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Problem Difficulty Level Topic Classification Tags

Intersecting cylinders

 Moderate Upper secondary education Problem Tag: problem of the month, geometry, Trigonometry, constructions
Prerequisites: trigonometry

Fruit cutting High School

 Perplexing Lower secondary education Problem Tag: problem of the month, 3D geometry, combinatorial geometry

Fruit cutting 6-8th grades

 Moderate Lower secondary education Problem Tag: problem of the month, sections of polytopes
Topics: Combinatorial Geometry

Pumpkin cutting

 Moderate Primary education Problem Tag: Pumpkin, problem of the month, topology
Topics: Topology, cutting

Fruit cutting preK-5th grade

 Easy Primary education Geometry, counting Problem Tag: Fruit cutting, geometry, counting, problem of the month
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Specifying a pixel

 Easy Pedagogy: warm-up

Distance between points in the plane

 Easy Pedagogy: warm-up

Why two-dimensional space?

 Easy Dimensions Pedagogy: warm-up

The Game of Criss Cross- 4 Point Outer Boundary Formula

 Moderate Euler characteristic, Graph Theory Problem Tag: Criss Cross Game
, Pedagogy: Further Problems

The Game of Criss Cross- 4 Point Outer Boundary

 Moderate Euler characteristic, Graph Theory Problem Tag: Criss-Cross game
, Pedagogy: Further Problems

Criss Cross Village

 Moderate Euler characteristic Pedagogy: Further Problems
Prerequisites: Criss-Cross

The Game of Criss Cross- Who will win?

 Easy Graph Theory, Euler characteristic Problem Tag: Game
, Pedagogy: warm-up

The Game of Criss Cross- How Many Moves

 Easy Graph Theory, Euler characteristic Problem Tag: Game
, Pedagogy: warm-up

Generating functions and binomial coefficients

 Perplexing Prerequisites: calculus
Topics: generating function, combinatorics, calculus, binomial coefficient

Sum of cubes of roots of a quadratic

 Moderate Prerequisites: algebra
Topics: polynomials, algebra

Sum of squares by derivative of generating function

 Challenging

Quadratic whose roots are cubes of another quadratic

 Challenging Prerequisites: algebra
Topics: algebra, polynomials, roots

Partitioning indistinguishable items

 Challenging Topics: combinations, combinatorics

Fibonacci sum by generating functions

 Challenging Prerequisites: calculus
Topics: generating function, combinatorics

Roots and coefficients of a cubic

 Moderate Prerequisites: algebra
Topics: symmetry, roots, polynomials, algebra

Generating function for number of ways to triangulate an $ n $-gon

 Challenging Prerequisites: calculus
Topics: generating function, combinatorics, enumeration, catalan numbers

Partitioning $ k $ items into $ n $ indistinguishable boxes

 Challenging Topics: combinatorics, enumeration

Arbelos area

 Easy Pedagogy: warm-up
Prerequisites: geometry
Topics: tangent

Divisibility test for 8

 Easy Pedagogy: warm-up
Topics: divisibility

Sum of cubes of roots of a cubic

 Challenging Prerequisites: algebra
Topics: symmetry, roots, polynomials, algebra

Coloring book covers

 Challenging Topics: enumeration, combinatorics

Counting colorings of basic graphs

 Challenging Prerequisites: algebra
Topics: polynomial, graph theory

Sum of powers of the roots of a cubic

 Challenging Prerequisites: algebra
Topics: polynomials, roots, symmetry, algebra

Number of ways for $ M $ people to disembark a train at one of $ N $ stops

 Challenging Topics: combinatorics, enumeration

Coloring a graph with -1 colors

 Perplexing Prerequisites: algebra
Topics: graph theory, polynomial

Finding roots of a cubic in terms of sums of powers of the roots

 Challenging Prerequisites: algebra
Topics: algebra, polynomials, roots, symmetry

Arranging objects with a nonadjacency restriction

 Challenging Topics: combinatorics, enumeration

Existence of 2x2 magic squares

 Easy Pedagogy: warm-up
Topics: magic squares

Factoring $ x^3 + a^3 + b^3 - 3abx $

 Challenging Prerequisites: algebra
Topics: algebra, polynomials, roots

Tower of Hanoi with cyclic moves

 Challenging Topics: recursion, induction, special cases

Representations of 100000 as the product of 3 factors

 Challenging Topics: combinatorics, enumeration

Counting magic 3x3 squares

 Easy Pedagogy: warm-up
Topics: enumeration, magic squares

Relation between coefficients and sums of powers of roots of a polynomial

 Perplexing Prerequisites: algebra
Topics: algebra, polynomials, roots, symmetry

Choosing books with no two adjacent

 Challenging Topics: combinatorics, enumeration

Magic sum for $ n \times n $ square

 Moderate Pedagogy: warm-up
Prerequisites: algebra
Topics: combinatorics, magic squares

