{"app":{"locale":"en-GB"},"problem":{"content":"

Let $a_k$, $b_k$, and $c_k$ represent the three solutions (real or complex numbers) to the equation$\\frac 1 x = (\\frac k x)^2(k+x^2)-k x$.

For instance, for $k=5$, we see that $\\{a_5, b_5, c_5 \\}$ is approximately $\\{5.727244, -0.363622+2.057397i, -0.363622-2.057397i\\}$.

Let $\\displaystyle S(n) = \\sum_{p=1}^n\\sum_{k=1}^n(a_k+b_k)^p(b_k+c_k)^p(c_k+a_k)^p$.

Interestingly, $S(n)$ is always an integer. For example, $S(4) = 51160$.

Find $S(10^6)$ modulo $1\\,000\\,000\\,007$.

","id":479,"title":"Roots on the Rise"},"problems":[{"id":1,"title":"Multiples of 3 and 5"},{"id":2,"title":"Even Fibonacci numbers"},{"id":3,"title":"Largest prime factor"},{"id":4,"title":"Largest palindrome product"},{"id":5,"title":"Smallest multiple"},{"id":6,"title":"Sum square difference"},{"id":7,"title":"10001st prime"},{"id":8,"title":"Largest product in a series"},{"id":9,"title":"Special Pythagorean triplet"},{"id":10,"title":"Summation of primes"},{"id":11,"title":"Largest product in a grid"},{"id":12,"title":"Highly divisible triangular number"},{"id":13,"title":"Large sum"},{"id":14,"title":"Longest Collatz sequence"},{"id":15,"title":"Lattice paths"},{"id":16,"title":"Power digit sum"},{"id":17,"title":"Number letter counts"},{"id":18,"title":"Maximum path sum I"},{"id":19,"title":"Counting Sundays"},{"id":20,"title":"Factorial digit sum"},{"id":21,"title":"Amicable numbers"},{"id":22,"title":"Names scores"},{"id":23,"title":"Non-abundant sums"},{"id":24,"title":"Lexicographic permutations"},{"id":25,"title":"1000-digit Fibonacci number"},{"id":26,"title":"Reciprocal cycles"},{"id":27,"title":"Quadratic primes"},{"id":28,"title":"Number spiral diagonals"},{"id":29,"title":"Distinct powers"},{"id":30,"title":"Digit fifth powers"},{"id":31,"title":"Coin sums"},{"id":32,"title":"Pandigital products"},{"id":33,"title":"Digit cancelling fractions"},{"id":34,"title":"Digit factorials"},{"id":35,"title":"Circular primes"},{"id":36,"title":"Double-base palindromes"},{"id":37,"title":"Truncatable primes"},{"id":38,"title":"Pandigital multiples"},{"id":39,"title":"Integer right triangles"},{"id":40,"title":"Champernowne's constant"},{"id":41,"title":"Pandigital prime"},{"id":42,"title":"Coded triangle numbers"},{"id":43,"title":"Sub-string divisibility"},{"id":44,"title":"Pentagon numbers"},{"id":45,"title":"Triangular, pentagonal, and hexagonal"},{"id":46,"title":"Goldbach's other conjecture"},{"id":47,"title":"Distinct primes factors"},{"id":48,"title":"Self powers"},{"id":49,"title":"Prime permutations"},{"id":50,"title":"Consecutive prime sum"},{"id":51,"title":"Prime digit replacements"},{"id":52,"title":"Permuted multiples"},{"id":53,"title":"Combinatoric selections"},{"id":54,"title":"Poker hands"},{"id":55,"title":"Lychrel numbers"},{"id":56,"title":"Powerful digit sum"},{"id":57,"title":"Square root convergents"},{"id":58,"title":"Spiral primes"},{"id":59,"title":"XOR decryption"},{"id":60,"title":"Prime pair sets"},{"id":61,"title":"Cyclical figurate numbers"},{"id":62,"title":"Cubic permutations"},{"id":63,"title":"Powerful digit counts"},{"id":64,"title":"Odd period square roots"},{"id":65,"title":"Convergents of e"},{"id":66,"title":"Diophantine equation"},{"id":67,"title":"Maximum path sum II"},{"id":68,"title":"Magic 5-gon ring"},{"id":69,"title":"Totient maximum"},{"id":70,"title":"Totient permutation"},{"id":71,"title":"Ordered fractions"},{"id":72,"title":"Counting fractions"},{"id":73,"title":"Counting fractions in a range"},{"id":74,"title":"Digit factorial chains"},{"id":75,"title":"Singular integer right triangles"},{"id":76,"title":"Counting summations"},{"id":77,"title":"Prime summations"},{"id":78,"title":"Coin partitions"},{"id":79,"title":"Passcode derivation"},{"id":80,"title":"Square root digital expansion"},{"id":81,"title":"Path sum: two ways"},{"id":82,"title":"Path sum: three ways"},{"id":83,"title":"Path