Math and counting

Number theory, combinatorics, and geometry

Under construction

Still being built. Things may change

Total results13,153 problems

Greatest Chicken Dish
Hard9
Dynamic programming, Number theory+2
No attempts yet
Time limit
15s
Memory limit
512 MB
Judgeable
The Halfwitters
Hard9
Dynamic programming, Graph+2
No attempts yet
Time limit
5s
Memory limit
512 MB
Judgeable
Lighthouses
Hard9
Dynamic programming, Geometry
No attempts yet
Time limit
15s
Memory limit
512 MB
Judgeable
Div
Hard9
Math, Number theory
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Ineq
Hard9
Geometry, Math+2
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Data Structure Quiz
Hard9
Segment tree, Divide and conquer+1
No attempts yet
Time limit
8s
Memory limit
512 MB
Judgeable
Knowledge-Oriented Problem
Hard9
Matrix, Math+1
No attempts yet
Time limit
5s
Memory limit
512 MB
Judgeable
LCM Sum
Hard9
Number theory, Math+1
No attempts yet
Time limit
7s
Memory limit
512 MB
Judgeable
Delightful (Hard)
Hard9
Number theory, Implementation+2
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
Xorshift32
Hard9
Math, Bit manipulation+2
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
Help Yourself (Platinum)
Hard9
Combinatorics, Dynamic programming+2
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Chameleon's Love
Hard9
Graph, Combinatorics+2
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Sprinklers 2: Return of the Alfalfa
Hard9
Dynamic programming, Prefix sum+2
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Exercise
Hard9
Combinatorics, Number theory+2
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Cow Exercise
Hard9
Math, Number theory+2
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
Alternative Permutations
Hard9
Combinatorics, Tree+2
No attempts yet
Time limit
8s
Memory limit
256 MB
Judgeable
Laser Intensification
Hard9
Math, Combinatorics+2
No attempts yet
Time limit
2s
Memory limit
64 MB
Judgeable
Easiest Sum
Hard9
Prefix sum, Binary search+2
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
Histogram and Blue Rectangles
Hard9
Stack, Dynamic programming+1
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Integer Equation Checker
Hard9
Brute force, Implementation+2
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
Yet Another Problem on Empodia
Hard9
Combinatorics, Dynamic programming+2
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Curious Ball Game
Hard9
Number theory, Math+2
No attempts yet
Time limit
0.5s
Memory limit
256 MB
Judgeable
Rigged ㄱ Bomb Game
Hard9
Game theory, Segment tree+1
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
Tree Average Weight
Hard9
Combinatorics, Tree+2
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
Continue the Sequence
Hard9
Math, Number theory+2
No attempts yet
Time limit
4s
Memory limit
256 MB
Judgeable
Little Q and Big Integers
Hard9
Combinatorics, Dynamic programming+1
No attempts yet
Time limit
1.5s
Memory limit
512 MB
Judgeable
Alice and Bob (and string): Double Menace
Hard9
Game theory, String matching+1
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Counting Different Summands
Hard9
Combinatorics, Math+2
No attempts yet
Time limit
2s
Memory limit
256 MB
Judgeable
Fast Travel Coloring
Hard9
Graph, Math+1
No attempts yet
Time limit
2s
Memory limit
256 MB
Judgeable
Gnutella Chessmaster
Hard9
Combinatorics, Math+2
No attempts yet
Time limit
3s
Memory limit
512 MB
Judgeable
Interpolate
Hard9
Math, Bit manipulation+2
No attempts yet
Time limit
4s
Memory limit
256 MB
Judgeable
Kingdom Connectivity
Hard9
Graph, Minimum spanning tree+2
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Period Sequence
Hard9
Math, Segment tree+1
No attempts yet
Time limit
6s
Memory limit
256 MB
Judgeable
Chiaki Sequence Revisited
Hard9
Math, Combinatorics+2
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
RMQ Similar Sequence
Hard9
Tree, Combinatorics+2
No attempts yet
Time limit
2s
Memory limit
256 MB
Judgeable
Rikka with Proper Fractions
Hard9
Number theory, Math+2
No attempts yet
Time limit
10s
Memory limit
512 MB
Judgeable
Rikka with Tree Game
Hard9
Game theory, Tree+2
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Rikka with Equation
Hard9
Math, Number theory
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Unseen Segments
