B3. Veronika has a sheet of graph paper with $78 \times 78$ squares. She wants to cut the sheet into smaller pieces, each of which will have either 14 or 15 squares, with each cut dividing one piece of paper into two along one of the lines on the paper. What is the minimum number of times Veronika must cut the paper?

## Problems for 3rd Grade

Time for solving: 180 minutes. Each problem in set A has exactly one correct answer. In set A, we will award two points for a correct answer and deduct one point for an incorrect answer. Write the answers for set A in the left table, leave the right table blank. The committee will consider only the answers written in the table for set A.
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B3. With each cut, the number of paper pieces increases by 1. At the end, the number of paper pieces will therefore be 1 greater than the number of cuts Veronika has made. To perform as few cuts as possible, she must end up with as few paper pieces as possible. Suppose she ends up with $k$ pieces of 14 squares and $n$ pieces of 15 squares, for a total of $n+k$ pieces of paper. Then it must be true that $14 k+15 n=78^{2}$. We can rewrite this equation as $15(n+k)=78^{2}+k$ and express $n+k=\frac{78^{2}+k}{15}$. From this, it follows that the number $78^{2}+k$ must be divisible by 15. To minimize $n+k$, $k$ must be as small as possible. Since the number $78^{2}=6084$ has a remainder of 9 when divided by 15, $k \geq 6$. When $k=6$, $n+k=406$, so Veronika must cut the paper at least 405 times.

Let's verify that Veronika can indeed cut the paper into pieces of 14 or 15 squares with 405 cuts. Veronika first cuts the sheet into 5 strips of size $15 \times 78$ with 5 cuts. Each of these strips is then cut into pieces of size $15 \times 1$ with 77 cuts. This leaves her with a piece of size $3 \times 78$. With 10 cuts, she cuts 10 pieces of size $3 \times 5$ from this piece, leaving a piece of size $3 \times 28$. With 1 cut, she cuts this piece into 2 pieces of size $3 \times 14$, and finally, each of these pieces is cut into pieces of size $1 \times 14$ with 2 cuts. Thus, all the resulting pieces have 14 or 15 squares, and it took $5+5 \cdot 77+10+1+2 \cdot 2=405$ cuts.

The equation $14 k+15 n=78^{2} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

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Justification that $k$ must be as small as possible, or $n$ as large as possible, and estimation of $n$ National Competition, May 15, 2021  Solutions for 3rd grade problems

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