作者： chenchen

2025年1月的比赛以算法编程问题为特色，涵盖了广泛的技术和难度水平。

在为期4天的比赛中，共有11565名不同的用户登录。来自100多个不同国家的9450名参与者提交了至少一个解决方案。4276名参与者来自美国，其中中国、加拿大、韩国、罗马尼亚、马来西亚、印度和新加坡的参与程度也很高。

总计有23508份已评级的提交材料，按语言细分如下：

14190 C++17
3324 Python-3.6.9
3069 C++11
2763 Java
127 C
35 Python-2.7.17

以下是白金、金牌、银牌和铜牌比赛的详细结果。您还可以找到每个问题的解决方案和测试数据。

USACO 2025年 1 月竞赛，白金奖

白金组共有 352 名参与者，其中 254 名是大学预科生。得分最高的成员的成绩在这里。祝贺所有最优秀的参赛者取得优异的成绩！

1 DFS Order
查看问题 | 测试数据 | 解决方案
2 Shock Wave
查看问题 | 测试数据 | 解决方案
3 Watering the Plants
查看问题 | 测试数据 | 解决方案

USACO 2025 年 1 月比赛，金奖

黄金组共有 1032 名参与者，其中 738 名是大学预科生。所有在本次比赛中获得 700 分或更高认证分数的参赛者将自动晋升到白金组。所有晋级者的详细结果将在我们完成学术诚信检查后很快公布。

1 Median Heap
查看问题 | 测试数据 | 解决方案
2 Reachable Pairs
查看问题 | 测试数据 | 解决方案
3 Photo Op
查看问题 | 测试数据 | 解决方案

USACO 2025 年 1 月比赛，银奖

白银组共有 4070 名参与者，其中 3072 名是大学预科生。所有在本次比赛中获得 700 分或更高分的参赛者将自动晋级金牌组。

1 Cow Checkups
查看问题 | 测试数据 | 解决方案
2 Farmer John's Favorite Operation
查看问题 | 测试数据 | 解决方案
3 Table Recovery
查看问题 | 测试数据 | 解决方案

USACO 2025 年 1 月比赛，铜奖

铜牌组共有 6735 名参与者，其中 5060 名是大学预科生。所有在本次比赛中获得 700 分或更高分的参赛者将自动晋升到银牌组。

1 Astral Superposition
查看问题 | 测试数据 | 解决方案
2 It's Mooin' Time II
查看问题 | 测试数据 | 解决方案
3 Cow Checkups
查看问题 | 测试数据 | 解决方案

结语

又是一场具有挑战性的比赛！从技术角度来看，1 月份的比赛进行得很顺利。我们看到了很高的参与度，有相当数量的参与者得到了晋升。

对于那些尚未晋升的人， 请记住，练习得越多，你的算法编码技能就会越强——请坚持下去！USACO比赛旨在挑战最优秀的学生，要想脱颖而出，需要付出大量努力。为了帮助你修复代码中的任何错误，你现在可以使用“分析模式”重新提交你的解决方案，并从评判服务器获取反馈。

众多人士为USACO竞赛的质量和成功做出了贡献。本次竞赛的协助人员包括 Alex Liang, Tina Wang, Chongtian Ma, Haokai Ma, Richard Qi, Thomas Liu, Claire Zhang, Larry Xing, Benjamin Chen, Weiming Zhou, Daniel Zhang, Eric Yang, Spencer Compton, William Lin, Danny Mittal, David Hu, Suhas Nagar, Nathan Wang, Brandon Wang, Michelle Wei, Nick Wu, Ho Tin Fan, Andi Qu, and Benjamin Qi，也感谢我们的翻译帮助我们扩大比赛的范围。最后，我们非常 感谢 USACO 赞助商的慷慨支持：Citadel， Ansatz、X-Camp、TwoSigma、VPlanet Coding、EasyFunCoding、Orijtech 和 Jump Trading。

我们期待在二月再次见到大家 比赛。

祝您编码愉快！

**Note: We suggest using a language other than Python to earn full credit on this problem.**

Farmer John's N (1≤N≤7500) cows are standing in a line, with cow 1 at the front of the line and cow N at the back of the line. FJ's cows also come in many different species. He denotes each species with an integer from 1 to N. The i'th cow from the front of the line is of species ai (1≤ a_i ≤N).

