Math and counting

Number theory, combinatorics, and geometry

Under construction

Still being built. Things may change

Total results13,153 problems

SolveMe
Hard9
Combinatorics, Math+1
No attempts yet
Time limit
8s
Memory limit
512 MB
Judgeable
Webby Subway
Hard9
Geometry, Graph+2
No attempts yet
Time limit
8s
Memory limit
512 MB
Judgeable
Exact Arithmetic
Hard9
Implementation, Simulation+2
No attempts yet
Time limit
8s
Memory limit
512 MB
Judgeable
Cut out
Hard9
Geometry, Math+2
No attempts yet
Time limit
8s
Memory limit
512 MB
Judgeable
Strange Key
Hard9
Graph, Geometry+2
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Bridge Crossing Game
Hard9
Graph, Game theory+2
No attempts yet
Time limit
1s
Memory limit
1024 MB
Judgeable
Pillar and Wall Defense Game
Hard9
Geometry, Brute force+2
No attempts yet
Time limit
1s
Memory limit
1024 MB
Judgeable
Game of the Retired
Hard9
Game theory, String+1
No attempts yet
Time limit
1s
Memory limit
1024 MB
Judgeable
Slide Circuits
Hard9
Graph, Union-find+2
No attempts yet
Time limit
Not set
Memory limit
1024 MB
Judgeable
Permutation CFG
Hard9
Recursion, Combinatorics+2
No attempts yet
Time limit
4s
Memory limit
2048 MB
Judgeable
Sinchon Defense Headquarters
Hard9
Geometry, Math+2
No attempts yet
Time limit
1.5s
Memory limit
1024 MB
Judgeable
Master of String Manipulation
Hard9
Dynamic programming, Combinatorics+2
No attempts yet
Time limit
2.5s
Memory limit
1024 MB
Judgeable
Binary Operator
Hard9
Math, Hash map+2
No attempts yet
Time limit
20s
Memory limit
1024 MB
Judgeable
Pizzo Collectors
Hard9
Number theory, Math+2
No attempts yet
Time limit
2s
Memory limit
1024 MB
Judgeable
Tobacco Growing
Hard9
Dynamic programming, Combinatorics+2
No attempts yet
Time limit
2s
Memory limit
1024 MB
Judgeable
How to Move the Beans
Hard9
Game theory, Graph+1
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Kingdoms and Quarantine
Hard9
Math, Graph+1
No attempts yet
Time limit
8s
Memory limit
512 MB
Judgeable
Crab's Cannon
Hard9
String, String matching+1
No attempts yet
Time limit
3s
Memory limit
512 MB
Judgeable
Intellectual Implementation
Hard9
Math, Segment tree+1
No attempts yet
Time limit
6s
Memory limit
512 MB
Judgeable
Elephants
Hard9
Graph, Math
No attempts yet
Time limit
3s
Memory limit
256 MB
Judgeable
Magic Box
Hard9
String matching, String+1
No attempts yet
Time limit
5s
Memory limit
512 MB
Judgeable
Swapping Inversions
Hard9
Probability, Combinatorics+2
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
Minimal Cyclic Shift
Hard9
String, Number theory+2
No attempts yet
Time limit
1.5s
Memory limit
256 MB
Judgeable
Interval
Hard9
Divide and conquer, Segment tree+2
No attempts yet
Time limit
4s
Memory limit
512 MB
Judgeable
Nondeterministic Finite Automaton
Hard9
Graph, Number theory+1
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
The Struggle
Hard9
Bit manipulation, Math+1
No attempts yet
Time limit
4.5s
Memory limit
512 MB
Judgeable
Power Station of Art
Hard9
Graph, Union-find+1
No attempts yet
Time limit
3s
Memory limit
512 MB
Judgeable
Game
Hard9
Game theory, Intervals+1
No attempts yet
Time limit
8s
Memory limit
512 MB
Judgeable
Goldberg Machine 2
Hard9
Math, Segment tree+1
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Rational Dimasik
Hard9
Number theory, Math+2
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Lamb's Respite
Hard9
Segment tree, Implementation+2
No attempts yet
Time limit
3s
Memory limit
1024 MB
Judgeable
Deer-Proof Fence
Hard9
Geometry, Brute force+2
No attempts yet
Time limit
2s
Memory limit
1024 MB
Judgeable
Jail or Joyride
Hard9
Graph, Shortest path+1
No attempts yet
Time limit
4s
Memory limit
1024 MB
Judgeable
Kinking Cables
Hard9
Geometry, Implementation+2
No attempts yet
Time limit
1s
Memory limit
1024 MB
Judgeable
Shooting
Hard9
Geometry, Math+2
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
Black and White
Hard9
Combinatorics, Math
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
Dirichlet k-th root
Hard9
Number theory, Math+2
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
King
Hard9
Number theory, Math+1
No attempts yet
Time limit
2s
Memory limit
256 MB
Judgeable
Moon
Hard9
Geometry, Probability+1
No attempts yet
Time limit
2s
Memory limit
256 MB
Judgeable
Permutation
Hard9
Dynamic programming, Combinatorics