Hard9
Geometry, Sorting+2
No attempts yet
Time limit
2s
Memory limit
256 MB
Judgeable
Convolution
Hard9
Divide and conquer, Combinatorics+1
No attempts yet
Time limit
3s
Memory limit
512 MB
Judgeable
Good Game
Hard9
Combinatorics, Dynamic programming+2
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Jump Jump Jump
Hard9
Probability, Dynamic programming+1
No attempts yet
Time limit
5s
Memory limit
512 MB
Judgeable
Entanglement
Hard9
Combinatorics, Graph+1
No attempts yet
Time limit
3s
Memory limit
256 MB
Judgeable
Salaj
Hard9
Dynamic programming, Combinatorics+1
No attempts yet
Time limit
2s
Memory limit
256 MB
Judgeable
Taxi
Hard9
Tree, Dynamic programming+2
No attempts yet
Time limit
1.5s
Memory limit
256 MB
Judgeable
Candies
Hard9
Number theory, Math+2
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
Nice Numbers
Hard9
Combinatorics, Number theory+2
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
In the End
Hard9
Probability, Dynamic programming+1
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Array Challenge
Hard9
Math, Number theory+2
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
Function Counting
Hard9
Combinatorics, Dynamic programming+1
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Master Zhu and Magic Numbers
Hard9
Bit manipulation, Implementation+2
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
Master Zhu and Candies
Hard9
Game theory, Math+2
No attempts yet
Time limit
3s
Memory limit
512 MB
Judgeable
Master Zhu and Rikka
Hard9
Tree, DFS+2
No attempts yet
Time limit
3s
Memory limit
512 MB
Judgeable
Master Zhu and Root
Hard9
Number theory, Math+2
No attempts yet
Time limit
3s
Memory limit
512 MB
Judgeable
Master Zhu and the Leaper
Hard9
Dynamic programming, Combinatorics+2
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
Random Points
Hard9
Geometry, Combinatorics+2
No attempts yet
Time limit
5s
Memory limit
512 MB
Judgeable
GCD vs LCM
Hard9
Number theory, Math+2
No attempts yet
Time limit
2.5s
Memory limit
512 MB
Judgeable
Power of XOR
Hard9
Bit manipulation, Math+2
No attempts yet
Time limit
6s
Memory limit
512 MB
Judgeable
Game
Hard9
Game theory, Dynamic programming+1
No attempts yet
Time limit
8s
Memory limit
512 MB
Judgeable
Bermutation
Hard9
Combinatorics, Math+1
No attempts yet
Time limit
2s
Memory limit
256 MB
Judgeable
Strange Sequence
Hard9
Matrix, Math+1
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Catalan Combinatorial Objects
Hard9
Combinatorics, Math+1
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Masterpiece
Hard9
Combinatorics, Math+1
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Airport Check-in
Hard9
Probability, Math+1
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
Short Random Problem
Hard9
Tree, Probability+2
No attempts yet
Time limit
6s
Memory limit
512 MB
Judgeable
Earthquake
Hard9
Probability, Dynamic programming+2
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
Window XOR
Hard9
Math, Bit manipulation+2
No attempts yet
Time limit
2s
Memory limit
1024 MB
Judgeable
Conic Section
Hard9
Segment tree, Geometry+1
No attempts yet
Time limit
3s
Memory limit
256 MB
Judgeable
Dogs
Hard9
Graph, Combinatorics+1
No attempts yet
Time limit
2s
Memory limit
256 MB
Judgeable
Flip a Coin
Hard9
Probability, Dynamic programming+2
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
Multi-stage Marathon
Hard9
Matrix, Dynamic programming+1
No attempts yet
Time limit
3s
Memory limit
512 MB
Judgeable
Compressed LCS
Hard9
Dynamic programming, Greedy+1
No attempts yet
Time limit
3s
Memory limit
512 MB
Judgeable
K-th String
Hard9
String, Combinatorics+2
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
Kitamasa's Counterattack
Hard9
Graph, Bit manipulation+1
No attempts yet
Time limit
2s
Memory limit
256 MB
Judgeable
Those Russian Hackers
Hard9
Dynamic programming, Probability+2
No attempts yet
Time limit
3s
Memory limit
256 MB
Judgeable
Palindrome
Hard9
String, String matching+1
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Territory Game
Hard9
Tree, Game theory+1
No attempts yet
Time limit
3s
Memory limit
512 MB
Judgeable
Ascent Sequences Avoiding Pattern 201
Hard9
Dynamic programming, Combinatorics+2
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Inversions in Lexicographical Order
Hard9