FJ is taking his cows to a checkup at a local bovine hospital. However, the bovine veterinarian is very picky and wants to perform a checkup on the i'th cow in the line, only if it is of species b_i (1≤b_i≤N).

FJ is lazy and does not want to completely reorder his cows. He will perform the following operation exactly once.

Select two integers l and r such that 1≤l≤r≤N. Reverse the order of the cows that are between the l-th cow and the r-th cow in the line, inclusive.

FJ wants to measure how effective this approach is. For each c=0…N, help FJ find the number of distinct operations (l,r) that result in exactly c cows being checked. Two operations (l₁,r₁) and (l₂,r₂) are different if l₁≠l₂ or r₁≠r₂.

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains an integer N.

The second line contains a₁,a₂,...,a_N.

The third line contains b₁,b₂,…,b_N.

OUTPUT FORMAT (print output to the terminal / stdout):

Output N+1 lines with the i-th line containing the number of distinct operations (l,r) that result in i−1 cows being checked.

SAMPLE INPUT:

3
1 3 2
3 2 1

SAMPLE OUTPUT:

3
3
0
0

If FJ chooses (l=1,r=1), (l=2,r=2), or (l=3,r=3) then no cows will be checked. Note that those operations do not modify any of the cows' locations.

The following operations result in one cow being checked.

l=1,r=2: FJ reverses the order of the first and second cows so the species of each cow in the new lineup will be [3,1,2]. The first cow will be checked.
l=2,r=3: FJ reverses the order of the second and third cows so the species of each cow in the new lineup will be [1,2,3]. The second cow will be checked.
l=1,r=3: FJ reverses the order of the first, second, and third cows so the species of each cow in the new lineup will be [2,3,1]. The third cow will be checked.

SAMPLE INPUT:

3
1 2 3
1 2 3

SAMPLE OUTPUT:

0
3
0
3

The three possible operations that cause 3 cows to be checked are (l=1,r=1),(l=2,r=2), and (l=3,r=3).

SAMPLE INPUT:

7
1 3 2 2 1 3 2
3 2 2 1 2 3 1

SAMPLE OUTPUT:

0
6
14
6
2
0
0
0

The two possible operations that cause 4 cows to be checked are (l=4,r=5) and (l=5,r=7).

SCORING:

Inputs 4-6: N≤100
Inputs 7-13: No additional constraints

Problem credits: Chongtian Ma and Haokai Ma

扫码领取USACO试题答案+详细解析

咨询一对一备赛规划

Farmer John is trying to describe his favorite USACO contest to Elsie, but she is having trouble understanding why he likes it so much. He says "My favorite part of the contest was when Bessie said 'It's Mooin' Time' and mooed all over the contest."

Elsie still doesn't understand, so Farmer John downloads the contest as a text file and tries to explain what he means. The contest is defined as an array of N
(1≤N≤10⁶) integers a₁,a₂,...,a_N (1≤ a_i ≤N). Farmer John defines a moo as an array of three integers where the second integer equals the third but not the first. A moo is said to occur in the contest if it is possible to remove integers from the array until only the moo remains.

As Bessie allegedly "mooed all over the contest", help Elsie count the number of distinct moos that occur in the contest! Two moos are distinct if they do not consist of the same integers in the same order.

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains N.

The second line contains N space-separated integers a₁,a₂,...,a_N .

OUTPUT FORMAT (print output to the terminal / stdout):

Output the number of distinct moos that occur in the contest.

