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 Match Editorials
TCHS08 Online Round 1
Saturday, January 12, 2008

## Match summary

The 2008 TCHS tournament started with Round 1 with 191 competitors. This giving all the posibility to advance with a positive score, which most of the competitors were able to do. For high rated highschoolers the first round was a warm up round. With a pretty easy set of problems 77 of competitors sent solution to all 3 problems but only 51 ended up with corect solution to all 3 problems. Only after 5 minutes solutions started to be submitted, and after 15 minutes some competitors were done with all problems. After a succesful challenge Loner managed to be 1st, fallowed by Zuza on second and gurugan1 on third.

# The Problems

StudentEnrollment
Used as: Division One - Level One:
 Value 250 Submission Rate 183 / 189 (96.83%) Success Rate 178 / 183 (97.27%) High Score LittlePig for 248.49 points (2 mins 13 secs) Average Score 229.94 (for 178 correct submissions)

A pretty strightforward problem. Take each query in order, and check if the class is available in names, and check if there are any spaces left for that class left. Each time we find a class we decrement the number of spaces available for that class, so each element of spaces will still represent the number of free spaces left. An example of java solution follows:

```for (i = 0; i < queries.length; ++i) {
int available = -1;  // the number of spaces available in class queries[i]
for (j = 0; j < names.length; ++j)
if (names[j].equals(queries[i])) {available = spaces[j]; break;} //we found the class
if (available == -1) {ret[i] = "DOES NOT EXIST"; continue;}; //if available == -1 there is no class queries[i]
if (available == 0) {ret[i] = "NOT GOOD"; continue;}; // if available == 0 there is no more room in that class
spaces[j]--; //decrement the number of spaces available in class
ret[i] = "GOOD";
};
return ret;
```

TaliluluCoffee
Used as: Division One - Level Two:
 Value 500 Submission Rate 186 / 189 (98.41%) Success Rate 177 / 186 (95.16%) High Score mosrecki for 497.69 points (1 mins 56 secs) Average Score 443.20 (for 177 correct submissions)

Most of the coders used the intuitive greedy solution. It was pretty easy to see by intuition that if you serve the clients in tips orders, from the largest tip to the lowest one, you would get the right answer. Let's prove this solution is really optimal. Each second we pick a customer, take all coins from him, and our rival takes one coin from every other customer which is still present. If we minimize the number of coins our rival gets, we maximize the number of coins we get. The number of coins our rival takes each turn is equal to the number of customers with at least one coin. If we always pick the customer with the maximal amount of coins available, we maximize the number of customers without money after each second. So, our rival gets the smallest possible amount, and we get the biggest.

DecorationDay
Used as: Division One - Level Three:
 Value 1000 Submission Rate 77 / 189 (40.74%) Success Rate 51 / 77 (66.23%) High Score Zuza for 983.85 points (3 mins 39 secs) Average Score 686.43 (for 51 correct submissions)

For some this problem was a classic one, but for others it was something new. For those who had trouble with it or never saw problems like this, I recommend to take a look on Dynamic Programming Tutorial from TopCoder and practice as much as they can, TCHS final will be soon.

Consider the table DP[i][X] with i = 1, N and X = 0, 100000, where N is the number of groups from the problem. DP[i][X] has the meaning what is the number of subsets that are made using only first i groups and such that the greatest common dividor (GCD) of each of those subsets is equal to X. Easy to see that the answer for the problem will be equal to DP[N][1]. If at some point we know DP[i][X], we can go further and count the number of subsets built using the (i + 1)-st group as well. In details, since each subset built using first i groups can be also built using first i+1 groups, we need to add DP[i][X] to element DP[i + 1][X]. If we want to use the (i+1)-st element, the GCD of the new group will be equal to GCD(X, groups[i + 1]), so we need to add DP[i][X] to DP[i + 1][GCD(X, groups[i + 1])] as well.

A simple java solution follows. The reference solution used longs instead of ints, and was taking the modulo at the very end.

```    long ret = 0;
int n, i, j;
n = groups.length;
long [][] DP = new long[n+3][100002];
for (i = 1; i <= n; ++i) {
DP[i][groups[i-1]]++;
for (j = 1; j <= 100000; ++j) if (DP[i-1][j] != 0) DP[i][gcd(j, groups[i-1])]+=DP[i-1][j];