Difference between revisions of "1971 IMO Problems/Problem 6"

Revision as of 13:58, 6 March 2025

Problem

Let $A = (a_{ij})(i, j = 1, 2, \cdots, n)$ be a square matrix whose elements are non-negative integers. Suppose that whenever an element $a_{ij} = 0$ , the sum of the elements in the $i$ th row and the $j$ th column is $\geq n$ . Prove that the sum of all the elements of the matrix is $\geq n^2 / 2$ .

Solution 1

Take the the row or column (without loss of generality, a row) with the minimum possible sum $S$ of its elements. Let $z$ be the number of zeros in this row. We will assume $S < \frac{n}{2}$ once the other case is obvious. Clearly $S \ge n - z \Rightarrow z \ge n - S$ . The total sum $T$ of all elements of the matrix is at least the number of zeros in this row multiplicated by $n - S$ (because the sum of the elements of a row and a column meeting in a zero is $\ge n$ ) plus the number of nonzero elements times $S$ (that is the minimum possible sum), it is, $z(n - S) + (n - z)S.$ Note that, being $z \ge n - S$ and $n - S \ge S$ , we can put $z - 1$ instead of $z$ and $n - z + 1$ instead of $n - z$ until we get $z = n - S$ , what makes the sum become smaller. So we have $T \ge (n - S)^2 + S^2 \ge 2(\frac{n - S + S}{2})^2 = \frac{n^2}{2}$ by AM - QM inequality.

The above solution was posted and copyrighted by Renan. The original thread for this problem can be found here: [1]

Solution 2

Denote $\ell_i,c_i$ the $i$ th line and column and $s(\ell_i), s(c_i)$ the sum of the elements on that row. Let $M=\max_{\sigma \in S_n} | \{ 1\le i\le n| a_{i\sigma(i)}=0\} |$ . For each permutation $\sigma\in S_n$ , assign to it the number $\sum\limits_{a_{i\sigma(i)=0}} \left ( s(\ell_i)+s(c_{\sigma(i)}) \right )$ Now pick a permutation $\sigma$ such that it generates $M$ and its assigned number is minimal. Suppose wlog that $\{ 1\le i\le n| a_{i\sigma(i)}=0\}=\{1,2,..,k\}$ . If $k=n$ , we are done as twice the sum of all elements is $\sum\limits_{i=1}^{n} s(\ell_i) +\sum\limits_{i=1}^{n} s(c_i)=\sum\limits_{i=1}^{n} (s(\ell_i)+s(c_{\sigma(i)}))\ge n^2$ So suppose $k<n$ and look at a $j$ such that $k+1\le j\le n$ . By the maximality of $M$ , $\ell_j$ can have $0$ only on the columns $\sigma(1),...,\sigma(k)$ . Note that if $\ell_j$ has a $0$ on $c_{\sigma(r)}$ with $r\le k$ , i.e. $a_{j\sigma(r)}=0$ , by the minimality of the assigned number of $\sigma$ , we have that $s(\ell_j)\ge s(\ell_r)$ . However, if the column $c_{\sigma(j)}$ has a $0$ on $\ell_r$ , i.e. $a_{r\sigma(j)}=0$ , by the same argument we have $s(c_\sigma(j))\ge s(c_\sigma(r))$ , hence $s(\ell_j)+s(c_{\sigma(j)})\ge s(r)+s(c_{\sigma(r)})\ge n$ .