**Note that the large size of integers involved in this problem may require the use of 64-bit integer data types (e.g., a "long" in Java, a "long long" in C/C++).**

SAMPLE INPUT:

6
1 2 3 4 4 4

SAMPLE OUTPUT:

3
This contest has three distinct moos: "1 4 4", "2 4 4", and "3 4 4".

SCORING:

• Inputs 2-4: N≤10²
• Inputs 5-7: N≤10⁴
• Inputs 8-11: No additional constraints.

Problem credits: Benjamin Qi

扫码领取USACO试题答案+详细解析

咨询一对一备赛规划

**Note: The time limit for this problem is 4s, twice the default.**

Bessie is using her nifty telescope to take photos of all the stars in the night sky. Her telescope can capture an N×N(1≤N≤1000) photo of the stars where each pixel is either a star or empty sky. Each star will be represented by exactly one pixel, and no two distinct stars share the same pixel.

Overnight, something strange happens to the stars in the sky. Every star either disappears or moves A pixels to the right, and B pixels downwards (0≤A,B≤N
). If a star disappears or moves beyond the photo boundary, it no longer appears in the second photo.

Bessie took photos before and after the shifting positions, but after experimenting in Mootoshop, she accidentally superimposed one photo onto the other. Now, she can see white pixels where both photos were empty, gray pixels where stars existed in exactly one photo, and black pixels where there was a star in both photos. Bessie also remembers that no new stars moved into the frame of the second photo, so her first photo contains all of the stars in the night sky.

Given the final photo, determine the minimum possible number of stars in the sky before the shifting incident for T (1≤T≤1000) independent test cases. If no arrangement of stars can produce the given final photo, output −1.

INPUT FORMAT (input arrives from the terminal / stdin):

The first line of input contains T and T test cases will follow.

The first line of each test case will contain N A B .

Then follow N lines each representing one row of the superimposed photo. The ith row from the top is represented by a string c_i,1c_i,2…c_i,N, where each c_i,j∈{W,G,B}, representing the colors white, gray, and black respectively.

It is guaranteed that the sum of N² over all test cases will not exceed 10⁷.

OUTPUT FORMAT (print output to the terminal / stdout):

For each test case, output the minimum number of stars that existed before the shifting, or −1 if impossible.

SAMPLE INPUT:

1
3 0 0
WWB
BBB
GGG

SAMPLE OUTPUT:

7

In this example, there is no shifting. The first photo is as follows: (. for sky, * for star)

..*
***
***

The second photo, where the stars on the bottom row disappeared, is as follows:

..*
***
...

This is the only way to produce the superimposed photo, so the minimum possible number of initial stars is 7.

SAMPLE INPUT:

3
5 1 2
GWGWW
WGWWW
WBWGW
WWWWW
WWGWW
3 1 1
WWW
WBW
WWW
3 1 0
GGB
GGW
WWW

SAMPLE OUTPUT:

4
-1
4

In the first case, there were at least 4 stars at the start. If we let (r,c) denote the intersection of the rth row from the top and cth column from the left, one possibility is that they were initially at (1,1),(3,2),(2,2), and (1,3). All the stars shifted, except for the one at (2,2)which disappeared.

In the second case, there is no arrangement of stars in the initial photo that can produce the middle black pixel given the shift.

In the third case, there were at least 4 stars at the start. One possibility is that they were initially at (1,1),(1,2),(1,3),and (2,1). In the second photo, the star originally at (1,1) disappeared and the star originally at (1,3) moved off frame. The other two stars shifted to the right by 1.

SCORING:

Input 3: A=B=0
Inputs 4-7: A=1,B=0,N≤10
Inputs 8-9: A=1,B=0
Inputs 10-12: No additional constraints.