sum: four ways"},{"id":84,"title":"Monopoly odds"},{"id":85,"title":"Counting rectangles"},{"id":86,"title":"Cuboid route"},{"id":87,"title":"Prime power triples"},{"id":88,"title":"Product-sum numbers"},{"id":89,"title":"Roman numerals"},{"id":90,"title":"Cube digit pairs"},{"id":91,"title":"Right triangles with integer coordinates"},{"id":92,"title":"Square digit chains"},{"id":93,"title":"Arithmetic expressions"},{"id":94,"title":"Almost equilateral triangles"},{"id":95,"title":"Amicable chains"},{"id":96,"title":"Su Doku"},{"id":97,"title":"Large non-Mersenne prime"},{"id":98,"title":"Anagramic squares"},{"id":99,"title":"Largest exponential"},{"id":100,"title":"Arranged probability"},{"id":101,"title":"Optimum polynomial"},{"id":102,"title":"Triangle containment"},{"id":103,"title":"Special subset sums: optimum"},{"id":104,"title":"Pandigital Fibonacci ends"},{"id":105,"title":"Special subset sums: testing"},{"id":106,"title":"Special subset sums: meta-testing"},{"id":107,"title":"Minimal network"},{"id":108,"title":"Diophantine reciprocals I"},{"id":109,"title":"Darts"},{"id":110,"title":"Diophantine reciprocals II"},{"id":111,"title":"Primes with runs"},{"id":112,"title":"Bouncy numbers"},{"id":113,"title":"Non-bouncy numbers"},{"id":114,"title":"Counting block combinations I"},{"id":115,"title":"Counting block combinations II"},{"id":116,"title":"Red, green or blue tiles"},{"id":117,"title":"Red, green, and blue tiles"},{"id":118,"title":"Pandigital prime sets"},{"id":119,"title":"Digit power sum"},{"id":120,"title":"Square remainders"},{"id":121,"title":"Disc game prize fund"},{"id":122,"title":"Efficient exponentiation"},{"id":123,"title":"Prime square remainders"},{"id":124,"title":"Ordered radicals"},{"id":125,"title":"Palindromic sums"},{"id":126,"title":"Cuboid layers"},{"id":127,"title":"abc-hits"},{"id":128,"title":"Hexagonal tile differences"},{"id":129,"title":"Repunit divisibility"},{"id":130,"title":"Composites with prime repunit property"},{"id":131,"title":"Prime cube partnership"},{"id":132,"title":"Large repunit factors"},{"id":133,"title":"Repunit nonfactors"},{"id":134,"title":"Prime pair connection"},{"id":135,"title":"Same differences"},{"id":136,"title":"Singleton difference"},{"id":137,"title":"Fibonacci golden nuggets"},{"id":138,"title":"Special isosceles triangles"},{"id":139,"title":"Pythagorean tiles"},{"id":140,"title":"Modified Fibonacci golden nuggets"},{"id":141,"title":"Investigating progressive numbers, n, which are also square"},{"id":142,"title":"Perfect Square Collection"},{"id":143,"title":"Investigating the Torricelli point of a triangle"},{"id":144,"title":"Investigating multiple reflections of a laser beam"},{"id":145,"title":"How many reversible numbers are there below one-billion?"},{"id":146,"title":"Investigating a Prime Pattern "},{"id":147,"title":"Rectangles in cross-hatched grids"},{"id":148,"title":"Exploring Pascal's triangle"},{"id":149,"title":"Searching for a maximum-sum subsequence"},{"id":150,"title":"Searching a triangular array for a sub-triangle having minimum-sum"},{"id":151,"title":"Paper sheets of standard sizes: an expected-value problem"},{"id":152,"title":"Writing 1/2 as a sum of inverse squares"},{"id":153,"title":"Investigating Gaussian Integers"},{"id":154,"title":"Exploring Pascal's pyramid"},{"id":155,"title":"Counting Capacitor Circuits"},{"id":156,"title":"Counting Digits"},{"id":157,"title":"Solving the diophantine equation 1/a+1/b= p/10n "},{"id":158,"title":"Exploring strings for which only one character comes lexicographically after its neighbour to the left"},{"id":159,"title":"Digital root sums of factorisations"},{"id":160,"title":"Factorial trailing digits"},{"id":161,"title":"Triominoes"},{"id":162,"title":"Hexadecimal numbers"},{"id":163,"title":"Cross-hatched triangles"},{"id":164,"title":"Numbers for which no three consecutive digits have a sum greater than a given