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
Travel
Hard9
Graph, Dynamic programming+1
No attempts yet
Time limit
2.5s
Memory limit
256 MB
Judgeable
Ternary String Revolution
Hard9
String, Math+2
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
Stone Game
Hard9
Game theory, Number theory+2
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
Fraction Reduction
Hard9
Math, Number theory+2
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
Notebook
Hard9
Math, Bit manipulation+2
No attempts yet
Time limit
1s
Memory limit
128 MB
Judgeable
Very Simple Sum
Hard9
Math, Combinatorics+2
No attempts yet
Time limit
3s
Memory limit
256 MB
Judgeable
String Theory
Hard9
String, String matching+2
No attempts yet
Time limit
4s
Memory limit
512 MB
Judgeable
Grid Triangle
Hard9
Math, Number theory+2
No attempts yet
Time limit
0.5s
Memory limit
1024 MB
Judgeable
Fibonaccis' vouchers
Hard9
Math, Combinatorics+2
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
Road
Hard9
Segment tree, Math+1
No attempts yet
Time limit
4s
Memory limit
512 MB
Judgeable
Subsequences
Hard9
Dynamic programming, Combinatorics+2
No attempts yet
Time limit
5s
Memory limit
512 MB
Judgeable
Modular Knapsack
Hard9
Dynamic programming, Number theory+1
No attempts yet
Time limit
1.5s
Memory limit
256 MB
Judgeable
Handsome multisets
Hard9
Number theory, Combinatorics+2
No attempts yet
Time limit
15s
Memory limit
256 MB
Judgeable
Polynomial
Hard9
Math, Combinatorics+2
No attempts yet
Time limit
3s
Memory limit
256 MB
Judgeable
Heretical Möbius
Hard9
Number theory, String matching+1
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
Immortal Universe
Hard9
Dynamic programming, Combinatorics
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Philosophical Balance
Hard9
String, Game theory+2
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
Eventual Journey
Hard9
Graph, Shortest path+2
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
Labeled Connected Graphs
Hard9
Combinatorics, Dynamic programming+2
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Kunyavskiy Pavel
Hard9
Game theory, Dynamic programming+2
No attempts yet
Time limit
3s
Memory limit
512 MB
Judgeable
A Simple Tree Problem
Hard9
Tree, DFS+2
No attempts yet
Time limit
8s
Memory limit
1024 MB
Judgeable
Server Expansion
Hard9
Combinatorics, Dynamic programming+2
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
Kućice
Hard9
Geometry, Combinatorics+2
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
Yet Another Minimax Problem
Hard9
Geometry, Binary search+1
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Bracelet Crossings
Hard9
Geometry, Implementation+2
No attempts yet
Time limit
2s
Memory limit
1024 MB
Judgeable
Tearing Barcodes
Hard9
String, Greedy+2
No attempts yet
Time limit
1s
Memory limit
1024 MB
Judgeable
Linear Algebra
Hard9
Geometry, Segment tree+1
No attempts yet
Time limit
2s
Memory limit
1024 MB
Judgeable
Sequence
Hard9
Bit manipulation, Math+2
No attempts yet
Time limit
1s
Memory limit
1024 MB
Judgeable
Abduction
Hard9
Dynamic programming, Combinatorics+1
No attempts yet
Time limit
1s
Memory limit
1024 MB
Judgeable
Rabbit
Hard9
Game theory, Simulation
No attempts yet
Time limit
Not set
Memory limit
1024 MB
Judgeable
Pretty sequences
Hard9
Combinatorics, Math+2
No attempts yet
Time limit
0.4s
Memory limit
1024 MB
Judgeable
Game on two heaps
Hard9
Game theory, Math+2
No attempts yet
Time limit
2s
Memory limit
1024 MB
Judgeable
Uniform String
Hard9
String, Implementation+2
No attempts yet
Time limit
1s
Memory limit
1024 MB
Judgeable
Avoiding Asteroids
Hard9
Geometry, Math
No attempts yet
Time limit
1s
Memory limit
1024 MB
Judgeable
Integer Graph
Hard9
Number theory, Graph+2
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
Even Substrings
Hard9
Segment tree, Bit manipulation+2
No attempts yet
Time limit
7s
Memory limit
1024 MB
Judgeable
Problem Set Construction
Hard9
Probability, Combinatorics+2
No attempts yet
Time limit
1s
Memory limit
1024 MB
Judgeable
Super Nine
Hard9
Combinatorics, Backtracking+2
No attempts yet
Time limit
1s
Memory limit
512 MB
Judgeable
Notability Index
Hard9
Number theory, Prefix sum+2
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
First to Solve
Hard9
Probability, Combinatorics+2
No attempts yet