Problem credits: Suhas Nagar

扫码领取USACO试题答案+详细解析

咨询一对一备赛规划

Bessie has an N×N (1≤N≤1000) addition table where the integer in the cell at row r and column c is r+c, for all 1≤r,c≤N. For example, for N=3, the table would look like this:

2 3 4
3 4 5
4 5 6

Unfortunately, Elsie got ahold of the table and permuted it by performing the following three types of operations as many times as she wanted.

1. Swap two rows
2. Swap two columns
3. Select two values a and b that are both present in the table, then simultaneously change every occurrence of a to b and every occurrence of b to a.

Elsie will always perform operations in increasing order of type; that is, she performs as many operations as she likes (possibly none) first of type 1, then of type 2, and finally of type 3.

Help Bessie recover a possible state of the table after Elsie finished applying all of her operations of types 1 and 2, but before applying any operations of type 3.There may be multiple possible answers, in which case you should output the lexicographically smallest one.

To compare two tables lexicographically, compare the first entries at which they differ, when reading both tables in the natural order (rows from top to bottom, left to right within a row).

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains N.

The next N lines each contain N integers, representing Bessie's addition table after Elsie has permuted it.

OUTPUT FORMAT (print output to the terminal / stdout):

The lexicographically minimum possible state of the table after all operations of types 1 and 2, but before any operations of type 3. It is guaranteed that the answer exists.

SAMPLE INPUT:

1
2

SAMPLE OUTPUT:

2
Regardless of what operations Elsie performs, the table won't change.

SAMPLE INPUT:

3
3 4 2
5 2 3
6 3 5

SAMPLE OUTPUT:

4 2 3
5 3 4
6 4 5

Here is a possible sequence of operations Elsie might have performed.

2 3 4
3 4 5
4 5 6
-> (op 1: swap columns 2 and 3)
2 4 3
3 5 4
4 6 5
-> (op 1: swap columns 1 and 2)
4 2 3
5 3 4
6 4 5
-> (op 3: swap values 2 and 3)
4 3 2
5 2 4
6 4 5
-> (op 3: swap values 3 and 4)
3 4 2
5 2 3
6 3 5

Note: the following is also a possible state of the table after operations of types 1 and 2, but it is not the lexicographically smallest because the second entry of the first row is larger than in the correct answer.

4 6 5
3 5 4
2 4 3

SAMPLE INPUT:

6
8 10 5 6 7 4
12 11 10 4 8 2
5 4 6 7 9 8
10 2 4 8 5 12
6 8 7 9 3 5
4 12 8 5 6 10

SAMPLE OUTPUT:

7 5 8 9 10 6
4 2 5 6 7 3
8 6 9 10 11 7
5 3 6 7 8 4
9 7 10 11 12 8
6 4 7 8 9 5

SCORING:

Inputs 4-5: N≤6
Inputs 6-7: N≤8
Inputs 8-11: N≤100
Inputs 12-15: No additional constraints.

Problem credits: Benjamin Qi

扫码领取USACO试题答案+详细解析

咨询一对一备赛规划

It is another cold and boring day on Farmer John's farm. To pass the time, Farmer John has invented a fun leisure activity involving performing operations on an integer array.

Farmer John has an array a of N (1≤N≤2⋅10⁵) non-negative integers and an integer M(1≤M≤10⁹). Then, FJ will ask Bessie for an integer x. In one operation, FJ can pick an index i and subtract or add 1 to a_i. FJ's boredom value is the minimum number of operations he must perform so that a_i−x is divisible by M for all 1≤i≤N.

Among all possible x, output FJ's minimum possible boredom value.

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains T (1≤T≤10), the number of independent test cases to solve.

The first line of each test case contains N and M.

The second line of each test case contains a₁,a₂,...,a_N (0≤a_i≤10⁹).

It is guaranteed that the sum of N over all test cases does not exceed 5⋅10⁵.

OUTPUT FORMAT (print output to the terminal / stdout):

For each test case, output an integer on a new line containing FJ's minimum possible boredom value among all possible values of x.