value"},{"id":165,"title":"Intersections"},{"id":166,"title":"Criss Cross"},{"id":167,"title":"Investigating Ulam sequences"},{"id":168,"title":"Number Rotations"},{"id":169,"title":"Exploring the number of different ways a number can be expressed as a sum of powers of 2"},{"id":170,"title":"Find the largest 0 to 9 pandigital that can be formed by concatenating products"},{"id":171,"title":"Finding numbers for which the sum of the squares of the digits is a square"},{"id":172,"title":"Investigating numbers with few repeated digits"},{"id":173,"title":"Using up to one million tiles how many different \"hollow\" square laminae can be formed?"},{"id":174,"title":"Counting the number of \"hollow\" square laminae that can form one, two, three, ... distinct arrangements"},{"id":175,"title":"Fractions involving the number of different ways a number can be expressed as a sum of powers of 2"},{"id":176,"title":"Right-angled triangles that share a cathetus"},{"id":177,"title":"Integer angled Quadrilaterals"},{"id":178,"title":"Step Numbers"},{"id":179,"title":"Consecutive positive divisors"},{"id":180,"title":"Rational zeros of a function of three variables"},{"id":181,"title":"Investigating in how many ways objects of two different colours can be grouped"},{"id":182,"title":"RSA encryption"},{"id":183,"title":"Maximum product of parts"},{"id":184,"title":"Triangles containing the origin"},{"id":185,"title":"Number Mind"},{"id":186,"title":"Connectedness of a network"},{"id":187,"title":"Semiprimes"},{"id":188,"title":"The hyperexponentiation of a number"},{"id":189,"title":"Tri-colouring a triangular grid"},{"id":190,"title":"Maximising a weighted product"},{"id":191,"title":"Prize Strings"},{"id":192,"title":"Best Approximations"},{"id":193,"title":"Squarefree Numbers"},{"id":194,"title":"Coloured Configurations"},{"id":195,"title":"Inscribed circles of triangles with one angle of 60 degrees"},{"id":196,"title":"Prime triplets"},{"id":197,"title":"Investigating the behaviour of a recursively defined sequence"},{"id":198,"title":"Ambiguous Numbers"},{"id":199,"title":"Iterative Circle Packing"},{"id":200,"title":"Find the 200th prime-proof sqube containing the contiguous sub-string \"200\""},{"id":201,"title":"Subsets with a unique sum"},{"id":202,"title":"Laserbeam"},{"id":203,"title":"Squarefree Binomial Coefficients"},{"id":204,"title":"Generalised Hamming Numbers"},{"id":205,"title":"Dice Game"},{"id":206,"title":"Concealed Square"},{"id":207,"title":"Integer partition equations"},{"id":208,"title":"Robot Walks"},{"id":209,"title":"Circular Logic"},{"id":210,"title":"Obtuse Angled Triangles"},{"id":211,"title":"Divisor Square Sum"},{"id":212,"title":"Combined Volume of Cuboids"},{"id":213,"title":"Flea Circus"},{"id":214,"title":"Totient Chains"},{"id":215,"title":"Crack-free Walls"},{"id":216,"title":"Investigating the primality of numbers of the form 2n2-1"},{"id":217,"title":"Balanced Numbers"},{"id":218,"title":"Perfect right-angled triangles"},{"id":219,"title":"Skew-cost coding"},{"id":220,"title":"Heighway Dragon"},{"id":221,"title":"Alexandrian Integers"},{"id":222,"title":"Sphere Packing"},{"id":223,"title":"Almost right-angled triangles I"},{"id":224,"title":"Almost right-angled triangles II"},{"id":225,"title":"Tribonacci non-divisors"},{"id":226,"title":"A Scoop of Blancmange"},{"id":227,"title":"The Chase"},{"id":228,"title":"Minkowski Sums"},{"id":229,"title":"Four Representations using Squares"},{"id":230,"title":"Fibonacci Words"},{"id":231,"title":"The prime factorisation of binomial coefficients"},{"id":232,"title":"The Race"},{"id":233,"title":"Lattice points on a circle"},{"id":234,"title":"Semidivisible numbers"},{"id":235,"title":"An Arithmetic Geometric sequence"},{"id":236,"title":"Luxury Hampers"},{"id":237,"title":"Tours on a 4 x n playing board"},{"id":238,"title":"Infinite string tour"},{"id":239,"title":"Twenty-two Foolish Primes"},{"id":240,"title":"Top Dice"},{"id":241,"title":"Perfection Quotients"},{"id":242,"title":"Odd Triplets"},{"id":243,"title":"Resilience"},{"id":244,"title":"Sliders"},{"id":245,"title":"Coresilience"},{"id":246,"title":"Tangents to an ellipse"},{"id":247,"title":"Squares under a hyperbola"},{"id":248,"title":"Numbers for which Euler’s totient function equals 13!"