Combinatorics, Dynamic programming+1
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Connected Spanning Subgraph
Hard9
Graph, Combinatorics+2
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
Power of Power Partition Function
Hard9
Number theory, Divide and conquer+2
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Line Counting
Hard9
Number theory, Geometry+1
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Binary Neural Network
Hard9
Implementation, Greedy+2
No attempts yet
Time limit
2s
Memory limit
256 MB
Judgeable
Simple Graph
Hard9
Combinatorics, Dynamic programming+2
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
Parentheses
Hard9
Combinatorics, Math+1
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
Values on a Tree
Hard9
Tree, Dynamic programming+2
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
Generator
Hard9
Probability, String matching+2
No attempts yet
Time limit
3s
Memory limit
256 MB
Judgeable
Less Time, More Profit
Hard9
Dynamic programming, Graph+2
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
Pet Tree
Hard9
Dynamic programming, Tree+2
No attempts yet
Time limit
8s
Memory limit
1024 MB
Judgeable
Matrix and Queries
Hard9
Math, Matrix+2
No attempts yet
Time limit
5s
Memory limit
512 MB
Judgeable
Rectangle
Hard9
Math, Combinatorics+2
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
Navigation Game
Hard9
Greedy, Implementation+2
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
Fence
Hard9
Geometry, Binary search+2
No attempts yet
Time limit
1s
Memory limit
1024 MB
Judgeable
Human Error
Hard9
Game theory, Bit manipulation+2
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
Wrong Answer
Hard9
Greedy, Dynamic programming+2
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
Sequence and Queries 39
Hard9
Segment tree, Math+2
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Landlords
Hard9
Probability, Math+1
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
Holy cow, Vim! (Easy)
Hard9
Implementation, Simulation+2
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
Holy cow, Vim! (Hard)
Hard9
Implementation, Stack+2
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
Star Trek
Hard9
Game theory, Tree+2
No attempts yet
Time limit
1s
Memory limit
32 MB
Judgeable

Title	Level	Topics	Solved	Time limit	Memory limit	Judge
Greatest Chicken DishCount, for each query range [L, R] and value D, the number of contiguous subarrays inside [L, R] whose GCD equals D.	Hard9	Dynamic programmingNumber theory+2	No attempts yet	15s	512 MB	Judgeable
The HalfwittersFor each start permutation, compute the minimum expected time to reach the identity order using adjacent swaps (cost a), a full reversal (cost b), or a random reshuffle (cost c).	Hard9	Dynamic programmingGraph+2	No attempts yet	5s	512 MB	Judgeable
LighthousesFind the longest simple path along given tram segments between vertices of a convex polygon, where the path must never cross itself.	Hard9	Dynamic programmingGeometry	No attempts yet	15s	512 MB	Judgeable
DivCount positive integers x for which a signed sum of powers of x is divisible by 1 + x + ... + x^(m-1), or report -1 if there are infinitely many.	Hard9	MathNumber theory	No attempts yet	2s	512 MB	Judgeable
IneqGiven a finite set S of integer lattice points, decide whether S can be cut out as exactly the integer points lying strictly below every line of some finite family of half-planes.	Hard9	GeometryMath+2	No attempts yet	2s	512 MB	Judgeable
Data Structure QuizApply m1 weighted rectangle additions to an n by n grid, then answer m2 queries for the maximum cell value inside each query rectangle.	Hard9	Segment treeDivide and conquer+1	No attempts yet	8s	512 MB	Judgeable
Knowledge-Oriented ProblemGiven a graph G with n vertices and k copies stacked in a chain with matching vertex edges, count the spanning trees of the result modulo 10^9+7.	Hard9	MatrixMath+1	No attempts yet	5s	512 MB	Judgeable
LCM SumGiven n up to 10^18 and k up to 30, compute the sum of lcm(x, x+1, ..., x+k) for x from 1 to n, modulo 10^9+7.	Hard9	Number theoryMath+1	No attempts yet	7s	512 MB	Judgeable
Delightful (Hard)Write a program of at most 5000 commands for a ternary computer with 26 forty-trit registers that sets Y to 1 if the input X (0 to 109) is prime and 0 otherwise.	Hard9	Number theoryImplementation+2	No attempts yet	1s	512 MB	Judgeable