Time limit
5s
Memory limit
512 MB
Judgeable
High Powers
Hard9
Math, Combinatorics+2
No attempts yet
Time limit
2s
Memory limit
1024 MB
Judgeable
Inversions
Hard9
Math, Combinatorics
No attempts yet
Time limit
3s
Memory limit
256 MB
Judgeable
Mountains
Hard9
Dynamic programming, Combinatorics+1
No attempts yet
Time limit
1s
Memory limit
256 MB
Judgeable
Casual Dancers
Hard9
Probability, Dynamic programming+1
No attempts yet
Time limit
4s
Memory limit
512 MB
Judgeable
First Occurrence
Hard9
String matching, Bit manipulation+2
No attempts yet
Time limit
2s
Memory limit
512 MB
Judgeable
Mismatch
Hard9
Bit manipulation, Combinatorics+1
No attempts yet
Time limit
4s
Memory limit
512 MB
Judgeable
Find the Magic Bead
Hard9
Dynamic programming, Divide and conquer+1
No attempts yet
Time limit
2s
Memory limit
1024 MB
Judgeable
Security System
Hard9
Graph, Geometry
No attempts yet
Time limit
2.5s
Memory limit
1024 MB
Judgeable
Tree GCD
Hard9
Tree, Divide and conquer+1
No attempts yet
Time limit
2.5s
Memory limit
1024 MB
Judgeable
Leaderboard Effect
Hard9
Dynamic programming, Probability+2
No attempts yet
Time limit
3s
Memory limit
1024 MB
Judgeable
Uplifting Excursion
Hard9
Dynamic programming, Math
No attempts yet
Time limit
4s
Memory limit
512 MB
Judgeable
Race for the Galaxy
Hard9
Dynamic programming, Combinatorics+1
No attempts yet
Time limit
4s
Memory limit
1024 MB
Judgeable
Gravity Hackenbush
Hard9
Graph, Game theory+1
No attempts yet
Time limit
2s
Memory limit
1024 MB
Judgeable
Hide-and-Seek with a Spirit
Hard9
Implementation, Bit manipulation+1
No attempts yet
Time limit
1s
Memory limit
1024 MB
Judgeable
Nieun Forest Artist
Hard9
Combinatorics, Dynamic programming+1
No attempts yet
Time limit
2s
Memory limit
1024 MB
Judgeable
Simple Problem (Large)
Hard9
Number theory, Math+1
No attempts yet
Time limit
2.4s
Memory limit
1024 MB
Judgeable
Lord of the Characteristic Polynomials (1)
Hard9
Matrix, Number theory+1
No attempts yet
Time limit
5s
Memory limit
1024 MB
Judgeable
Younghee's Errand
Hard9
Combinatorics, Dynamic programming+1
No attempts yet
Time limit
2s
Memory limit
1024 MB
Judgeable
Make Numbers with Division
Hard9
Math, Combinatorics+1
No attempts yet
Time limit
1s
Memory limit
1024 MB
Judgeable
Floor Tiles in a Park
Hard9
Combinatorics, Math+1
No attempts yet
Time limit
1s
Memory limit
1024 MB
Judgeable

Title	Level	Topics	Solved	Time limit	Memory limit	Judge
SolveMeCount pairs of maps A and B on N rooms where following A X times, then B, A Y times, B, and A Z times returns every room to its start, modulo 1,000,000,007.	Hard9	CombinatoricsMath+1	No attempts yet	8s	512 MB	Judgeable
Webby SubwayGiven N polyline subway lines, each a chain of segments, find the minimum number of floors so that lines placed on the same floor have no intersecting or touching segments.	Hard9	GeometryGraph+2	No attempts yet	8s	512 MB	Judgeable
Exact ArithmeticSimulate a stack calculator over numbers of the form a/b plus c/d times sqrt(r), maintaining exact arithmetic and canonical output formatting.	Hard9	ImplementationSimulation+2	No attempts yet	8s	512 MB	Judgeable
Cut outGiven a convex polyhedron as polygonal faces, find the maximum area of its cross section by a plane perpendicular to the z-axis.	Hard9	GeometryMath+2	No attempts yet	8s	512 MB	Judgeable
Strange KeyDecide whether two descriptions of a 3D polyline key, each a set of paths of unit axis-parallel bars, define congruent shapes under rotation and translation.	Hard9	GraphGeometry+2	No attempts yet	2s	512 MB	Judgeable
Bridge Crossing GameTwo players own private sets of directed bridges; from each start island, decide who wins or whether the game never ends, with optimal play.	Hard9	GraphGame theory+2	No attempts yet	1s	1024 MB	Judgeable
Pillar and Wall Defense GameGiven pillars, the wall is the convex hull; each expansion ticket adds rotated pillars at angle a, and we choose the ticket order to maximize the final hull perimeter.	Hard9	GeometryBrute force+2	No attempts yet	1s	1024 MB	Judgeable
Game of the RetiredGiven K grid boards fenced by two non-touching R/D paths, decide who wins the token game where Jeyeon moves right and Deokin moves down.	Hard9	Game theoryString+1	No attempts yet	1s	1024 MB	Judgeable