},{"id":249,"title":"Prime Subset Sums"},{"id":250,"title":"250250"},{"id":251,"title":"Cardano Triplets"},{"id":252,"title":"Convex Holes"},{"id":253,"title":"Tidying up"},{"id":254,"title":"Sums of Digit Factorials"},{"id":255,"title":"Rounded Square Roots"},{"id":256,"title":"Tatami-Free Rooms"},{"id":257,"title":"Angular Bisectors"},{"id":258,"title":"A lagged Fibonacci sequence"},{"id":259,"title":"Reachable Numbers"},{"id":260,"title":"Stone Game"},{"id":261,"title":"Pivotal Square Sums"},{"id":262,"title":"Mountain Range"},{"id":263,"title":"An engineers' dream come true"},{"id":264,"title":"Triangle Centres"},{"id":265,"title":"Binary Circles"},{"id":266,"title":"Pseudo Square Root"},{"id":267,"title":"Billionaire"},{"id":268,"title":"Counting numbers with at least four distinct prime factors less than 100"},{"id":269,"title":"Polynomials with at least one integer root"},{"id":270,"title":"Cutting Squares"},{"id":271,"title":"Modular Cubes, part 1"},{"id":272,"title":"Modular Cubes, part 2"},{"id":273,"title":"Sum of Squares"},{"id":274,"title":"Divisibility Multipliers"},{"id":275,"title":"Balanced Sculptures"},{"id":276,"title":"Primitive Triangles"},{"id":277,"title":"A Modified Collatz sequence"},{"id":278,"title":"Linear Combinations of Semiprimes"},{"id":279,"title":"Triangles with integral sides and an integral angle "},{"id":280,"title":"Ant and seeds"},{"id":281,"title":"Pizza Toppings"},{"id":282,"title":"The Ackermann function"},{"id":283,"title":"Integer sided triangles for which the area/perimeter ratio is integral"},{"id":284,"title":"Steady Squares"},{"id":285,"title":"Pythagorean odds"},{"id":286,"title":"Scoring probabilities"},{"id":287,"title":"Quadtree encoding (a simple compression algorithm)"},{"id":288,"title":"An enormous factorial"},{"id":289,"title":"Eulerian Cycles"},{"id":290,"title":"Digital Signature"},{"id":291,"title":"Panaitopol Primes"},{"id":292,"title":"Pythagorean Polygons"},{"id":293,"title":"Pseudo-Fortunate Numbers"},{"id":294,"title":"Sum of digits - experience #23"},{"id":295,"title":"Lenticular holes"},{"id":296,"title":"Angular Bisector and Tangent"},{"id":297,"title":"Zeckendorf Representation"},{"id":298,"title":"Selective Amnesia"},{"id":299,"title":"Three similar triangles"},{"id":300,"title":"Protein folding"},{"id":301,"title":"Nim"},{"id":302,"title":"Strong Achilles Numbers"},{"id":303,"title":"Multiples with small digits"},{"id":304,"title":"Primonacci"},{"id":305,"title":"Reflexive Position"},{"id":306,"title":"Paper-strip Game"},{"id":307,"title":"Chip Defects"},{"id":308,"title":"An amazing Prime-generating Automaton"},{"id":309,"title":"Integer Ladders"},{"id":310,"title":"Nim Square"},{"id":311,"title":"Biclinic Integral Quadrilaterals"},{"id":312,"title":"Cyclic paths on Sierpiński graphs"},{"id":313,"title":"Sliding game"},{"id":314,"title":"The Mouse on the Moon"},{"id":315,"title":"Digital root clocks"},{"id":316,"title":"Numbers in decimal expansions"},{"id":317,"title":"Firecracker"},{"id":318,"title":"2011 nines"},{"id":319,"title":"Bounded Sequences"},{"id":320,"title":"Factorials divisible by a huge integer"},{"id":321,"title":"Swapping Counters"},{"id":322,"title":"Binomial coefficients divisible by 10"},{"id":323,"title":"Bitwise-OR operations on random integers"},{"id":324,"title":"Building a tower"},{"id":325,"title":"Stone Game II"},{"id":326,"title":"Modulo Summations"},{"id":327,"title":"Rooms of Doom"},{"id":328,"title":"Lowest-cost Search"},{"id":329,"title":"Prime Frog"},{"id":330,"title":"Euler's Number"},{"id":331,"title":"Cross flips"},{"id":332,"title":"Spherical