Xorshift32Given a starting value x and a target t, find the index at which t first appears in the Xorshift32 pseudorandom sequence.	Hard9	MathBit manipulation+2	No attempts yet	1s	256 MB	Judgeable
Help Yourself (Platinum)Sum the K-th power of the number of connected components of the union over all 2^N subsets of N intervals, modulo 1e9+7.	Hard9	CombinatoricsDynamic programming+2	No attempts yet	2s	512 MB	Judgeable
Chameleon's LoveUsing at most 20000 queries that return the number of distinct skin colors in a chosen subset, recover every pair of chameleons sharing an original color.	Hard9	GraphCombinatorics+2	No attempts yet	2s	512 MB	Judgeable
Sprinklers 2: Return of the AlfalfaCount the ways to place sweet corn and alfalfa sprinklers on an N by N grid, with some blocked squares, so that every square is covered by exactly one sprinkler type.	Hard9	Dynamic programmingPrefix sum+2	No attempts yet	2s	512 MB	Judgeable
ExerciseGiven N and prime M, compute modulo M the product over all N! permutations of the order (number of iterations until identity) of each permutation.	Hard9	CombinatoricsNumber theory+2	No attempts yet	2s	512 MB	Judgeable
Cow ExerciseGiven N and prime M, sum every positive integer K for which some permutation of length N has order K, then output the sum modulo M.	Hard9	MathNumber theory+2	No attempts yet	1s	512 MB	Judgeable
Alternative PermutationsGiven k, find the smallest n for which some BST on 1 to n is produced by exactly k permutations, and print the lexicographically smallest one.	Hard9	CombinatoricsTree+2	No attempts yet	8s	256 MB	Judgeable
Laser IntensificationFind probability p so that one photon entering the lower-left of a w by h grid, with nodes independently faulty with probability 1-p except n known faulty cells, yields k photons in expectation at the upper-right; print -1 if impossible.	Hard9	MathCombinatorics+2	No attempts yet	2s	64 MB	Judgeable
Easiest SumGiven an array, each coin decreases one element by 1, so find the minimum possible maximum subarray sum for each budget from 1 to k and sum them modulo 998244353.	Hard9	Prefix sumBinary search+2	No attempts yet	1s	512 MB	Judgeable
Histogram and Blue RectanglesFor each prefix of j columns of a histogram with heights a_i, find the largest axis-aligned rectangle whose interior lies entirely within those columns.	Hard9	StackDynamic programming+1	No attempts yet	2s	512 MB	Judgeable
Integer Equation CheckerClassify an equation string as correct, format error, math error, or a typo fixable by replacing at most two characters with a valid correct equation.	Hard9	Brute forceImplementation+2	No attempts yet	1s	512 MB	Judgeable
Yet Another Problem on EmpodiaCount, for each length i up to N, the number of equivalence classes of permutations of length i under the relation that agrees on which intervals are framed (max minus min equals length minus one), modulo a prime.	Hard9	CombinatoricsDynamic programming+2	No attempts yet	2s	512 MB	Judgeable
Curious Ball GameFor each query (N, M), find the M-th smallest bag size A such that drawing two balls gives probability exactly 1/N^2 that at least one is not red, outputting A and B mod 1e9+7.	Hard9	Number theoryMath+2	No attempts yet	0.5s	256 MB	Judgeable
Rigged ㄱ Bomb GameBoards hold A, B, and C bombs, and each query asks who wins the combined partizan game on a range of boards after swapping A and B on K of them.	Hard9	Game theorySegment tree+1	No attempts yet	1s	256 MB	Judgeable
Tree Average WeightGiven a degree sequence with some free entries, pick a labeled tree uniformly at random and find the integer part of its expected weight, where weight sums usz(u)+vsz(v) over edges.	Hard9	CombinatoricsTree+2	No attempts yet	1s	256 MB	Judgeable
Continue the SequenceGiven n values, extend the sequence by m terms so that it agrees with a polynomial of the smallest possible degree modulo 998244353.	Hard9	MathNumber theory+2	No attempts yet	4s	256 MB	Judgeable
Little Q and Big IntegersCount positive base-k integers with no zero digit whose digit-count pattern is allowed by a 0/1 matrix, summed over m single-bit flips modulo 786433.	Hard9	CombinatoricsDynamic programming+1	No attempts yet	1.5s	512 MB	Judgeable
