Increasing CP5 Lower Bound (Day 5)


2020-12-13 18:00 AKST

Increasing CP5 Lower Bound (Day 5)


2020-12-13 23:00 AKST
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Problem B
Pyramid Construction

A finished and an unfinished pyramid of height $4$.

Kim likes building Lego pyramids and now wants to build one of height $H$. How many more bricks are needed if Kim already has $N$ bricks of size $2 \times 2$ and $M$ bricks of size $4 \times 2$? A pyramid is hollow and consists of $H$ layers, as shown in the image. The $k$th layer has width $2k$ for $1\leq k\leq H$.


Three integers $1 \leq H \leq 1\, 000$, $0 \leq N \leq 10^6$, and $0 \leq M \leq 10^6$.


Print two integers $A$ and $B$ on a single line, where $A$ is the number of extra $2 \times 2$-bricks needed and $B$ is the number of extra $4 \times 2$-bricks. You must use as few extra bricks as possible, so you must minimise $A+B$. If there is more than one such solution, print the one maximising $A$.

Explanation of Sample 2

Sample input $2$ corresponds to the situation shown in the image. Starting with the bricks in the unfinished pyramid to the right, Kim can build the left pyramid using an extra $2 \times 2$-brick and $4$ extra $4 \times 2$-bricks. There is no way to finish a height-$4$ pyramid with fewer than $5$ extra bricks. Note that it is also possible to build a height-$4$ pyramid using $5$ extra bricks of size $4\times 2$ (and no extra $2\times 2$-bricks). However, the sample output is correct because it uses more extra $2 \times 2$-bricks.

Sample Input 1 Sample Output 1
1 1 0
0 0
Sample Input 2 Sample Output 2
4 2 7
1 4
Sample Input 3 Sample Output 3
3 0 0
1 6
Sample Input 4 Sample Output 4
1 0 5
1 0