Slide CircuitsAfter each range-and-multiple enable or disable operation on slides, report the single disabled slide whose activation makes every building lie on exactly one directed cycle, or X if none does.	Hard9	GraphUnion-find+2	No attempts yet	Not set	1024 MB	Judgeable
Permutation CFGStarting from n and applying a permutation-defined CFG expansion s times, answer queries counting occurrences of integer k in a prefix of the resulting list.	Hard9	RecursionCombinatorics+2	No attempts yet	4s	2048 MB	Judgeable
Sinchon Defense HeadquartersGiven N missile points, count the lattice points that lie in the convex hull of the missiles (including boundary) but are not shield locations.	Hard9	GeometryMath+2	No attempts yet	1.5s	1024 MB	Judgeable
Master of String ManipulationCount distinct strings reachable from S after exactly M single-letter increments (z stays z), modulo 1e9+7, with N up to 300 and M up to 1e18.	Hard9	Dynamic programmingCombinatorics+2	No attempts yet	2.5s	1024 MB	Judgeable
Binary OperatorGroup fully parenthesized expressions with +, * and an arbitrary binary operator # into equivalence classes that agree for every possible meaning of #, then output the lexicographically smallest class numbering.	Hard9	MathHash map+2	No attempts yet	20s	1024 MB	Judgeable
Pizzo CollectorsAssign a category to each '?' house on a cycle of length N (a prime power) and choose non-conflicting collectors, each walking a fixed stride, to maximize total pizzo.	Hard9	Number theoryMath+2	No attempts yet	2s	1024 MB	Judgeable
Tobacco GrowingConstruct a configuration of planted, grass, and flower tiles so that after at most 100 days the harvestable total equals a given N up to 10^18.	Hard9	Dynamic programmingCombinatorics+2	No attempts yet	2s	1024 MB	Judgeable
How to Move the BeansOn a cylindrical grid of dishes, Alice and Bob alternately move beans that may never return to their own previous squares; the task is to find the winner under optimal play.	Hard9	Game theoryGraph+1	No attempts yet	2s	512 MB	Judgeable
Kingdoms and QuarantineGiven a bipartite graph of roads between two kingdoms, find the most roads that can be closed under parity rules and print one valid closing order.	Hard9	MathGraph+1	No attempts yet	8s	512 MB	Judgeable
Crab's CannonGiven a subset of the palindromic prefix lengths of an unknown string of length l, find the minimum number of palindromic prefixes such a string can have.	Hard9	StringString matching+1	No attempts yet	3s	512 MB	Judgeable
Intellectual ImplementationCount the triples of axis-parallel rectangles, with all coordinates distinct, where no two rectangles in a triple share any point.	Hard9	MathSegment tree+1	No attempts yet	6s	512 MB	Judgeable
ElephantsColor each of n elephants black or white so that every group's black and white counts differ by at most 1, or report -1 if no coloring exists.	Hard9	GraphMath	No attempts yet	3s	256 MB	Judgeable
Magic BoxGiven a string, count pairs of intervals that spell the same text, grouped by how many squares the two intervals activate, for each count from 0 to n.	Hard9	String matchingString+1	No attempts yet	5s	512 MB	Judgeable
Swapping InversionsSort a permutation by repeatedly swapping a uniformly random adjacent inversion, where each swap costs the absolute difference of the two values; find the expected total cost modulo 1e9+7.	Hard9	ProbabilityCombinatorics+2	No attempts yet	1s	256 MB	Judgeable
Minimal Cyclic ShiftGiven n lengths of random lowercase strings, compute the expected number of answers that still match after the answers are written in a shifted order, modulo 998244353.	Hard9	StringNumber theory+2	No attempts yet	1.5s	256 MB	Judgeable
IntervalGiven n intervals and m queries [A,B], find the expected union length of intervals over a uniformly random sub-range [L,R] within [A,B], modulo 998244353.	Hard9	Divide and conquerSegment tree+2	No attempts yet	4s	512 MB	Judgeable
Nondeterministic Finite AutomatonBuild an n-state NFA over 0 and 1 so that the shortest string it cannot recognize is longer than 18 for n=6 and longer than 400 for n=20.	Hard9	GraphNumber theory+1	No attempts yet	1s	256 MB	Judgeable
The StruggleGiven an ellipse with integer coefficients, compute the sum of (x XOR y)^33 / (x^2 * y) modulo 1e9+7 over all lattice points inside it.	Hard9	Bit manipulationMath+1	No attempts yet	4.5s	512 MB	Judgeable