Alice and Bob (and string): Double MenaceGiven string s and k, find the k-th lexicographically smallest substring t of s from which Alice wins the two-sided extension game, or report that fewer than k exist.	Hard9	Game theoryString matching+1	No attempts yet	2s	512 MB	Judgeable
Counting Different SummandsSum f(a) over all ordered partitions of n into m positive parts, where f counts distinct part values, modulo 998244353, with n up to 1e18 and m up to 500.	Hard9	CombinatoricsMath+2	No attempts yet	2s	256 MB	Judgeable
Fast Travel ColoringColor the edges of a complete graph with 7n vertices using n colors so that every vertex pair is joined by an edge of each color or by a two-edge path of that color.	Hard9	GraphMath+1	No attempts yet	2s	256 MB	Judgeable
Gnutella ChessmasterCount the non-attacking bishop placements on an n by n board for each bishop count k from 1 to 2n-1, modulo 998244353.	Hard9	CombinatoricsMath+2	No attempts yet	3s	512 MB	Judgeable
InterpolateGiven m distinct boolean input vectors with outputs, build a Zhegalkin (XOR-of-ANDs) polynomial matching all of them using at most 9000 terms, or report that none exists.	Hard9	MathBit manipulation+2	No attempts yet	4s	256 MB	Judgeable
Kingdom ConnectivityGiven a planar straight-line graph with weighted edges, choose a minimum-weight set of edges whose removal makes both sides of every wall reachable from the exterior.	Hard9	GraphMinimum spanning tree+2	No attempts yet	2s	512 MB	Judgeable
Period SequenceGiven a periodic-plus-growing sequence built from n integers, compute the sum over all subarrays inside index range [a, b] of sum of x times count of x squared, modulo 1e9+7.	Hard9	MathSegment tree+1	No attempts yet	6s	256 MB	Judgeable
Chiaki Sequence RevisitedSum the first n terms of the self-referential Chiaki sequence, where each term depends on earlier terms via a_{n-a_{n-1}} + a_{n-1-a_{n-2}}, modulo 1e9+7.	Hard9	MathCombinatorics+2	No attempts yet	1s	256 MB	Judgeable
RMQ Similar SequenceGiven sequence A, count the expected sum of a random real sequence B in [0,1] that has identical RMQ answers to A for every subarray, modulo 1e9+7.	Hard9	TreeCombinatorics+2	No attempts yet	2s	256 MB	Judgeable
Rikka with Proper FractionsCount reduced proper fractions e/f with f at most n lying between two given fractions a/b and c/d, answering modulo 998244353.	Hard9	Number theoryMath+2	No attempts yet	10s	512 MB	Judgeable
Rikka with Tree GameOn a rooted tree where players alternately move a token to a child and the score is the final depth, repeatedly attach leaves and find the limit of f(k)/k, where f(k) is the fewest additions making the optimal score exactly k.	Hard9	Game theoryTree+2	No attempts yet	2s	512 MB	Judgeable
Rikka with EquationFor each n, count tuples (a, b, m) with m at most n such that some integers x and y satisfy x^2 + y^2 is congruent to a and xy is congruent to b modulo m.	Hard9	MathNumber theory	No attempts yet	2s	512 MB	Judgeable
Unseen SegmentsGiven n vertical segments and queries of (west power, east power), find the total length of parts no observer sees through at most that many segments.	Hard9	GeometrySorting+2	No attempts yet	2s	256 MB	Judgeable
ConvolutionGiven sequences a and b of length n+1, compute c_k as the sum of C(k,i)·a_i·b_(k-i) modulo 2^32 for every k from 0 to n.	Hard9	Divide and conquerCombinatorics+1	No attempts yet	3s	512 MB	Judgeable
Good GameCount monotone lattice paths in n dimensions from the origin to a target, avoiding m forbidden points, modulo 1e9+7.	Hard9	CombinatoricsDynamic programming+2	No attempts yet	2s	512 MB	Judgeable
Jump Jump JumpGiven k jump vectors with coordinates up to 3, find for each x up to n the probability that a random walk from (0, 0) first lands on the diagonal point (x, x).	Hard9	ProbabilityDynamic programming+1	No attempts yet	5s	512 MB	Judgeable
EntanglementCount pairs of arrays A (length N) and B (length M) with values from 1 to K such that every cell C[i][j] equals A[i] or B[j].	Hard9	CombinatoricsGraph+1	No attempts yet	3s	256 MB	Judgeable
SalajFor each edge count j up to MAX, count the valid connectivity history arrays on V vertices with j edges, modulo MOD.	Hard9	Dynamic programmingCombinatorics+1	No attempts yet	2s	256 MB	Judgeable