Sum of cubes of roots of a quartic

 Challenging Prerequisites: algebra
Topics: algebra, polynomials, roots

Counting distinct necklaces

 Challenging Topics: combinatorics, enumeration

Constraints on weak $ 3 \times 3 $ magic squares

 Moderate Prerequisites: algebra
Topics: combinatorics, magic squares

Sum of 1000th powers of roots of a 1000th degree polynomial

 Challenging Prerequisites: algebra
Topics: algebra, polynomials, roots

Counting words with at least one ``A"

 Moderate Topics: combinatorics, enumeration, complement

Choosing officers

 Easy Topics: combinatorics, permutations

Last digit of fourth powers, and its generalizations

 Moderate Prerequisites: algebra
Topics: modular arithmetic

Factor theorem and remainders

 Challenging Prerequisites: algebra
Topics: algebra, polynomials, roots, division algorithm

Paths in a complete graph

 Moderate Topics: combinatorics, permutations

Diffie-Hellman introduction

 Challenging Prerequisites: modular arithmetic
Topics: modular arithmetic, cryptography

Factor theorem and remainders

 Challenging Prerequisites: algebra
Topics: algebra, polynomials, roots, division algorithm

Counting rectangles in the grid

 Challenging Topics: combinatorics, enumeration

Euler phi function introduction

 Challenging Prerequisites: modular arithmetic
Topics: modular arithmetic

Paths through a cube

 Challenging Topics: combinatorics, multinomial coefficients

Introduction to RSA cryptosystem

 Challenging Prerequisites: modular arithmetic, euler's phi-function
Topics: modular arithmetic, cryptography

Counting numbers with adjacent equal digits

 Challenging Topics: combinatorics, enumeration

Arbelos construction

 Perplexing Prerequisites: geometry
Topics: circles, tangents

Counting (small) Sudoku

 Moderate Topics: combinatorics, enumeration

Partitioning a deck of cards

 Challenging Topics: combinatorics, enumeration

Partitioning colored balls

 Challenging Topics: combinatorics, enumeration

Lotto winners with at least one adjacent pair

 Challenging Topics: combinatorics, enumeration

Combinatorial approach to computing \sum_{k=1}^n {n \choose k}k^3

 Perplexing Topics: combinatorics, enumeration

Choosing objects when some are distinguishable and some are not

 Challenging Topics: combinatorics, enumeration

Choosing an odd number of objects

 Challenging Topics: combinatorics, enumeration

Distinct ways to sit around a circular table

 Challenging Topics: combinatorics, enumeration

Connecting points in pairs with nonintersecting segments

 Perplexing Topics: combinatorics, enumeration, catalan numbers

Ways to triangulate an $ n $-gon

 Perplexing Prerequisites: geometry
Topics: combinatorics, enumeration, catalan numbers

Divisibility test for $ 2^n $

 Easy Pedagogy: warm-up
Topics: divisibility

Ways to nest parentheses

 Perplexing Topics: combinatorics, enumeration, catalan numbers

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?

 Challenging Topics: combinatorics

Choosing two distinct subsets

 Perplexing Topics: combinatorics, enumeration

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.

 Challenging Topics: combinatorics, enumeration

Subsets with no consecutive elements

 Challenging Topics: combinatorics, enumeration

Distributing billiard balls into pockets

 Challenging Topics: combinatorics, enumeration

Rearrangements with a maximum distance of 1

 Challenging Topics: combinatorics, enumeration

Polyhedra with odd faces and odd edges

 Challenging Prerequisites: geometry
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?

