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Problem List

Problem Name Tags
Find $ \displaystyle\sum_{n=1}^{\infty} \frac{n^3}{3^n} $. Topics: generalization, geometric, infinite, series
$ \cos x\cdot\cos 2x\cdot\cos 4x\cdots<br /> \cos 2^{n-1}x = \frac{\sin 2^nx}{2^n \sin x} $ Topics: induction
Prerequisites: trigonometry
$ \sum_{k=0}^n {n+k \choose k}\frac{1}{2^k} = 2^n $ Topics: induction
All horses are the same color fallacy Topics: fallacy, induction
Alternating sum of binomial coefficients Topics: binomial coefficients
AM-GM inequality Topics: induction, inequalities
Prerequisites: algebra
Arbelos area Topics: area, circles, tangent
Prerequisites: geometry
Pedagogy: warm-up
Arbelos area Topics: area, circles, tangent
Prerequisites: geometry
Pedagogy: warm-up
Arbelos construction Topics: circles, tangents
Prerequisites: geometry
Arbelos radii Topics: analytic geometry, see heath's "archimedes" for synthetic geometry solution.
Prerequisites: algebra, geometry
Arbelos tangent radius Topics: analytic geometry, circles, tangents
Prerequisites: geometry
Archimedes lemma for tangent circles Topics: circles, collinearity, geometry, tangents
Prerequisites: geometry
Arctan identities and pi approximations Topics: trigonometry
Prerequisites: trigonometry
Arctan identity Topics: trigonometry
Prerequisites: trigonometry
Area of a flower Topics: area, circles, geometry
Prerequisites: geometry
Arrange $ n $ lines to give 100 points of intersection Topics: concurrent, geometry, intersection
Pedagogy: warm-up
Arranging objects with a nonadjacency restriction Topics: combinatorics, enumeration
Bijection between closed ray and open ray Topics: bijection, geometry, ray
Pedagogy: warm-up
Binomial theorem and approximating cube roots Topics: binomial theorem, infinite, series
Prerequisites: algebra 2
Breaking chocolate game Topics: induction
Can you arrange the numbers $ 1, 2, \ldots, 9 $ along a circle
such that the sum of two neighbors is never divisible by 3, 5, or
7?		 Topics: combinatorics
Can you partition the set of positive integers into infinitely
many infinite subsets such that each subset is generated from
any other by adding the same positive integer to each element
of the subset?		 Topics: combinatorics, infinity
Choosing an odd number of objects Topics: combinatorics, enumeration
Choosing books with no two adjacent Topics: combinatorics, enumeration
Choosing objects when some are distinguishable and some are not Topics: combinatorics, enumeration
Choosing officers Topics: combinatorics, permutations
Choosing team captains Topics: combinatorics, permutations
Choosing teams with specified roles Topics: combinations, combinatorics, enumeration
Choosing two distinct subsets Topics: combinatorics, enumeration
Circles and Pythagoras Topics: circles, geometry, pythagorean theorem
Prerequisites: geometry
Coloring a graph with -1 colors Topics: graph theory, polynomial
Prerequisites: algebra
Coloring book covers Topics: combinatorics, enumeration
Combinations and forming teams Topics: combinations, combinatorics
Combinations and pairings Topics: combinations, combinatorics
Combinations exercise Topics: combinations, combinatorics
Combinatorial approach to computing \sum_{k=1}^n {n \choose k}k^3 Topics: combinatorics, enumeration
Combinatorics of assigning people to rooms Topics: combinations, combinatorics
Connecting points in pairs with nonintersecting segments Topics: catalan numbers, combinatorics, enumeration
Consecutive Fibonacci are relatively prime Topics: induction
Constraints on weak $ 3 \times 3 $ magic squares Topics: combinatorics, magic squares
Prerequisites: algebra
Constructible numbers and the 17-gon Topics: construction
Prerequisites: algebra, geometry
Counting (small) Sudoku Topics: combinatorics, enumeration
Counting colorings of basic graphs Topics: graph theory, polynomial
Prerequisites: algebra
Counting diagonals Topics: combinatorics, enumeration, geometry