TaxiSum over all ways to place M taxis and M customers on a weighted tree of the maximum-weight perfect matching cost, modulo 1e9+7.	Hard9	TreeDynamic programming+2	No attempts yet	1.5s	256 MB	Judgeable
CandiesFor each query k, count modulo 2 the pairs (child, candy kind) where buying only her favorite candy at that price leaves exactly k dollars.	Hard9	Number theoryMath+2	No attempts yet	1s	512 MB	Judgeable
Nice NumbersCount numbers in [L, R] that are pandigital (a permutation of all digits) in some base d, modulo 998244353, with L and R up to 5000-digit integers.	Hard9	CombinatoricsNumber theory+2	No attempts yet	1s	512 MB	Judgeable
In the EndEach column holds one cake at a random row, and the robot moves right with a row change of at most 1; find the limit of f(m)/m modulo 1e9+7.	Hard9	ProbabilityDynamic programming+1	No attempts yet	2s	512 MB	Judgeable
Array ChallengeGiven a linear recurrence h and closed-form arrays b and a, compute floor(sqrt(a_n)) modulo 1e9+7 for n up to 1e15.	Hard9	MathNumber theory+2	No attempts yet	1s	512 MB	Judgeable
Function CountingCount maps f on integers in [-n, n] where k applications of f send every x to -x and \| \|f(x)\| - \|x\| \| is at most 2. Print the count modulo 1e9+7.	Hard9	CombinatoricsDynamic programming+1	No attempts yet	2s	512 MB	Judgeable
Master Zhu and Magic NumbersGiven sparse positions of ones in a triangular binary layout, simulate a bitwise dependency program to get values b_j, then answer queries asking for the popcount of ORs of selected b_j.	Hard9	Bit manipulationImplementation+2	No attempts yet	1s	512 MB	Judgeable
Master Zhu and CandiesGiven n candy heaps, players may remove any positive number from one heap or split one heap into three non-empty heaps; decide the winner under optimal play.	Hard9	Game theoryMath+2	No attempts yet	3s	512 MB	Judgeable
Master Zhu and RikkaGiven a rooted tree with values on vertices, answer queries that ask for the GCD of sums of values occurring exactly a times and exactly b times in a subtree or on a path.	Hard9	TreeDFS+2	No attempts yet	3s	512 MB	Judgeable
Master Zhu and RootFor each query (x, y), find the smallest k >= 0 with x^k congruent to y modulo some prime divisor p of n, or report -1.	Hard9	Number theoryMath+2	No attempts yet	3s	512 MB	Judgeable
Master Zhu and the LeaperCount monotone leaper paths from (1,1) to (n,m) on a huge board with at most 100 blocked cells, modulo 110119.	Hard9	Dynamic programmingCombinatorics+2	No attempts yet	1s	512 MB	Judgeable
Random PointsGiven n points in general position, output 2^n times the expected number of vertices of the convex hull of a uniformly random subset, modulo 1e9+7.	Hard9	GeometryCombinatorics+2	No attempts yet	5s	512 MB	Judgeable
GCD vs LCMFor each of q queries with n, m, a up to 1e5, sum lcm(i,j) over all i<=n, j<=m with gcd(i,j)<=a, modulo 1e9+7.	Hard9	Number theoryMath+2	No attempts yet	2.5s	512 MB	Judgeable
Power of XORGiven n integers, compute the sum over all 2^n subsets of the k-th power of the popcount of the XOR of the subset, modulo 1e9+7.	Hard9	Bit manipulationMath+2	No attempts yet	6s	512 MB	Judgeable
GamePlayers alternately remove an end element from a sequence, and the mover who reaches a terminal sequence wins. Decide whether the first player, the second, or neither wins.	Hard9	Game theoryDynamic programming+1	No attempts yet	8s	512 MB	Judgeable
BermutationGiven a permutation and b, find the lexicographic rank of p among all permutations reachable by swapping the two halves of any 2b-length segment.	Hard9	CombinatoricsMath+1	No attempts yet	2s	256 MB	Judgeable
Strange SequenceGiven n up to 10^18, find the length of the n-th term of the look-and-say sequence starting from "2", modulo 7340033.	Hard9	MatrixMath+1	No attempts yet	2s	512 MB	Judgeable
Catalan Combinatorial ObjectsGiven k from 120 to 139, print a construction from lists, multisets, cycles, and pairs over one unit object that has exactly k objects of weight 6.	Hard9	CombinatoricsMath+1	No attempts yet	2s	512 MB	Judgeable
MasterpieceCount the brush paths (a right/down route and a return left/up route) on an n by n grid whose painted cells match the given row and column counts, modulo 1e9+7.	Hard9	CombinatoricsMath+1	No attempts yet	2s	512 MB	Judgeable
Airport Check-inEach counter has a random per-passenger time and a random remaining time for the current passenger; find the probability that the counter finishing first is also the one with the smallest per-passenger time.	Hard9	ProbabilityMath+1	No attempts yet	1s	256 MB	Judgeable