 Perplexing Topics: combinatorics, infinity

Tower of Hanoi with duplicate discs

 Challenging Topics: recursion, induction, special cases

Sum of products of reciprocals of subsets

 Perplexing Topics: combinatorics

Counting groupings (Stirling)

 Challenging Topics: combinatorics

Counting poker hands

 Perplexing Topics: combinatorics, combinations, enumeration, permutations, complement

Counting zeroes used in counting to 333,333,333

 Moderate Topics: combinatorics

Circles and Pythagoras

 Challenging Prerequisites: geometry
Topics: geometry, circles, pythagorean theorem

Diagonals in a dodecahedron

 Challenging Prerequisites: geometry
Topics: geometry, combinatorics, combinations

Pool table bounces

 Challenging Prerequisites: geometry
Topics: geometry, reflection

Area of a flower

 Challenging Prerequisites: geometry
Topics: geometry, circles, area

Radius of circle circumscribed around three smaller ones

 Challenging Prerequisites: geometry
Topics: geometry, circles, area

Radius of circles in a tangent ring

 Moderate Prerequisites: trigonometry
Topics: geometry, trigonometry

Counting diagonals

 Moderate Prerequisites: geometry
Topics: geometry, combinatorics, enumeration

Alternating sum of binomial coefficients

 Moderate Topics: binomial coefficients

Show that

\[ {r \choose m}{m \choose k} = {r \choose k} {r-k \choose m-k}. \]
Moderate Topics: binomial coefficients

Show that

\[ \sum_{k=0}^n {k \choose m} = {n+1 \choose m+1}. \]
Moderate Topics: binomial coefficients

Show that

\[ \sum_{r=1}^n r{n \choose r} = n2^{n-1}. \]
Moderate Topics: binomial coefficients

Find a nice formula for the sum:

\[ {n \choose 0}^2 + {n \choose 1}^2 + {n \choose 2}^2 + \cdots + {n \choose n}^2. \]
Challenging Topics: binomial coefficients

Show that if $ m > k $ and $ n > k $ then:

\[\sum_{j=0}^k {n \choose j}{m \choose k-j} = {n+m \choose k}. \]
Challenging Topics: binomial coefficients

Show that

\[ \sum_{k=0}^n {r+k \choose k} = {r+n+1 \choose n}.\]
Challenging Topics: binomial coefficients

Show that $ 3^n \ge 2^n $.

 Moderate Topics: induction

Prove that for any $ n > 0 $:

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

Arbelos tangent radius

 Perplexing Prerequisites: geometry
Topics: circles, tangents, analytic geometry

Regions formed by lines in a plane

 Moderate Topics: induction

Numbers that equal the sum of their digits plus the sum of the cubes of their digits

 Easy Pedagogy: warm-up, excercise
Prerequisites: algebra
Topics: patterns, polynomial equations

Sum of cubes - proof without words

 Easy Pedagogy: warm-up
Prerequisites: algebra
Topics: patterns, proof without words, polynomial equations

Sets of numbers such that the sum of their cubes equals the square of their sum

 Easy Pedagogy: warm-up, excercise
Prerequisites: algebra
Topics: patterns, polynomial equations

Nontransitive dice

 Easy Pedagogy: warm-up
Prerequisites: probability
Topics: probability

Arrange $ n $ lines to give 100 points of intersection

 Easy Pedagogy: warm-up
Topics: geometry, intersection, concurrent

Locus of midpoint of sliding ladder

 Easy Pedagogy: warm-up
Topics: trigonometry

Regions formed by intersecting circles

 Moderate Topics: geometry, circles, enumeration

Bijection between closed ray and open ray

 Easy Pedagogy: warm-up
Topics: geometry, ray, bijection

Tile space with circles

 Easy Pedagogy: warm-up
Prerequisites: geometry
Topics: tangent, intersection, tilings, non-degenerate circles, sphere

Recognizing decimals

 Easy Pedagogy: warm-up
Topics: mathematical constants

Tiling chessboard with corners removed

 Moderate Topics: tilings, invariant

Filling a grid

 Moderate Topics: tilings, invariant, cases, proof without words

Show that if five points lie on the surface of a sphere, then there is some hemisphere containing four of them.