Suppose that $0$ appears exactly $t$ times on $\ell_j$ . This means that $s(\ell_j)\ge n-t$ . Suppose that the $0$ s of $\ell_j$ appear on the columns $c_{\sigma(i_1)},...,c_{\sigma(i_t)}$ . We have seen earlier that if $c_\sigma(j)$ has a $0$ on $i_1,..,i_t$ we have that $s(\ell_j)+s(c_{\sigma(j)})\ge n$ . So suppose that it doesn't have $0$ on these lines. However, $c_{\sigma(j)}$ has $0$ s only on the lines $\ell_1,..,\ell_k$ by the maximality of $M$ , hence $c_{\sigma(j)}$ has at most $k-t$ zeroes, so $s(c_{\sigma(j)})\ge n-k+t$ . We thus infer that $s(\ell_j)+s(c_{\sigma(j)})\ge (n-t)+(n-k+t)=2n-k\ge n$ so $s(\ell_j)+s(c_{\sigma(j)})\ge n,\ \forall j\ge k+1$ . By the hypothesis this is also true for $j\le k$ so summing over all values of $j$ we get that the sum is at least $\dfrac{n^2}{2}$ .

The above solution was posted and copyrighted by Aiscrim. The original thread for this problem can be found here: [2]

Solution 3

If the main diagonal contains all zeroes, we can immediately deduce from the condition that the sum $S$ of the elements is at least $n^2/2$ . This motivates us to consider the largest possible subset of elements, all zero, which are pairwise in different rows of columns - let the size of this subset be $k$ . (If there is no such subset, $S\ge n^2$ .)

Since the condition is invariant under the operation of switching rows or columns, we may rearrange the matrix such that these $k$ entries are on the main diagonal of the matrix, i.e. $a_{11}=a_{22}=\dots=a_{kk}=0$ . Let $X=\sum_{1\le i,j\le k}a_{ij}$ , $Y=\sum_{i\le k<j}a_{ij}+\sum_{j\le k<i}a_{ij}$ and $Z=\sum_{k<i,j\le n}a_{ij}$ . Then applying the condition to $a_{11},\dots,a_{kk}$ , we get $2X+Y\ge kn$ .

If $i\le k<j$ , then if $a_{ij}=a_{ji}=0$ , then we may toss $a_{ii}$ out of our set and replace it with $a_{ij}$ and $a_{ji}$ , to form a larger feasible set of elements, contradiction. Consequently, $a_{ij}+a_{ji}\ge 1$ for $i\le k<j$ , which upon summation yields $Y\ge k(n-k)$ . Finally, $Z\ge (n-k)^2$ , trivially. Therefore, $2S=2(X+Y+Z)\ge 2X+Y+Y+Z\ge kn+k(n-k)+(n-k)^2=k^2+2k(n-k)+(n-k)^2=n^2.$

The above solution was posted and copyrighted by randomusername. The original thread for this problem can be found here: [3]

Remarks (added by pf02, March 2025)

Each of the three solutions above is incorrect.

Solution 1 is plainly wrong, but it can be salvaged.

Solution 2 is hopelessly incorrect. An attempt to salvage it would be along the lines of Solution 3, in the context of the unnecessarily complicated setup of Solution 2.

Solution 3 is based on a very good idea, and it is easy to make the necessary corrections to make it robust.

The discussion page https://artofproblemsolving.com/community/c6h60831s1_sum_of_all_the_elements_in_the_matrix_is_at_least_n22 contains a fourth solution, by Saucepan_man02, which I will add below.

Below I will point out the errors in the three solutions given above, and when I can, the corrected solutions. Then I will give some examples. The examples will show that the inequality $\sum_{i, j = 1}^n a_{ij} \ge \frac{n^2}{2}$ is optimal.

Solution 1 analysis and correction

The error in Solution 1

I will use the notation of the solution. At some point in the solution the author writes:

"Note that, being $z \ge n - S$ and $n - S \ge S$ , we can put $z - 1$ instead of $z$ and $n - z + 1$ instead of $n - z$ until we get $z = n - S$ "

(To put this in context: we chose a row or column with minimal sum of its elements (we can assume it was a row), $S$ was the sum of the elements in this row, and $z$ was the number of zeros in it.)

This is wrong! If $z$ was chosen to represent a chosen quantity, we can not change its value in the middle of the proof.