SAMPLE INPUT:

2
5 9
15 12 18 3 8
3 69
1 988244353 998244853

SAMPLE OUTPUT:

10
21

In the first test case, one optimal choice of x is 3. FJ can perform 10 operations to make a=[12,12,21,3,12].

SCORING:

Input 2: N≤1000 and M≤1000.
Input 3: N≤1000.
Inputs 4-5: M≤10⁵.
Inputs 6-16: No additional constraints.

Problem credits: Chongtian Ma

扫码领取USACO试题答案+详细解析

咨询一对一备赛规划

Farmer John's N (1≤N≤5⋅10⁵) cows are standing in a line, with cow 1
at the front of the line and cow N at the back of the line. FJ's cows also come in many different species. He denotes each species with an integer from 1 to N. The i'th cow from the front of the line is of species a_i (1≤a_i ≤N).

FJ is taking his cows to a checkup at a local bovine hospital. However, the bovine veterinarian is very picky and wants to perform a checkup on the i'th cow in the line, only if it is species b_i (1≤b_i≤N).

FJ is lazy and does not want to completely reorder his cows. He will perform the following operation exactly once.

Select two integers l and r such that 1≤l≤r≤N. Reverse the order of the cows that are between the l-th cow and the r-th cow in the line, inclusive.

FJ wants to measure how effective this approach is. Find the sum of the number of cows that are checked by the veterinarian over all N(N+1)/2 possible operations.

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains an integer N.

The second line contains a₁,a₂,…,a_N.

The third line contains b₁,b₂,…,b_N.

OUTPUT FORMAT (print output to the terminal / stdout):

Output one line with the sum of the number of cows that are checked by the veterinarian over all possible operations.

SAMPLE INPUT:

3
1 3 2
3 2 1

SAMPLE OUTPUT:

3

If FJ chooses (l=1,r=1), (l=2,r=2), or (l=3,r=3) then no cows will be checked. Note that those operations do not modify the location of the cows.

The following operations result in one cow being checked.

• l=1,r=2: FJ reverses the order of the first and second cows so the species of each cow in the new lineup will be [3,1,2]. The first cow will be checked.
• l=2,r=3: FJ reverses the order of the second and third cows so the species of each cow in the new lineup will be [1,2,3]. The second cow will be checked.
• l=1,r=3: FJ reverses the order of the first, second, and third cows so the species of each cow in the new lineup will be [2,3,1]. The third cow will be checked.

The total number of cows checked over all six operations is 0+0+0+1+1+1=3.

SAMPLE INPUT:

3
1 2 3
1 2 3

SAMPLE OUTPUT:

12

There are three possible operations that cause 3 cows to be checked: (l=1,r=1), (l=2,r=2), and (l=3,r=3). The remaining operations each result in 1 cow being checked. The total number of cows checked over all six operations is 3+3+3+1+1+1=12.

SAMPLE INPUT:

7
1 3 2 2 1 3 2
3 2 2 1 2 3 1

SAMPLE OUTPUT:

60

SCORING:

Input 4: N≤100
Input 5: N≤5000
Inputs 6-9:a_i,b_i  are all generated uniformly at random in the range [1,N]
Inputs 10-15:a_i,b_i are all generated uniformly at random in the range [1,2]
Inputs 16-23: No additional constraints.

Problem credits: Chongtian Ma, Haokai Ma, and Alex Liang

扫码领取USACO试题答案+详细解析

咨询一对一备赛规划

Farmer John's farm is full of lush vegetation and every cow wants a photo of its natural beauty. Unfortunately, Bessie still has places to be, but she doesn't want to disrupt any photo ops.

Bessie is currently standing at (X,0) on the XY-plane and she wants to get to (0,Y)
(1≤X,Y≤106). Unfortunately, N (1≤N≤3⋅105 ) other cows have decided to pose on the X axis. More specifically, cow i will be positioned at (xi,0) with a photographer at (0,yi) where (1≤xi,yi≤106) ready to take their picture. They will begin posing moments before time si (1≤si<T) and they will keep posing for a very long time (they have to get their picture just right). Here, 1≤T≤N+1.