 Moderate Prerequisites: geometry
Topics: sphere, pigeonhole principle

Maximum number of nonattacking bishops

 Moderate Topics: pigeonhole principle

Finding counterfeit coin in 2 weighings

 Moderate Topics: pigeonhole principle

Two rooks on a chessboard

 Moderate Topics: combinatorics, permutations

Arbelos area

 Easy Pedagogy: warm-up
Prerequisites: geometry
Topics: circles, tangent, area

Measuring using 13 and 21

 Easy Topics: modular arithmetic, prime numbers

Inaccessible squares for three-dimensional knight

 Moderate Topics: modular arithmetic, prime numbers

Triangulation of $ n $-gon

 Challenging Topics: induction

Sum of geometric series by induction

 Challenging Topics: induction

Sum of cubes formula by induction

 Moderate Topics: induction

Breaking chocolate game

 Moderate Topics: induction

Show that $ 3^{n+1} $ divides evenly into $ 2^{3^n} +1 $ for all $ n \ge 0 $.

 Moderate Topics: induction

Consecutive Fibonacci are relatively prime

 Challenging Topics: induction

Diameters of circles in an arbelos chain (Pappus)

 Perplexing Prerequisites: geometry
Topics: circles, tangents, inversion

$ \cos x\cdot\cos 2x\cdot\cos 4x\cdots<br /> \cos 2^{n-1}x = \frac{\sin 2^nx}{2^n \sin x} $

 Challenging Prerequisites: trigonometry
Topics: induction

If there's enough gas in total, then there's enough gas for one car to make it around the circle

 Challenging Topics: induction

$ \sum_{k=0}^n {n+k \choose k}\frac{1}{2^k} = 2^n $

 Moderate Topics: induction

Divisibility test for 5

 Easy Pedagogy: warm-up
Topics: divisibility

Counting regions when the plane is cut by lines

 Moderate Topics: geometry, enumeration, lines

Two kings on a chessboard

 Moderate Topics: combinatorics, enumeration

Trigonometry and the 17-gon

 Perplexing Prerequisites: trigonometry, geometry, algebra 2, number theory
Topics: roots of unity, construction of 17-gon

Divisibility test for 10

 Easy Pedagogy: warm-up
Topics: divisibility

Counting regions when the plane is cut by "zig-zags"

 Challenging Topics: geometry, enumeration

Counting distinct words

 Moderate Topics: combinatorics, enumeration

Constructible numbers and the 17-gon

 Moderate Prerequisites: geometry, algebra
Topics: construction

Divisibility test for 3

 Moderate Topics: divisibility, modular arithmetic

Divisibility test for 2

 Easy Pedagogy: warm-up
Topics: divisibility

Josephus recursion for even $ n $

 Challenging Topics: recursion

Rearranging letters

 Moderate Topics: combinatorics, permutations

Primes dividing $ a^2 + 1 $

 Challenging Prerequisites: algebra 2
Topics: divisibility, number theory, factoring, fermat's little theory

Divisibility test for 9

 Moderate Topics: divisibility, modular arithmetic

Josephus recursion for odd $ n $

 Challenging Topics: recursion

Eight rooks on a chessboard

 Moderate Topics: combinatorics, permutations, factorial

Divisibility of Fermat numbers

 Moderate Prerequisites: number theory
Topics: number theory, factoring, divisiibility, fermat's little theorem

Divisibility test for 11

 Moderate Topics: divisibility, modular arithmetic

Josephus problem, sequence of differences

 Challenging Topics: recursion

Combinatorics of assigning people to rooms

 Moderate Topics: combinatorics, combinations

Tower of Hanoi with special peg

 Moderate Pedagogy: hands-on
Topics: combinatorics, recursion, induction

Summing geometric series, inverse problem

 Moderate Prerequisites: algebra
Topics: series, geometric

Divisibility test for 7

 Moderate Topics: divisibility, modular arithmetic

Counting numbers with repeated digits

 Moderate Topics: combinatorics, complement

Finite geometric series exercise

 Moderate Prerequisites: algebra
Topics: series, geometric

Divisibility test for 11

 Moderate Topics: divisibility

Josephus problem, penultimate survivor

 Topics: recursion, special cases

Two queens on a chessboard

 Moderate Topics: combinatorics, enumeration

Infinite geometric series exercise

 Moderate Pedagogy: warm-up
Prerequisites: algebra
Topics: series, infinite, geometric