Short Random ProblemGiven a tree whose edge lengths are independent uniform random reals in [0, 1], find the expected diameter modulo 10^9+7.	Hard9	TreeProbability+2	No attempts yet	6s	512 MB	Judgeable
EarthquakeEach route is usable only if all its bridges survive; pick an adaptive inspection order of bridges to minimize the expected number of inspections before deciding whether any route connects the two lands.	Hard9	ProbabilityDynamic programming+2	No attempts yet	1s	512 MB	Judgeable
Window XORApply the operation that replaces each element with the XOR of K consecutive elements (cyclically) exactly T times, where T can be as large as 10^18.	Hard9	MathBit manipulation+2	No attempts yet	2s	1024 MB	Judgeable
Conic SectionPoints (i, y) receive single-point updates and range reflections of y, and each query asks for the max of ax + by + cxy over an x-range.	Hard9	Segment treeGeometry+1	No attempts yet	3s	256 MB	Judgeable
DogsGiven a directed inspection graph on n villagers, sum over all sick-dog configurations the day of the first shooting and the total number of dogs shot, modulo 998244353.	Hard9	GraphCombinatorics+1	No attempts yet	2s	256 MB	Judgeable
Flip a CoinTwo players each pick a heads/tails string of length up to 20; a fair coin is flipped until one or both strings appear, and we must output the probabilities of Alice winning, Bob winning, and a tie.	Hard9	ProbabilityDynamic programming+2	No attempts yet	1s	256 MB	Judgeable
Multi-stage MarathonGiven a directed graph of up to 70 checkpoints and players entering at set times, compute the XOR over all moments 1 to T of the expected number of players at checkpoint n, modulo 1e9+7.	Hard9	MatrixDynamic programming+1	No attempts yet	3s	512 MB	Judgeable
Compressed LCSGiven two run-length encoded sequences with up to 2000 runs each and total lengths up to 1e9, find the length of their LCS.	Hard9	Dynamic programmingGreedy+1	No attempts yet	3s	512 MB	Judgeable
K-th StringCount permutations t of n distinct letters whose k-th smallest non-empty substring equals s, modulo 1e9+7.	Hard9	StringCombinatorics+2	No attempts yet	1s	256 MB	Judgeable
Kitamasa's CounterattackKitamasa raises key prices by paying per-shop costs to maximize the gap between Akiba's purchase cost and his payment, so the answer is a min-cut or max-flow value, or -1 if unbounded.	Hard9	GraphBit manipulation+1	No attempts yet	2s	256 MB	Judgeable
Those Russian HackersGiven check-time and hacking-time distributions over k frames of length n, find the maximum probability of logging in and finishing the hack before any security check hits your session.	Hard9	Dynamic programmingProbability+2	No attempts yet	3s	256 MB	Judgeable
PalindromeGiven a string s, each query concatenates the length-l substrings starting at the listed positions and asks for the number of palindromic substrings of the result.	Hard9	StringString matching+1	No attempts yet	2s	512 MB	Judgeable
Territory GameGiven a tree and several games where Alice and Bob start at two vertices and color k moves each, find the optimal final value of A minus B.	Hard9	TreeGame theory+1	No attempts yet	3s	512 MB	Judgeable
Ascent Sequences Avoiding Pattern 201Count ascent sequences of length n that avoid the pattern 201, modulo a prime p, with n up to 500.	Hard9	Dynamic programmingCombinatorics+2	No attempts yet	2s	512 MB	Judgeable
Inversions in Lexicographical OrderGiven n with up to 250000 digits, count the inversions in the permutation formed by sorting 1 to n as strings.	Hard9	CombinatoricsDynamic programming+1	No attempts yet	2s	512 MB	Judgeable
Connected Spanning SubgraphGiven a connected undirected graph, count the non-empty edge subsets whose chosen edges form a connected spanning subgraph, and print the count modulo 2.	Hard9	GraphCombinatorics+2	No attempts yet	1s	512 MB	Judgeable
Power of Power Partition FunctionGiven n up to 1e18, m up to 1e18, and k up to 10, compute the sum of the k-th convolution power of the m-ary partition function up to n, modulo 1e9+7.	Hard9	Number theoryDivide and conquer+2	No attempts yet	2s	512 MB	Judgeable