triangles"},{"id":333,"title":"Special partitions"},{"id":334,"title":"Spilling the beans"},{"id":335,"title":"Gathering the beans"},{"id":336,"title":"Maximix Arrangements"},{"id":337,"title":"Totient Stairstep Sequences"},{"id":338,"title":"Cutting Rectangular Grid Paper"},{"id":339,"title":"Peredur fab Efrawg"},{"id":340,"title":"Crazy Function"},{"id":341,"title":"Golomb's self-describing sequence"},{"id":342,"title":"The totient of a square is a cube"},{"id":343,"title":"Fractional Sequences"},{"id":344,"title":"Silver dollar game"},{"id":345,"title":"Matrix Sum"},{"id":346,"title":"Strong Repunits"},{"id":347,"title":"Largest integer divisible by two primes"},{"id":348,"title":"Sum of a square and a cube"},{"id":349,"title":"Langton's ant"},{"id":350,"title":"Constraining the least greatest and the greatest least"},{"id":351,"title":"Hexagonal orchards"},{"id":352,"title":"Blood tests"},{"id":353,"title":"Risky moon"},{"id":354,"title":"Distances in a bee's honeycomb "},{"id":355,"title":"Maximal coprime subset"},{"id":356,"title":"Largest roots of cubic polynomials"},{"id":357,"title":"Prime generating integers"},{"id":358,"title":"Cyclic numbers"},{"id":359,"title":"Hilbert's New Hotel"},{"id":360,"title":"Scary Sphere"},{"id":361,"title":"Subsequence of Thue-Morse sequence"},{"id":362,"title":"Squarefree factors"},{"id":363,"title":"Bézier Curves"},{"id":364,"title":"Comfortable distance"},{"id":365,"title":"A huge binomial coefficient"},{"id":366,"title":"Stone Game III"},{"id":367,"title":"Bozo sort"},{"id":368,"title":"A Kempner-like series"},{"id":369,"title":"Badugi"},{"id":370,"title":"Geometric triangles"},{"id":371,"title":"Licence plates"},{"id":372,"title":"Pencils of rays"},{"id":373,"title":"Circumscribed Circles"},{"id":374,"title":"Maximum Integer Partition Product"},{"id":375,"title":"Minimum of subsequences"},{"id":376,"title":"Nontransitive sets of dice"},{"id":377,"title":"Sum of digits, experience 13"},{"id":378,"title":"Triangle Triples"},{"id":379,"title":"Least common multiple count"},{"id":380,"title":"Amazing Mazes!"},{"id":381,"title":"(prime-k) factorial"},{"id":382,"title":"Generating polygons"},{"id":383,"title":"Divisibility comparison between factorials"},{"id":384,"title":"Rudin-Shapiro sequence"},{"id":385,"title":"Ellipses inside triangles"},{"id":386,"title":"Maximum length of an antichain"},{"id":387,"title":"Harshad Numbers"},{"id":388,"title":"Distinct Lines"},{"id":389,"title":"Platonic Dice"},{"id":390,"title":"Triangles with non rational sides and integral area"},{"id":391,"title":"Hopping Game"},{"id":392,"title":"Enmeshed unit circle"},{"id":393,"title":"Migrating ants"},{"id":394,"title":"Eating pie"},{"id":395,"title":"Pythagorean tree"},{"id":396,"title":"Weak Goodstein sequence"},{"id":397,"title":"Triangle on parabola"},{"id":398,"title":"Cutting rope"},{"id":399,"title":"Squarefree Fibonacci Numbers"},{"id":400,"title":"Fibonacci tree game"},{"id":401,"title":"Sum of squares of divisors"},{"id":402,"title":"Integer-valued polynomials"},{"id":403,"title":"Lattice points enclosed by parabola and line"},{"id":404,"title":"Crisscross Ellipses"},{"id":405,"title":"A rectangular tiling"},{"id":406,"title":"Guessing Game"},{"id":407,"title":"Idempotents"},{"id":408,"title":"Admissible paths through a grid"},{"id":409,"title":"Nim Extreme"},{"id":410,"title":"Circle and tangent line"},{"id":411,"title":"Uphill paths"},{"id":412,"title":"Gnomon numbering"},{"id":413,"title":"One-child Numbers"},{"id":414,"title":"Kaprekar constant"},{"id":415,"title":"Titanic sets"},{"id":416,"title":"A frog's trip"},{"id":417,"title":"Reciprocal cycles II"},{"id":418,"title":"Factorisation triples"},{"id":419,"title":"Look and say sequence"},{"id":420,"title":"2x2 positive integer matrix"},{"id":421,"title":"Prime factors of n15+1"},{"id":422,"title":"Sequence of points on a hyperbola"},{"id":423,"title":"Consecutive die