge 51$ then By Claim 1 there must be at least $Y-1$ 1s. But then $Y$ plus the $Y-1$ 1s sums to over 100 for all such $Y$. $\surd$
So now sort the numbers from largest to smallest, and let $\ell$ be the largest integer such that the the $\ell$ largest integers in the list sum to something no greater than 50 i.e., the sum is $50-L$ for some nonnegative $L$. Then Claim 2 implies that $\ell>0$. Then if $L=0$ we are done, so we may assume that $L>0$.
We may also assume that the $\ell$ largest integers do not include any 1s, otherwise all the remaining $50+L$ numbers in the list would be 1 and so add $L$ of these 1s to the $\ell$ largest numbers to get a list that sums to 50 exactly. Thus we may assume that there are no 1s in the $\ell$ largest numbers. However, note that $L \le K$ and that by Claim 1 there are at least $K \ge L$ 1s, and as we just observed, none of these $\ge K$ 1s are in the first $\ell$ numbers. So add in $L$ of the ones to the $\ell$ largest numbers and they will sum to exactly 50.
Try a greedy approach: Let $a_1\ge a_2\ge\ldots \ge a_{51}\ge 1$ be the given natural numbers and assume that it is not possible to find a subsequence that sums up to $50$. Note that $a_1\le 50$ because already $$51+\underbrace{1+1+\cdots+1}_{50}>100.$$ If $a_1=50$, we are done, hence $a_1\le 49.$ As already $$\underbrace{2+2+\cdots+2}_{51}>100,$$ we see that $a_{51}=1$. Let $r$ be the numbre of summands that are $=1$. As just seen $r\ge1$. Let $m$ be maximal with $$\tag1\sum_{i=1}^{m} a_i\le 49.$$ From the above, $1\le m\le 50$. Then $\sum_{i=1}^{m+1} a_i\ge 51$. In particular, $a_{m+1}\ge2$. It follows that all numbers $=1$ are among $a_{m+2},\ldots, a_{51}$. Hence $$ 49\ge \sum_{i=m+2}^{51}a_i\ge (50-m)\cdot 2-r$$ or equivalently $$\tag2r\ge51-2m .$$ If $\sum_{i=1}^{m} a_i\ge 50-r$, we can fill up with summands that are $=1$ to reach a sum of $50$. Hence $\sum_{i=1}^{m} a_i\le 49-r$. We conclude that $a_{m+1}\ge r+2$. Then each summand in $(1)$ is $\ge r+2$ and hence $$\tag3m\le \left\lfloor\frac{49}{r+2}\right\rfloor.$$ Knowing that $r\ge1$, $(3)$ gives us $m\le 16$. Then $(2)$ gives us $r\ge19$, then $(3)$ gives $m\le 2$, then $(2)$ gives $r\ge47$, then $(3)$ gives $m\le 1$, in the next round we get $r\ge49$ and finally $m\le 0$, contradicting $m\ge1$.
Here's another variation, I'll make use of the statement Mike proved: if $K$ is the largest of the numbers and $J$ is the number of $1$'s, then $J\geq K$: the sum of all numbers is at least as big as $K + J + 2(50-J)$, so $K-J+100\leq 100$.
Suppose now that we have some numbers $a_1,a_2,...,a_{51}$ for which the statement "there exists an appropriate split" holds. Thus we can split the $a_i$ into sets $X$ and $Y$ such that the sums over elements of $X$ and $Y$ are $50$. We can construct another group of numbers by adding $1$ to one number and subtracting $1$ from another: $a_1,...,a_i+1,...,a_j-1,...,a_{51}$. Suppose $a_i$ and $a_j$ both belong to either $X$ or $Y$. Then the old split is still valid. Otherwise WLOG suppose $a_i\in X$ and $a_j\in Y$. Then the sum over $X$ will be $51$ and the sum over $Y$ will be $49$. Suppose there is at least one $1$ in $X$. We can move this $1$ to $Y$ to obtain a valid split. If not, all $J$ $1$'s are in $Y$. Pick any element $b\in X$ and move it to $Y$. Since $b\leq K\leq J$, we can move enough $1$'s from $Y$ to $X$ to obtain a valid split.
In all cases, we've found a valid split so the statement holds for this new group of numbers as well.
The statement obviously holds for the solution $1,1,...,1,50$. All other solutions are constructible from this one by the above increment/decrement procedure. Thus the statement holds for all solutions.