Power Station of ArtGiven two numbered, red-or-black colored copies of the same graph, decide whether repeated edge swaps can make both copies identical.	Hard9	GraphUnion-find+1	No attempts yet	3s	512 MB	Judgeable
GameKoishi and Satori take turns removing the left or right end of an array, with special sub-segments set as wins or losses; decide the winner of each query [a, b].	Hard9	Game theoryIntervals+1	No attempts yet	8s	512 MB	Judgeable
Goldberg Machine 2After each flip of one arrow in either grid, report the fewest tokens needed for both grids to reach the same arrow configuration, or -1.	Hard9	MathSegment tree+1	No attempts yet	2s	512 MB	Judgeable
Rational DimasikGiven n fractions, compute the product, over all pairs, of the denominator of their difference written in lowest terms, modulo 998244353.	Hard9	Number theoryMath+2	No attempts yet	2s	512 MB	Judgeable
Lamb's RespiteFor each query, a champion starts at health x with Lamb's Respite clamping health to ceil(x/10) during actions l..r; report final health after all n actions, with point updates to a_i.	Hard9	Segment treeImplementation+2	No attempts yet	3s	1024 MB	Judgeable
Deer-Proof FenceGiven up to 9 points and a margin M, partition them into groups and fence each group with the shortest curve keeping distance at least M from every point, minimizing total length.	Hard9	GeometryBrute force+2	No attempts yet	2s	1024 MB	Judgeable
Jail or JoyridePolice chase teenagers who jump to the farthest reachable junction whenever the police arrive. Find the minimum distance for guaranteed capture, or impossible.	Hard9	GraphShortest path+1	No attempts yet	4s	1024 MB	Judgeable
Kinking CablesBuild a non-self-intersecting polyline from (0,0) to (n,m) inside the room, with marked points at least 1 apart, whose total length is exactly a given value.	Hard9	GeometryImplementation+2	No attempts yet	1s	1024 MB	Judgeable
ShootingFind the minimum launch speed for a projectile under gravity that starts at the first polyline vertex, ends at the last, and stays above every vertex in between.	Hard9	GeometryMath+2	No attempts yet	1s	256 MB	Judgeable
Black and WhiteCount monotone lattice paths from (0,0) to (n,m) whose white-minus-black cell score to the left equals k, modulo 998244353.	Hard9	CombinatoricsMath	No attempts yet	1s	256 MB	Judgeable
Dirichlet k-th rootGiven g on 1..n over F_p with g(1)=1, find f with f(1)=1 such that the k-fold Dirichlet convolution of f equals g, or report no solution.	Hard9	Number theoryMath+2	No attempts yet	1s	256 MB	Judgeable
KingGiven a prime p and a sequence B, find the longest subsequence where each term is the previous term times a fixed q mod p, or print -1 if it is under half of n.	Hard9	Number theoryMath+1	No attempts yet	2s	256 MB	Judgeable
MoonGiven n fixed points on a unit sphere, find the probability that a uniformly random extra point lets all points fit inside one hemisphere.	Hard9	GeometryProbability+1	No attempts yet	2s	256 MB	Judgeable
PermutationCount the distinct permutations reachable from a given one by rearranging elements next to an interval's minimum at either end, modulo 998244353.	Hard9	Dynamic programmingCombinatorics	No attempts yet	1s	256 MB	Judgeable
TravelCount pairs of paths covering every vertex of a directed graph with at most one cycle per vertex, each vertex used at most k times in total, mod 998244353.	Hard9	GraphDynamic programming+1	No attempts yet	2.5s	256 MB	Judgeable
Ternary String RevolutionGiven a rewrite system on ternary strings, count for each query string the substrings of s that can be transformed into it.	Hard9	StringMath+2	No attempts yet	1s	256 MB	Judgeable
Stone GameFor piles (a,b) with 1<=a<=n and 1<=b<=m, count how many positions are a first-player win, a draw, or a second-player win, modulo 1e9+7, with n and m given as binary strings.	Hard9	Game theoryNumber theory+2	No attempts yet	1s	256 MB	Judgeable
Fraction ReductionFor each fraction a/b, find the minimum number of steps (negate-and-reciprocal, or add 1) to reach 0, modulo 1e9+7, or -1 if impossible.	Hard9	MathNumber theory+2	No attempts yet	1s	256 MB	Judgeable
NotebookGiven a set of integers closed under doubling, halving when even, and xor, find the smallest reachable value; answer range queries with point updates.	Hard9	MathBit manipulation+2	No attempts yet	1s	128 MB	Judgeable