Line CountingFor each n, counts the distinct lines through at least two points of the triangular lattice {(x, y): 1 <= x <= y <= n}, modulo 1000000007.	Hard9	Number theoryGeometry+1	No attempts yet	2s	512 MB	Judgeable
Binary Neural NetworkGiven a truth table of an n-input Boolean function (n up to 10), build a layered sigmoid network with at most 25 layers, 10^4 neurons, and 3*10^4 edges whose single output neuron approximates f.	Hard9	ImplementationGreedy+2	No attempts yet	2s	256 MB	Judgeable
Simple GraphSum x^k over all simple graphs on n labeled vertices, where x counts the tree components, modulo 998244353.	Hard9	CombinatoricsDynamic programming+2	No attempts yet	1s	512 MB	Judgeable
ParenthesesCount parentheses strings of length n by their reversal-and-flip operation distance, then output the weighted sum of those counts modulo m.	Hard9	CombinatoricsMath+1	No attempts yet	1s	512 MB	Judgeable
Values on a TreeFor every K from 0 to n-1, count the non-empty vertex subsets of a tree whose maximum pairwise distance equals K, modulo 998244353.	Hard9	TreeDynamic programming+2	No attempts yet	1s	512 MB	Judgeable
GeneratorFind the expected number of random draws until all n given length-L sequences have each appeared as a contiguous block of the stream.	Hard9	ProbabilityString matching+2	No attempts yet	3s	256 MB	Judgeable
Less Time, More ProfitChoose a subset of plants to build; a shop pays off only when every plant it needs is built. Minimize the maximum build time, then maximize profit within that time.	Hard9	Dynamic programmingGraph+2	No attempts yet	1s	256 MB	Judgeable
Pet TreeCount assignments of edge lengths (each within its own interval) to a tree's edges so the resulting tree diameter falls between S and E, modulo 1e9+7.	Hard9	Dynamic programmingTree+2	No attempts yet	8s	1024 MB	Judgeable
Matrix and QueriesGiven an N x N integer matrix and Q values x, output det(A - xI) mod 998244353 for each query.	Hard9	MathMatrix+2	No attempts yet	5s	512 MB	Judgeable
RectangleConstruct a 30 by 30 table of integers from 0 to 10^6 so that among all axis-aligned sub-rectangles, every integer from 1 to 50000 appears as some rectangle's sum.	Hard9	MathCombinatorics+2	No attempts yet	1s	256 MB	Judgeable
Navigation GameConstruct a 100x100 arrangement of the numbers 1 to 10000 that maximizes a navigation score, where jumps off the current row or column cost points.	Hard9	GreedyImplementation+2	No attempts yet	1s	256 MB	Judgeable
FenceGiven up to 50,000 points, find the minimum perimeter of a rectangle with arbitrary orientation that contains all of them.	Hard9	GeometryBinary search+2	No attempts yet	1s	1024 MB	Judgeable
Human ErrorGiven a grid of Justin and Donald pieces where each turn a player must capture an adjacent piece, and each player may restrict their candidate moves to a set of fixed size, compute Justin's win probability under optimal play with random move choice.	Hard9	Game theoryBit manipulation+2	No attempts yet	1s	512 MB	Judgeable
Wrong AnswerConstruct a cost matrix that makes a greedy two-character solution as far from optimal as possible, maximizing the ratio of its output to the true minimum.	Hard9	GreedyDynamic programming+2	No attempts yet	1s	512 MB	Judgeable
Sequence and Queries 39Maintain an array under range updates that add an arithmetic progression, and answer queries for the longest arithmetic-progression subarray inside a range.	Hard9	Segment treeMath+2	No attempts yet	2s	512 MB	Judgeable
LandlordsGiven m riffle shuffles with split sizes A_i, find the expected value of f(i) for the card at each queried position c after all shuffles, modulo 998244353.	Hard9	ProbabilityMath+1	No attempts yet	1s	512 MB	Judgeable
Holy cow, Vim! (Easy)Construct a stack-language program that outputs x, but outputs 2x when its lines are reversed and -x when its lines are sorted lexicographically.	Hard9	ImplementationSimulation+2	No attempts yet	1s	512 MB	Judgeable
Holy cow, Vim! (Hard)Construct a stack-program whose lines, read normally, reversed, and lexicographically sorted, compute x, x squared, and negative x respectively.	Hard9	ImplementationStack+2	No attempts yet	1s	512 MB	Judgeable
Star TrekPlayers alternately claim unvisited planets in a tree copied into D universes and linked by D portals; count the portal placements where the first player wins.	Hard9	Game theoryTree+2	No attempts yet	1s	32 MB	Judgeable