Prerequisites: geometry
Counting distinct necklaces Topics: combinatorics, enumeration
Counting distinct words Topics: combinatorics, enumeration
Counting groupings (Stirling) Topics: combinatorics
Counting magic 3x3 squares Topics: enumeration, magic squares
Pedagogy: warm-up
Counting numbers by sum of digits Topics: combinatorics, enumeration
Counting numbers with adjacent equal digits Topics: combinatorics, enumeration
Counting numbers with repeated digits Topics: combinatorics, complement
Counting poker hands Topics: combinations, combinatorics, complement, enumeration, permutations
Counting rectangles in the grid Topics: combinatorics, enumeration
Counting regions when the plane is cut by "zig-zags" Topics: enumeration, geometry
Counting regions when the plane is cut by lines Topics: enumeration, geometry, lines
Counting routes Topics: combinatorics
Pedagogy: warm-up
Counting subsets - dinner party model Topics: combinatorics, enumeration
Counting subsets - staircase model Topics: combinatorics, enumeration
Counting words with at least one ``A" Topics: combinatorics, complement, enumeration
Counting zeroes used in counting to 333,333,333 Topics: combinatorics
Diagonals in a dodecahedron Topics: combinations, combinatorics, geometry
Prerequisites: geometry
Diameters of circles in an arbelos chain (Pappus) Topics: circles, inversion, tangents
Prerequisites: geometry
Difference of cubes, solving equations Topics: polynomials, symmetry
Prerequisites: advanced algebra
Difference of fourth powers, solving equations Topics: polynomials, symmetry
Prerequisites: advanced algebra
Diffie-Hellman introduction Topics: cryptography, modular arithmetic
Prerequisites: modular arithmetic
Distinct ways to sit around a circular table Topics: combinatorics, enumeration
Distributing billiard balls into pockets Topics: combinatorics, enumeration
Divisibility of Fermat numbers Topics: divisiibility, factoring, fermat's little theorem, number theory
Prerequisites: number theory
Divisibility test for $ 2^n $ Topics: divisibility
Pedagogy: warm-up
Divisibility test for 10 Topics: divisibility
Pedagogy: warm-up
Divisibility test for 11 Topics: divisibility, modular arithmetic
Divisibility test for 11 Topics: divisibility
Divisibility test for 13 Topics: divisibility, modular arithmetic
Divisibility test for 17 Topics: divisibility, modular arithmetic
Divisibility test for 19 Topics: divisibility, modular arithmetic
Divisibility test for 2 Topics: divisibility
Pedagogy: warm-up
Divisibility test for 3 Topics: divisibility, modular arithmetic
Divisibility test for 4 Topics: divisibility
Pedagogy: warm-up
Divisibility test for 5 Topics: divisibility
Pedagogy: warm-up
Divisibility test for 7 Topics: divisibility, modular arithmetic
Divisibility test for 7 Topics: divisibility, modular arithmetic
Divisibility test for 7 Topics: divisibility, modular arithmetic
Divisibility test for 8 Topics: divisibility
Pedagogy: warm-up
Divisibility test for 9 Topics: divisibility, modular arithmetic
Eight rooks on a chessboard Topics: combinatorics, factorial, permutations
Euler phi function introduction Topics: modular arithmetic
Prerequisites: modular arithmetic
Existence of 2x2 magic squares Topics: magic squares
Pedagogy: warm-up
Factor theorem and remainders Topics: algebra, division algorithm, polynomials, roots
Prerequisites: algebra
Factor theorem and remainders Topics: algebra, division algorithm, polynomials, roots
Prerequisites: algebra
Factoring $ x^3 + a^3 + b^3 - 3abx $ Topics: algebra, polynomials, roots
Prerequisites: algebra
Fibonacci generating function Topics: calculus, combinatorics, fibonacci, generating function, recursion
Prerequisites: calculus
Fibonacci sum by generating functions Topics: combinatorics, generating function
Prerequisites: calculus
Filling a grid Topics: cases, invariant, proof without words, tilings
Find a nice formula for the sum:

\[ {n \choose 0}^2 + {n \choose 1}^2 + {n \choose 2}^2 + \cdots + {n \choose n}^2. \]

Topics: binomial coefficients
Finding counterfeit coin in 2 weighings Topics: pigeonhole principle
Finding roots of a cubic in terms of sums of powers of the roots Topics: algebra, polynomials, roots, symmetry
Prerequisites: algebra
Finite geometric series exercise Topics: geometric, series
Prerequisites: algebra
Generating function for number of ways to triangulate an $ n $-gon Topics: catalan numbers, combinatorics, enumeration, generating function
Prerequisites: calculus
Generating functions and binomial coefficients Topics: binomial coefficient, calculus, combinatorics, generating function
Prerequisites: calculus
Generating functions and squaring Topics: calculus, combinatorics, generating functions
Prerequisites: algebra
Generating functions to solve recursion Topics: calculus, combinatorics, generating functions, recursion
Prerequisites: calculus
Harmonic series diverges Topics: convergent, divergent, infinite, series
Prerequisites: algebra
Harmonic series with prime divisors restricted Topics: generalization, infinite, series
Prerequisites: algebra
If there's enough gas in total, then there's enough gas for one car to make it around the circle Topics: induction
In how many ways can you choose two distinct subsets from a
collection of $ N $ items? The two subsets together need not
include all $ N $ items.
 Topics: combinatorics, enumeration
Inaccessible squares for three-dimensional knight Topics: modular arithmetic, prime numbers
Infinite geometric series exercise Topics: geometric, infinite, series
Prerequisites: algebra
Pedagogy: warm-up
Introduction to RSA cryptosystem Topics: cryptography, modular arithmetic
Prerequisites: euler's phi-function, modular arithmetic
Is it possible that $ ax^2 + bx + c $, $ bx^2 + cx + a $, and $ cx^2 + ax + b $ all have real roots? Topics: algebra, polynomials, roots
Prerequisites: algebra
Josephus problem, penultimate survivor Topics: recursion, special cases
Josephus problem, sequence of differences Topics: recursion
Josephus recursion for even $ n $ Topics: recursion
Josephus recursion for odd $ n $ Topics: recursion
Last digit of fourth powers, and its generalizations Topics: modular arithmetic
Prerequisites: algebra
Locus of midpoint of sliding ladder Topics: trigonometry
Pedagogy: warm-up
Lottery combinations Topics: combinations, combinatorics
Lotto winners with at least one adjacent pair Topics: combinatorics, enumeration
Magic sum for $ n \times n $ square Topics: combinatorics, magic squares
Prerequisites: algebra
Pedagogy: warm-up
Marriage problem Topics: combinatorics, factorial
Maximum number of nonattacking bishops Topics: pigeonhole principle
Measuring using 13 and 21 Topics: modular arithmetic, prime numbers
Nontransitive dice Topics: probability
Prerequisites: probability
Pedagogy: warm-up
Number of ways for $ M $ people to disembark a train at one of $ N $ stops Topics: combinatorics, enumeration
Number of ways to pair $ 2n $ people Topics: combinatorics
Numbers that equal the sum of their digits plus the sum of the cubes of their digits Topics: patterns, polynomial equations
Prerequisites: algebra
Pedagogy: excercise, warm-up
Partitioning $ k $ items into $ n $ indistinguishable boxes Topics: combinatorics, enumeration
Partitioning a deck of cards Topics: combinatorics, enumeration
Partitioning colored balls Topics: combinatorics, enumeration
Partitioning indistinguishable items Topics: combinations, combinatorics
Paths in a complete graph Topics: combinatorics, permutations
Paths through a cube Topics: combinatorics, multinomial coefficients
Placing checkers Topics: combinations, combinatorics
Polyhedra with odd faces and odd edges Topics: combinatorics
Prerequisites: geometry
Polynomials, sum of fifth powers Topics: polynomials, symmetry
Prerequisites: advanced algebra
Pool table bounces Topics: geometry, reflection
Prerequisites: geometry
Primes dividing $ a^2 + 1 $ Topics: divisibility, factoring, fermat's little theory, number theory
Prerequisites: algebra 2
Prove that for any $ n > 0 $:

\[1^2 + 4^2 + 7^2 + 10^2 + \cdots<br /> + (3n-2)^2 = n(6n^2 - 3n - 1)/2. \]

Topics: induction
Quadratic game Topics: algebra, games, polynomials
Prerequisites: advanced algebra
Quadratic whose roots are cubes of another quadratic Topics: algebra, polynomials, roots
Prerequisites: algebra
Radius of circle circumscribed around three smaller ones Topics: area, circles, geometry
Prerequisites: geometry
Radius of circles in a tangent ring Topics: geometry, trigonometry
Prerequisites: trigonometry
Rearrangements with a maximum distance of 1 Topics: combinatorics, enumeration
Rearranging letters Topics: combinatorics, permutations
Recognizing decimals Topics: mathematical constants
Pedagogy: warm-up
Recursive divisibility test for 7 Topics: divisibility, modular arithmetic
Regions formed by intersecting circles Topics: circles, enumeration, geometry
Regions formed by lines in a plane Topics: induction
Relation between coefficients and sums of powers of roots of a polynomial Topics: algebra, polynomials, roots, symmetry
Prerequisites: algebra
Representations of 100000 as the product of 3 factors Topics: combinatorics, enumeration
Roots and coefficients of a cubic Topics: algebra, polynomials, roots, symmetry
Prerequisites: algebra
Sets of numbers such that the sum of their cubes equals the square of their sum Topics: patterns, polynomial equations
Prerequisites: algebra
Pedagogy: excercise, warm-up
Show that

\[ \sum_{k=0}^n {k \choose m} = {n+1 \choose m+1}. \]

Topics: binomial coefficients
Show that

\[ \sum_{k=0}^n {r+k \choose k} = {r+n+1 \choose n}.\]

Topics: binomial coefficients
Show that

\[ \sum_{r=1}^n r{n \choose r} = n2^{n-1}. \]