Very Simple SumSum (a_x+a_y+a_z+a_w)^(b_x xor b_y xor b_z xor b_w) over all n^4 index quadruples, modulo 998244353, with a_i, b_i at most 500.	Hard9	MathCombinatorics+2	No attempts yet	3s	256 MB	Judgeable
String TheoryCount substrings of S that equal some nonempty string repeated exactly k times, counting each occurrence position separately.	Hard9	StringString matching+2	No attempts yet	4s	512 MB	Judgeable
Grid TriangleCount triangles with integer vertices, one at the origin, inside a box, where each pair is a diagonal of an X by Y by Z cuboid for three distinct positive integers X, Y, Z.	Hard9	MathNumber theory+2	No attempts yet	0.5s	1024 MB	Judgeable
Fibonaccis' vouchersGiven k and n, find the n-th smallest number expressible as a sum of exactly k Fibonacci values, or report that it exceeds 10^18.	Hard9	MathCombinatorics+2	No attempts yet	1s	256 MB	Judgeable
RoadProcess plow, salt, and blizzard events on a road of n segments and report the maximum snow depth on a range at a given minute, modulo 1e9+7.	Hard9	Segment treeMath+1	No attempts yet	4s	512 MB	Judgeable
SubsequencesGiven up to 20 short strings, count the permutations whose concatenation has an even number of distinct subsequences.	Hard9	Dynamic programmingCombinatorics+2	No attempts yet	5s	512 MB	Judgeable
Modular KnapsackGiven n items with weights and costs and a prime p, find for each remainder mod p the maximum total cost of a subset whose weight sum has that remainder.	Hard9	Dynamic programmingNumber theory+1	No attempts yet	1.5s	256 MB	Judgeable
Handsome multisetsCount, over all multisets of positive integers summing to n that uniquely represent every value 1 to n as a subset sum, the total number of elements, modulo a prime near 1e9.	Hard9	Number theoryCombinatorics+2	No attempts yet	15s	256 MB	Judgeable
PolynomialGiven a 0/1 polynomial of degree at most 20 and a huge n up to 1e16, count the odd coefficients in P(x)^n.	Hard9	MathCombinatorics+2	No attempts yet	3s	256 MB	Judgeable
Heretical MöbiusGiven a 200-character binary string t, find the first position p where t appears in the sequence \|mu(1)\|, \|mu(2)\|, ... up to 10^9, or report -1.	Hard9	Number theoryString matching+1	No attempts yet	1s	256 MB	Judgeable
Immortal UniverseCount the ways to fill the '?' characters in two P/V strings so the boy never goes bankrupt, modulo 998244353.	Hard9	Dynamic programmingCombinatorics	No attempts yet	2s	512 MB	Judgeable
Philosophical BalanceGiven a string, choose a probability distribution over its suffixes to maximize the minimum expected LCP against any suffix the guard picks.	Hard9	StringGame theory+2	No attempts yet	1s	256 MB	Judgeable
Eventual JourneyIn a connected graph whose vertices are colored 0 or 1, moving inside one color costs nothing and each edge costs one ticket; for every vertex, sum the ticket distances to all other vertices.	Hard9	GraphShortest path+2	No attempts yet	1s	256 MB	Judgeable
Labeled Connected GraphsSum the distance between vertices 1 and 2 over every labeled connected graph on n vertices, modulo a prime m.	Hard9	CombinatoricsDynamic programming+2	No attempts yet	2s	512 MB	Judgeable
Kunyavskiy PavelCount, over all assignments of integer pairs to the leaves of a binary tree, the total number of pure-strategy Nash equilibria of a two-player game where Carol picks children at even depth and David at odd depth.	Hard9	Game theoryDynamic programming+2	No attempts yet	3s	512 MB	Judgeable
A Simple Tree ProblemMaintain a weighted tree under point updates to vertex and edge weights, and after each update report the sum over all paths of (vertex-weight sum) times (edge-weight sum).	Hard9	TreeDFS+2	No attempts yet	8s	1024 MB	Judgeable
Server ExpansionCount the ways to choose some of a distinguishable disks of each size 2^0..2^(k-1) so the chosen capacities sum to exactly n, modulo 1048573.	Hard9	CombinatoricsDynamic programming+2	No attempts yet	1s	512 MB	Judgeable
KućiceGiven n points in general position, each chosen independently with probability 1/2, find the expected number of points inside the convex hull of the chosen subset, output as the numerator over 2^n modulo 1e9+7.	Hard9	GeometryCombinatorics+2	No attempts yet	1s	512 MB	Judgeable