throws"},{"id":424,"title":"Kakuro"},{"id":425,"title":"Prime connection"},{"id":426,"title":"Box-ball system"},{"id":427,"title":"n-sequences"},{"id":428,"title":"Necklace of circles"},{"id":429,"title":"Sum of squares of unitary divisors"},{"id":430,"title":"Range flips"},{"id":431,"title":"Square Space Silo"},{"id":432,"title":"Totient sum"},{"id":433,"title":"Steps in Euclid's algorithm"},{"id":434,"title":"Rigid graphs"},{"id":435,"title":"Polynomials of Fibonacci numbers"},{"id":436,"title":"Unfair wager"},{"id":437,"title":"Fibonacci primitive roots"},{"id":438,"title":"Integer part of polynomial equation's solutions"},{"id":439,"title":"Sum of sum of divisors"},{"id":440,"title":"GCD and Tiling"},{"id":441,"title":"The inverse summation of coprime couples"},{"id":442,"title":"Eleven-free integers"},{"id":443,"title":"GCD sequence"},{"id":444,"title":"The Roundtable Lottery"},{"id":445,"title":"Retractions A"},{"id":446,"title":"Retractions B"},{"id":447,"title":"Retractions C"},{"id":448,"title":"Average least common multiple"},{"id":449,"title":"Chocolate covered candy"},{"id":450,"title":"Hypocycloid and Lattice points"},{"id":451,"title":"Modular inverses"},{"id":452,"title":"Long Products"},{"id":453,"title":"Lattice Quadrilaterals"},{"id":454,"title":"Diophantine reciprocals III"},{"id":455,"title":"Powers With Trailing Digits"},{"id":456,"title":"Triangles containing the origin II"},{"id":457,"title":"A polynomial modulo the square of a prime"},{"id":458,"title":"Permutations of Project"},{"id":459,"title":"Flipping game"},{"id":460,"title":"An ant on the move"},{"id":461,"title":"Almost Pi"},{"id":462,"title":"Permutation of 3-smooth numbers"},{"id":463,"title":"A weird recurrence relation"},{"id":464,"title":"Möbius function and intervals"},{"id":465,"title":"Polar polygons"},{"id":466,"title":"Distinct terms in a multiplication table"},{"id":467,"title":"Superinteger"},{"id":468,"title":"Smooth divisors of binomial coefficients"},{"id":469,"title":"Empty chairs"},{"id":470,"title":"Super Ramvok"},{"id":471,"title":"Triangle inscribed in ellipse"},{"id":472,"title":"Comfortable Distance II"},{"id":473,"title":"Phigital number base"},{"id":474,"title":"Last digits of divisors"},{"id":475,"title":"Music festival"},{"id":476,"title":"Circle Packing II"},{"id":477,"title":"Number Sequence Game"},{"id":478,"title":"Mixtures"},{"id":479,"title":"Roots on the Rise"},{"id":480,"title":"The Last Question"},{"id":481,"title":"Chef Showdown"},{"id":482,"title":"The incenter of a triangle"},{"id":483,"title":"Repeated permutation"},{"id":484,"title":"Arithmetic Derivative"},{"id":485,"title":"Maximum number of divisors"},{"id":486,"title":"Palindrome-containing strings"},{"id":487,"title":"Sums of power sums"},{"id":488,"title":"Unbalanced Nim"},{"id":489,"title":"Common factors between two sequences"},{"id":490,"title":"Jumping frog"},{"id":491,"title":"Double pandigital number divisible by 11"},{"id":492,"title":"Exploding sequence"},{"id":493,"title":"Under The Rainbow"},{"id":494,"title":"Collatz prefix families"},{"id":495,"title":"Writing n as the product of k distinct positive integers"},{"id":496,"title":"Incenter and circumcenter of triangle"},{"id":497,"title":"Drunken Tower of Hanoi"},{"id":498,"title":"Remainder of polynomial division"},{"id":499,"title":"St. Petersburg Lottery"},{"id":500,"title":"Problem 500!!!"