Topics: binomial coefficients
Show that

\[ {r \choose m}{m \choose k} = {r \choose k} {r-k \choose m-k}. \]

Topics: binomial coefficients
Show that $ 3^n \ge 2^n $. Topics: induction
Show that $ 3^{n+1} $ divides evenly into $ 2^{3^n} +1 $ for all $ n \ge 0 $. Topics: induction
Show that if $ m > k $ and $ n > k $ then:

\[\sum_{j=0}^k {n \choose j}{m \choose k-j} = {n+m \choose k}. \]

Topics: binomial coefficients
Show that if five points lie on the surface of a sphere, then there is some hemisphere containing four of them.
 Topics: pigeonhole principle, sphere
Prerequisites: geometry
Solving a recurrence Topics: recursion, special cases
Prerequisites: algebra
Solving a recurrence with different rules for even and odd indexes Topics: recursion, special cases
Prerequisites: algebra
Solving recursion (inhomogeneous) Topics: combinatorics, recursion
Solving recursion for Fibonacci-like sequence Topics: calculus, combinatorics, generating function
Prerequisites: calculus
Subsets with no consecutive elements Topics: combinatorics, enumeration
Sum of $ \frac{n}{2^n} $ Topics: generalization, geometric, infinite, series
Prerequisites: algebra
Sum of 1000th powers of roots of a 1000th degree polynomial Topics: algebra, polynomials, roots
Prerequisites: algebra
Sum of cubes - proof without words Topics: patterns, polynomial equations, proof without words
Prerequisites: algebra
Pedagogy: warm-up
Sum of cubes formula by induction Topics: induction
Sum of cubes of roots of a cubic Topics: algebra, polynomials, roots, symmetry
Prerequisites: algebra
Sum of cubes of roots of a quadratic Topics: algebra, polynomials
Prerequisites: algebra
Sum of cubes of roots of a quadratic Topics: algebra, polynomials, symmetry
Prerequisites: algebra
Sum of cubes of roots of a quartic Topics: algebra, polynomials, roots
Prerequisites: algebra
Sum of geometric series by induction Topics: induction
Sum of powers of the roots of a cubic Topics: algebra, polynomials, roots, symmetry
Prerequisites: algebra
Sum of products of reciprocals of subsets Topics: combinatorics
Sum of reciprocals of squares of odd numbers Topics: infinite, series
Prerequisites: algebra
Sum of squares by derivative of generating function Topics: calculus, combinatorics, generating function
Prerequisites: calculus
Summing a Fibonacci-geometric series Topics: generalization, geometric, infinite, series
Prerequisites: algebra
Summing geometric series, inverse problem Topics: geometric, series
Prerequisites: algebra
Sums of powers of roots (quadratic) Topics: algebra, divisibility, polynomials, roots
Prerequisites: algebra, number theory
Symmetric polynomials of roots (quadratic) Topics: algebra, polynomials, roots
Prerequisites: algebra
Telescoping series and partial fractions (cubic) Topics: finite, infinite, partial fractions, series, telescoping
Prerequisites: algebra
Telescoping series and partial fractions (nonconsecutive terms) Topics: finite, infinite, partial fractions, series, telescoping
Prerequisites: algebra
Telescoping series and partial fractions (quadratic) Topics: finite, infinite, partial fractions, series, telescoping
Prerequisites: algebra
Tile space with circles Topics: intersection, non-degenerate circles, sphere, tangent, tilings
Prerequisites: geometry
Pedagogy: warm-up
Tiling chessboard with corners removed Topics: invariant, tilings
Tower of Hanoi with cyclic moves Topics: induction, recursion, special cases
Tower of Hanoi with duplicate discs Topics: induction, recursion, special cases
Tower of Hanoi with special peg Topics: combinatorics, induction, recursion
Pedagogy: hands-on
Tower of Hanoi with special peg is universal Topics: combinatorics, graph theory, induction, recursion
Triangles formed by 10 points in the plane Topics: combinations, combinatorics
Triangles formed by points on parallel lines Topics: combinations, combinatorics, enumeration
Prerequisites: geometry
Triangulation of $ n $-gon Topics: induction
Trigonometry and the 17-gon Topics: construction of 17-gon, roots of unity
Prerequisites: algebra 2, geometry, number theory, trigonometry
Two kings on a chessboard Topics: combinatorics, enumeration
Two polynomials whose coefficients are each others' roots Topics: algebra, polynomials, roots
Prerequisites: algebra
Two queens on a chessboard Topics: combinatorics, enumeration
Two rooks on a chessboard Topics: combinatorics, permutations
Ways to nest parentheses Topics: catalan numbers, combinatorics, enumeration
Ways to triangulate an $ n $-gon Topics: catalan numbers, combinatorics, enumeration
Prerequisites: geometry

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