We could try to make sense of the proof by saying that we replace the given matrix A by a matrix B, where the rows of B are equal to the rows of A, except for the "minimal" row we chose, in which we replace one of the zeros by a number $> 0$ (which would replace $z$ by $z - 1$ ). This does not work, because if we would show that the sum of the elements of $B$ is $\ge \frac{n^2}{2}$ , it would not follow that the same is true for $A$ .

Solution 1 corrected

As in the solution, choose a row or column such that the sum of its elements is minimal. We can assume this is a row (either notice that taking the transpose of $A$ changes neither the hypothesis, nor the conclusion; or notice that if we do the proof in the case of a row, the proof in the case of a column is analogous.) Let $S$ be the sum of the elements of this row, and $z$ be the number of zeros in this row.

For an element $a_{ij}$ of $A$ , we denote $r_i, c_j$ the row and the column containing $a_{ij}$ . We denote $sum(r_i), sum(c_j)$ the sum of the entries of the matrix in row $r_i$ and $c_j$ respectively.

If $S \ge \frac{n}{2}$ , then each row $r_i$ has its sum of elements $sum(r_i) \ge S \ge \frac{n}{2}$ , so $\sum_{i, j = 1}^n a_{ij} = \sum_{i=1}^n sum(r_i) \ge \frac{n^2}{2}$ .

So, let us assume $S < \frac{n}{2}$ . Let $r_i = \{a_{ij}\}$ be the row we chose (now $i$ is fixed). For each $j$ consider the sum of the elements of the column $c_j$ containing $a_{ij}$ . If $a_{ij} = 0$ , we know that $sum(r_i) + sum(c_j) \ge n$ (by hypothesis), so $S + sum(c_j) \ge n$ , so $sum(c_j) \ge n - S$ . We have $z$ such columns.

If $a_{ij} \ne 0$ , we know that $sum(c_j) \ge S$ because $S$ was minimal. We have $n - z$ such columns.

So, $\sum_{i, j = 1}^n a_{ij} = \sum_{j = 1}^n sum(c_j) \ge z(n - S) + (n - z)S$ .

Let $S = n - z + E$ . Here we split $S$ in two parts: $n - z$ represents the number of non-zeroes in $r_i$ , and $E$ represents the sum of the "excess" amounts over $1$ of the non-zero entries. More explicitly, $E = \sum_{\substack{j = 1 \\ a_{ij} \ne 0}}^n (a_{ij} - 1)$ . (To clarify: if all non-zero entries of $r_i$ would $= 1$ , we would have $S = n - z$ , and $E = 0$ .) In general, $E \ge 0$ .

Then, $\sum_{i, j = 1}^n a_{ij} \ge z(n - S) + (n - z)S = (n - S + E)(n - S) + (S - E)S = n^2 - 2nS + 2S^2 + nE - 2SE =$

$\frac{n^2}{2} + \frac{1}{2}\left(n^2 - 4nS + 4S^2\right) + E(n - 2S) = \frac{n^2}{2} + \frac{1}{2}(n - 2S)^2 + E(n - 2S)$ .

The last term is $\ge 0$ since $E \ge 0$ and we assumed we are in the case $S < \frac{n}{2}$ . It follows that $\sum_{i, j = 1}^n a_{ij} > \frac{n^2}{2}$ .

Note: The strict inequality is not true in general. We obtained it assuming that $S < \frac{n}{2}$ . When $S \ge \frac{n}{2}$ we have $\sum_{i, j = 1}^n a_{ij} \ge \frac{n^2}{2}$ .

TO BE CONTINUED. SAVING MID-WAY SO I DON'T LOSE TEXT WRITTEN SO FAR.

Revision as of 13:56, 6 March 2025 (view source) Pf02 (talk \| contribs) ← Older edit		Revision as of 13:58, 6 March 2025 (view source) Pf02 (talk \| contribs) m Newer edit →
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−	==Solution 1 analysis==	+	==Solution 1 analysis and correction==

	===The error in Solution 1===		===The error in Solution 1===

1971 IMO (Problems) • Resources
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