Yet Another Minimax ProblemGiven n integer points, pick a line with points on both sides so that the smallest distance from any point to the line is as large as possible, and print that distance.	Hard9	GeometryBinary search+1	No attempts yet	2s	512 MB	Judgeable
Bracelet CrossingsGiven vertical sweep readings of colors along M lines, decide whether the closed non-crossing polygonal chains could still exist.	Hard9	GeometryImplementation+2	No attempts yet	2s	1024 MB	Judgeable
Tearing BarcodesGiven a long bracket string (a pattern repeated many times) and a second string attached to each purchased drink, find the maximum number of drinks you can buy, where each drink costs K valid-barcode fragments and each purchase yields a new string to tear.	Hard9	StringGreedy+2	No attempts yet	1s	1024 MB	Judgeable
Linear AlgebraToggle 2D vectors in and out of a set, and report whether (a, b) is a sum of chosen vectors scaled by coefficients between 0 and 1.	Hard9	GeometrySegment tree+1	No attempts yet	2s	1024 MB	Judgeable
SequenceGiven the first m terms of a linear recurrence A_i = A_{i-1} + A_{i-m}, count how many of A_p through A_q are odd, with q up to 2^60.	Hard9	Bit manipulationMath+2	No attempts yet	1s	1024 MB	Judgeable
AbductionCount grid paths from the southwest corner to the northeast corner of a W by H street grid whose sequence of left and right turns matches a given string, with no U-turns and possible revisits, modulo 10^7.	Hard9	Dynamic programmingCombinatorics+1	No attempts yet	1s	1024 MB	Judgeable
RabbitGiven N cells and a rabbit that alternates S scared seconds and C curious seconds, output a fixed check sequence that finds the rabbit from any start.	Hard9	Game theorySimulation	No attempts yet	Not set	1024 MB	Judgeable
Pretty sequencesCount permutations of 1..N that contain at least one adjacent pair (x, x+1), modulo M, where N can reach 10^18 and M is up to 10^7 (not necessarily prime).	Hard9	CombinatoricsMath+2	No attempts yet	0.4s	1024 MB	Judgeable
Game on two heapsCount the first-player moves that win a subtraction game on two heaps, where a move taking x and y balls is illegal exactly when the ratio x:y equals the fixed starting ratio A:B.	Hard9	Game theoryMath+2	No attempts yet	2s	1024 MB	Judgeable
Uniform StringRepeatedly apply a reduction that maps zero-runs to 1 or 0 based on their lengths and count, and report when the string stops being uniform.	Hard9	StringImplementation+2	No attempts yet	1s	1024 MB	Judgeable
Avoiding AsteroidsDecide whether a ship with unknown speed can reach the base without hitting asteroids that move and tumble at unknown rates, printing Go or Surrender.	Hard9	GeometryMath	No attempts yet	1s	1024 MB	Judgeable
Integer GraphGiven n integers, find the minimum over removals of one value of the sum of pairwise shortest-path distances in a graph where each number connects to its smallest-prime-factor quotient.	Hard9	Number theoryGraph+2	No attempts yet	1s	512 MB	Judgeable
Even SubstringsGiven a string over six letters and point updates, count for each range query the substrings in which every distinct letter occurs an even number of times.	Hard9	Segment treeBit manipulation+2	No attempts yet	7s	1024 MB	Judgeable
Problem Set ConstructionFor each problem, compute the probability that a randomly chosen k-problem set contains it and the team solves it, given teams maximize count then minimize total time under limit t.	Hard9	ProbabilityCombinatorics+2	No attempts yet	1s	1024 MB	Judgeable
Super NineGiven some triples of players from 1 to 9, extend them to a Steiner triple system on 9 points, or report that none exists.	Hard9	CombinatoricsBacktracking+2	No attempts yet	1s	512 MB	Judgeable
Notability IndexGiven a prime P and a digit string T, answer queries that ask how many substrings of T[l..r] form numbers divisible by P.	Hard9	Number theoryPrefix sum+2	No attempts yet	2s	512 MB	Judgeable
First to SolveFor each contestant in a contest with random problem orderings, compute the expected number of problems where they are among the earliest solvers, modulo 998244353.	Hard9	ProbabilityCombinatorics+2	No attempts yet	5s	512 MB	Judgeable
High PowersGiven symmetric sums s, t, u of three distinct complex numbers, compute a symmetric alternating expression in a, b, c raised to huge powers n and m, modulo 998244353.	Hard9	MathCombinatorics+2	No attempts yet	2s	1024 MB	Judgeable