},{"id":501,"title":"Eight Divisors"},{"id":502,"title":"Counting Castles"},{"id":503,"title":"Compromise or persist"},{"id":504,"title":"Square on the Inside"},{"id":505,"title":"Bidirectional Recurrence"},{"id":506,"title":"Clock sequence"},{"id":507,"title":"Shortest Lattice Vector"},{"id":508,"title":"Integers in base i-1"},{"id":509,"title":"Divisor Nim"},{"id":510,"title":"Tangent Circles"},{"id":511,"title":"Sequences with nice divisibility properties"},{"id":512,"title":"Sums of totients of powers"},{"id":513,"title":"Integral median"},{"id":514,"title":"Geoboard Shapes"},{"id":515,"title":"Dissonant Numbers"},{"id":516,"title":"5-smooth totients"},{"id":517,"title":"A real recursion"},{"id":518,"title":"Prime triples and geometric sequences"},{"id":519,"title":"Tricoloured Coin Fountains"},{"id":520,"title":"Simbers"},{"id":521,"title":"Smallest prime factor"},{"id":522,"title":"Hilbert's Blackout"},{"id":523,"title":"First Sort I"},{"id":524,"title":"First Sort II"},{"id":525,"title":"Rolling Ellipse"},{"id":526,"title":"Largest prime factors of consecutive numbers"},{"id":527,"title":"Randomized Binary Search"},{"id":528,"title":"Constrained Sums"},{"id":529,"title":"10-substrings"},{"id":530,"title":"GCD of Divisors"},{"id":531,"title":"Chinese leftovers"},{"id":532,"title":"Nanobots on Geodesics"},{"id":533,"title":"Minimum values of the Carmichael function"},{"id":534,"title":"Weak Queens"},{"id":535,"title":"Fractal Sequence"},{"id":536,"title":"Modulo power identity "},{"id":537,"title":"Counting tuples"},{"id":538,"title":"Maximum quadrilaterals"},{"id":539,"title":"Odd elimination"},{"id":540,"title":"Counting primitive Pythagorean triples"},{"id":541,"title":"Divisibility of Harmonic Number Denominators"},{"id":542,"title":"Geometric Progression with Maximum Sum"},{"id":543,"title":"Prime-Sum Numbers"},{"id":544,"title":"Chromatic Conundrum"},{"id":545,"title":"Faulhaber's Formulas"},{"id":546,"title":"The Floor's Revenge"},{"id":547,"title":"Distance of random points within hollow square laminae"},{"id":548,"title":"Gozinta Chains"},{"id":549,"title":"Divisibility of factorials"},{"id":550,"title":"Divisor game"},{"id":551,"title":"Sum of digits sequence"},{"id":552,"title":"Chinese leftovers II"},{"id":553,"title":"Power sets of power sets"},{"id":554,"title":"Centaurs on a chess board"},{"id":555,"title":"McCarthy 91 function"},{"id":556,"title":"Squarefree Gaussian Integers"},{"id":557,"title":"Cutting triangles"},{"id":558,"title":"Irrational base"},{"id":559,"title":"Permuted Matrices"},{"id":560,"title":"Coprime Nim"},{"id":561,"title":"Divisor Pairs"},{"id":562,"title":"Maximal perimeter"},{"id":563,"title":"Robot Welders"},{"id":564,"title":"Maximal polygons"},{"id":565,"title":"Divisibility of sum of divisors"},{"id":566,"title":"Cake Icing Puzzle"},{"id":567,"title":"Reciprocal games I"},{"id":568,"title":"Reciprocal games II"},{"id":569,"title":"Prime Mountain Range"},{"id":570,"title":"Snowflakes"},{"id":571,"title":"Super Pandigital Numbers"},{"id":572,"title":"Idempotent matrices"},{"id":573,"title":"Unfair race"},{"id":574,"title":"Verifying Primes"},{"id":575,"title":"Wandering Robots"},{"id":576,"title":"Irrational jumps"},{"id":577,"title":"Counting hexagons"},{"id":578,"title":"Integers with decreasing prime powers"},{"id":579,"title":"Lattice points in lattice cubes"},{"id":580,"title":"Squarefree Hilbert numbers"},{"id":581,"title":"47-smooth triangular numbers"},{"id":582,"title":"Nearly isosceles 120 degree triangles"},{"id":583,"title":"Heron Envelopes"},{"id":584,"title":"Birthday Problem Revisited"},{"id":585,"title":"Nested square roots"},{"id":586,"title":"Binary Quadratic Form"},{"id":587,"title":"Concave triangle"},{"id":588,"title":"Quintinomial coefficients"},{"id":589,"title":"Poohsticks Marathon"},{"id":590,"title":"Sets with a given Least Common Multiple"},{"id":591,"title":"Best Approximations by Quadratic Integers"},{"id":592,"title":"Factorial trailing digits 2"},{"id":593,"title":"Fleeting Medians"},{"id":594,"title":"Rhombus Tilings"},{"id":595,"title":"Incremental Random Sort"},{"id":596,"title":"Number of lattice points in a hyperball"},{"id":597,"title":"Torpids"},{"id":598,"title":"Split Divisibilities"},{"id":599,"title":"Distinct Colourings of a Rubik's Cube"},{"id":600,"title":"Integer sided equiangular hexagons"},{"id":601,"title":"Divisibility streaks"},{"id":602,"title":"Product of Head Counts"},{"id":603,"title":"Substring sums of prime