Bessie knows the schedule for every cow's photo op, and she will take the shortest Euclidean distance to get to her destination, without crossing the line of sight from any photographer to their respective cow (her path will consist of one or more line segments).

If Bessie leaves at time t , she will avoid the line of sights for all photographer/cow pairs that started posing at time si≤t , and let the distance to her final destination be dt. Determine the values of ⌊dt⌋ for each integer t from 0 to T−1 inclusive.

INPUT FORMAT (input arrives from the terminal / stdin):

The first line of input contains N and T, representing the number of cows posing on the x-axis and the timeframe that Bessie could leave at.

The second line of input contains X and Y, representing Bessie's starting X coordinate and her target Y coordinate respectively.

The next N lines contain s_i x_i and y_i . It is guaranteed that all xi are distinct from each other and X, and all y_i are distinct from each other and Y. All s_i  will be given in increasing order, where s_i ≤s_i +1.

OUTPUT FORMAT (print output to the terminal / stdout):

Print T lines, with the t th (0-indexed) line containing ⌊dt⌋.

SAMPLE INPUT:

4 5
6 7
1 7 5
2 4 4
3 1 6
4 2 9

SAMPLE OUTPUT:

9
9
9
10
12

SAMPLE INPUT:

2 3
10 7
1 2 10
1 9 1

SAMPLE OUTPUT:

12
16
16

For t=0 the answer is =12.

For t=1 the answer is =16.

SAMPLE INPUT:

5 6
8 9
1 3 5
1 4 1
3 10 7
4 9 2
5 6 6

SAMPLE OUTPUT:

12
12
12
12
14
14

For t=5 the answer is . Path: (8,0)→(9,0)→(0,7)→(0,9)

SCORING:

Inputs 4-6: N≤100
Inputs 7-9: N≤3000
Inputs 10-12: T≤10
Inputs 13-18: No additional constraints

Problem credits: Suhas Nagar

扫码领取USACO试题答案+详细解析

咨询一对一备赛规划

Consider an undirected graph with N nodes labeled 1…N and M edges (1≤N≤2⋅10⁵,0≤M≤4⋅10⁵). You're given a binary string s1s2…sN. At time t
for each t∈[1,N],

If st=0, node t is removed from the graph.
If st=1, node t is removed from the graph, and edges are added between every pair of neighbors that node t had just before removal.

Note that in both cases, when a node is removed from the graph all of its incident edges are removed as well.

Count the number of pairs of nodes that can reach each other via some sequence of edges just before each of timesteps 1…N.

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains N and M.

The second line contains the bit string s of length N.

The next M lines each contain two integers denoting an edge of the graph.

OUTPUT FORMAT (print output to the terminal / stdout):

N lines, the number of pairs before each timestep.

SAMPLE INPUT:

3 2
111
1 2
1 3

SAMPLE OUTPUT:

3
1
0

Before any removals, all pairs of nodes are reachable from each other. After node 1 is removed, an edge is added between 2 and 3, so they can still reach each other.

SAMPLE INPUT:

3 2
000
1 2
1 3

SAMPLE OUTPUT:

3
0
0

Before any removals, all pairs of nodes are reachable from each other. After node 1 is removed, 2 and 3 can no longer reach each other.

SAMPLE INPUT:

7 8
1101101
6 2
1 2
2 3
6 3
1 3
1 7
4 5
2 7

SAMPLE OUTPUT:

11
7
4
2
1
1
0

SCORING:

Inputs 4-6: N≤100
Inputs 7-8: All s_i equal zero.
Inputs 9-11: All s_i equal one.
Inputs 12-23: No additional constraints.

Problem credits: Benjamin Qi