InversionsFor length-n permutations, the task is to sum inv(p)^k over all permutations modulo 998244353, with n up to 10^18 and k up to 1000.	Hard9	MathCombinatorics	No attempts yet	3s	256 MB	Judgeable
MountainsCount n by m grids of non-negative heights, each at most k, whose maximum-weight monotone path from (1,1) to (n,m) has total sum at most k.	Hard9	Dynamic programmingCombinatorics+1	No attempts yet	1s	256 MB	Judgeable
Casual DancersThree friends random-walk on a number line, one random friend moves each second (+1 with p percent); output the expected final span modulo 998244353.	Hard9	ProbabilityDynamic programming+1	No attempts yet	4s	512 MB	Judgeable
First OccurrenceGiven l and r up to 10^18, find the smallest index where the Thue-Morse substring t[l..r] first appears in the sequence.	Hard9	String matchingBit manipulation+2	No attempts yet	2s	512 MB	Judgeable
MismatchFor each k from 1 to n, count the size-k subsequences whose bitwise AND is zero, modulo 998244353.	Hard9	Bit manipulationCombinatorics+1	No attempts yet	4s	512 MB	Judgeable
Find the Magic BeadSplit k+1 beads into M pouches to minimize the worst-case cleanup cost of finding the magic bead, for every k from 0 to N-1.	Hard9	Dynamic programmingDivide and conquer+1	No attempts yet	2s	1024 MB	Judgeable
Security SystemChoose which laser sensors to activate so that their beams to the walls never meet, maximizing the total importance of the chosen sensors.	Hard9	GraphGeometry	No attempts yet	2.5s	1024 MB	Judgeable
Tree GCDGiven a tree with N vertices labeled 1 to N, compute the sum of gcd(i, j, dist(i, j)) over all pairs i less than j.	Hard9	TreeDivide and conquer+1	No attempts yet	2.5s	1024 MB	Judgeable
Leaderboard EffectCompute the limiting proportion of teams that solve each problem when teams choose unread problems according to leaderboard solution counts.	Hard9	Dynamic programmingProbability+2	No attempts yet	3s	1024 MB	Judgeable
Uplifting ExcursionGiven how many art pieces have each uplift from -M to M, find the maximum number you can pick so their uplifts sum to exactly L, or report impossible.	Hard9	Dynamic programmingMath	No attempts yet	4s	512 MB	Judgeable
Race for the GalaxyCount non-intersecting monotone grid paths for N runners with puddles and mud rows, grouped by how many runners cross the mud row S to S+1.	Hard9	Dynamic programmingCombinatorics+1	No attempts yet	4s	1024 MB	Judgeable
Gravity HackenbushGiven a graph of points grounded at y=0 with red, blue, and green segments, determine the winner when both players cut optimally.	Hard9	GraphGame theory+1	No attempts yet	2s	1024 MB	Judgeable
Hide-and-Seek with a SpiritThe spirit fills a 200x200 grid with values 1 to 24 so the seeker can find each of nine hiders' quadrants using one Find call and four Get calls.	Hard9	ImplementationBit manipulation+1	No attempts yet	1s	1024 MB	Judgeable
Nieun Forest ArtistCount the ways to tile an N by N square with L-shaped pieces of sizes 1 to N, where equal-town pieces never share an edge, up to rotation, modulo 998244353.	Hard9	CombinatoricsDynamic programming+1	No attempts yet	2s	1024 MB	Judgeable
Simple Problem (Large)Count triples (x, y, z) with x up to a, y up to b, z up to c where x mod y, y mod z, and z mod x are all equal, for up to 600,000 queries.	Hard9	Number theoryMath+1	No attempts yet	2.4s	1024 MB	Judgeable
Lord of the Characteristic Polynomials (1)Given an n by n integer matrix and a modulus M, output the coefficients of its characteristic polynomial modulo M.	Hard9	MatrixNumber theory+1	No attempts yet	5s	1024 MB	Judgeable
Younghee's ErrandGiven an N by N grid of candies (o) and chocolates (x), find the average expected candy and chocolate counts over all target cells after each label flip.	Hard9	CombinatoricsDynamic programming+1	No attempts yet	2s	1024 MB	Judgeable
Make Numbers with DivisionGiven how many copies of each number 1 to 9 are available, count the distinct values reachable with division and parentheses, modulo 998244353.	Hard9	MathCombinatorics+1	No attempts yet	1s	1024 MB	Judgeable
Floor Tiles in a ParkCount the ways to draw grid-line segments on a W by H grid so that it splits into exactly k subrectangles, modulo 998244353.	Hard9	CombinatoricsMath+1	No attempts yet	1s	1024 MB	Judgeable