concatenations"},{"id":604,"title":"Convex path in square"},{"id":605,"title":"Pairwise Coin-Tossing Game"},{"id":606,"title":"Gozinta Chains II"},{"id":607,"title":"Marsh Crossing"},{"id":608,"title":"Divisor Sums"},{"id":609,"title":"π sequences"},{"id":610,"title":"Roman Numerals II"},{"id":611,"title":"Hallway of square steps"},{"id":612,"title":"Friend numbers"},{"id":613,"title":"Pythagorean Ant"},{"id":614,"title":"Special partitions 2"},{"id":615,"title":"The millionth number with at least one million prime factors"},{"id":616,"title":"Creative numbers"},{"id":617,"title":"Mirror Power Sequence"},{"id":618,"title":"Numbers with a given prime factor sum"},{"id":619,"title":"Square subsets"},{"id":620,"title":"Planetary Gears"},{"id":621,"title":"Expressing an integer as the sum of triangular numbers"},{"id":622,"title":"Riffle Shuffles"},{"id":623,"title":"Lambda Count"},{"id":624,"title":"Two heads are better than one"},{"id":625,"title":"Gcd sum"},{"id":626,"title":"Counting Binary Matrices"},{"id":627,"title":"Counting products"},{"id":628,"title":"Open chess positions"},{"id":629,"title":"Scatterstone Nim"},{"id":630,"title":"Crossed lines"},{"id":631,"title":"Constrained Permutations"},{"id":632,"title":"Square prime factors"},{"id":633,"title":"Square prime factors II"},{"id":634,"title":"Numbers of the form $a^2b^3$"},{"id":635,"title":"Subset sums"},{"id":636,"title":"Restricted Factorisations"},{"id":637,"title":"Flexible digit sum"},{"id":638,"title":"Weighted lattice paths"},{"id":639,"title":"Summing a multiplicative function"},{"id":640,"title":"Shut the Box"},{"id":641,"title":"A Long Row of Dice"},{"id":642,"title":"Sum of largest prime factors"},{"id":643,"title":"2-Friendly"},{"id":644,"title":"Squares on the line"},{"id":645,"title":"Every Day is a Holiday"},{"id":646,"title":"Bounded Divisors"},{"id":647,"title":"Linear Transformations of Polygonal Numbers"},{"id":648,"title":"Skipping Squares"},{"id":649,"title":"Low-Prime Chessboard Nim"},{"id":650,"title":"Divisors of Binomial Product"},{"id":651,"title":"Patterned Cylinders"},{"id":652,"title":"Distinct values of a proto-logarithmic function"},{"id":653,"title":"Frictionless Tube"},{"id":654,"title":"Neighbourly Constraints"},{"id":655,"title":"Divisible Palindromes"},{"id":656,"title":"Palindromic sequences"},{"id":657,"title":"Incomplete words"},{"id":658,"title":"Incomplete words II"},{"id":659,"title":"Largest prime"},{"id":660,"title":"Pandigital Triangles"},{"id":661,"title":"A Long Chess Match"},{"id":662,"title":"Fibonacci paths"},{"id":663,"title":"Sums of subarrays"},{"id":664,"title":"An infinite game"},{"id":665,"title":"Proportionate Nim"},{"id":666,"title":"Polymorphic Bacteria"},{"id":667,"title":"Moving Pentagon"},{"id":668,"title":"Square root smooth Numbers"},{"id":669,"title":"The King's Banquet"},{"id":670,"title":"Colouring a Strip"},{"id":671,"title":"Colouring a Loop"},{"id":672,"title":"One more one"},{"id":673,"title":"Beds and Desks"},{"id":674,"title":"Solving $\\mathcal{I}$-equations"},{"id":675,"title":"$2^{\\omega(n)}$"},{"id":676,"title":"Matching Digit Sums"},{"id":677,"title":"Coloured Graphs"},{"id":678,"title":"Fermat-like Equations"},{"id":679,"title":"Freefarea"},{"id":680,"title":"Yarra Gnisrever"},{"id":681,"title":"Maximal Area"},{"id":682,"title":"5-Smooth Pairs"},{"id":683,"title":"The Chase II"},{"id":684,"title":"Inverse Digit Sum"},{"id":685,"title":"Inverse Digit Sum II"},{"id":686,"title":"Powers of Two"},{"id":687,"title":"Shuffling Cards"},{"id":688,"title":"Piles of Plates"},{"id":689,"title":"Binary Series"},{"id":690,"title":"Tom and Jerry"},{"id":691,"title":"Long substring with many repetitions"},{"id":692,"title":"Siegbert and Jo"},{"id":693,"title":"Finite Sequence Generator"},{"id":694,"title":"Cube-full Divisors"}],"page":{"Home":{"location":{"params":{"slug":"479"},"search":{},"pathname":"/479"},"preloaded":true}},"router":{"location":null}}