Divisibility test for 13

 Moderate Topics: divisibility, modular arithmetic

Counting routes

 Easy Pedagogy: warm-up
Topics: combinatorics

Solving a recurrence

 Moderate Prerequisites: algebra
Topics: recursion, special cases

Number of ways to pair $ 2n $ people

 Moderate Topics: combinatorics

Sum of $ \frac{n}{2^n} $

 Challenging Prerequisites: algebra
Topics: series, infinite, geometric, generalization

Recursive divisibility test for 7

 Moderate Topics: divisibility, modular arithmetic

Solving a recurrence with different rules for even and odd indexes

 Perplexing Prerequisites: algebra
Topics: recursion, special cases

Marriage problem

 Moderate Topics: combinatorics, factorial

Summing a Fibonacci-geometric series

 Challenging Prerequisites: algebra
Topics: series, infinite, geometric, generalization

Divisibility test for 7

 Moderate Topics: divisibility, modular arithmetic

All horses are the same color fallacy

 Challenging Topics: induction, fallacy

Combinations exercise

 Moderate Topics: combinatorics, combinations

Archimedes lemma for tangent circles

 Moderate Prerequisites: geometry
Topics: geometry, circles, tangents, collinearity

Find $ \displaystyle\sum_{n=1}^{\infty} \frac{n^3}{3^n} $.

 Challenging Topics: series, infinite, geometric, generalization

Divisibility test for 7

 Moderate Topics: divisibility, modular arithmetic

AM-GM inequality

 Challenging Prerequisites: algebra
Topics: induction, inequalities

Combinations and pairings

 Moderate Topics: combinatorics, combinations

Telescoping series and partial fractions (quadratic)

 Moderate Prerequisites: algebra
Topics: series, infinite, finite, telescoping, partial fractions

Divisibility test for 17

 Moderate Topics: divisibility, modular arithmetic

Quadratic game

 Challenging Prerequisites: advanced algebra
Topics: algebra, polynomials, games

Combinations and forming teams

 Moderate Topics: combinatorics, combinations

Telescoping series and partial fractions (cubic)

 Challenging Prerequisites: algebra
Topics: series, infinite, finite, telescoping, partial fractions

Divisibility test for 19

 Moderate Topics: divisibility, modular arithmetic

Divisibility test for 4

 Easy Pedagogy: warm-up
Topics: divisibility

Is it possible that $ ax^2 + bx + c $, $ bx^2 + cx + a $, and $ cx^2 + ax + b $ all have real roots?

 Moderate Prerequisites: algebra
Topics: algebra, polynomials, roots

Triangles formed by 10 points in the plane

 Moderate Topics: combinatorics, combinations

Telescoping series and partial fractions (nonconsecutive terms)

 Challenging Prerequisites: algebra
Topics: series, infinite, finite, telescoping, partial fractions

Two polynomials whose coefficients are each others' roots

 Moderate Prerequisites: algebra
Topics: algebra, polynomials, roots

Choosing teams with specified roles

 Moderate Topics: combinatorics, combinations, enumeration

Arctan identity

 Moderate Prerequisites: trigonometry
Topics: trigonometry

Generating functions and squaring

 Moderate Prerequisites: algebra
Topics: combinatorics, calculus, generating functions

Symmetric polynomials of roots (quadratic)

 Challenging Prerequisites: algebra
Topics: algebra, polynomials, roots

Triangles formed by points on parallel lines

 Moderate Prerequisites: geometry
Topics: combinatorics, combinations, enumeration

Tower of Hanoi with special peg is universal

 Challenging Topics: combinatorics, graph theory, recursion, induction

Arctan identities and pi approximations

 Challenging Prerequisites: trigonometry
Topics: trigonometry

Generating functions to solve recursion

 Challenging Prerequisites: calculus
Topics: combinatorics, calculus, recursion, generating functions

Sums of powers of roots (quadratic)

 Challenging Prerequisites: algebra, number theory
Topics: algebra, polynomials, roots, divisibility

Placing checkers

 Moderate Topics: combinatorics, combinations

Binomial theorem and approximating cube roots

 Moderate Prerequisites: algebra 2
Topics: series, infinite, binomial theorem

Solving recursion (inhomogeneous)

 Challenging Topics: combinatorics, recursion

Sum of cubes of roots of a quadratic

 Moderate Prerequisites: algebra
Topics: algebra, polynomials, symmetry

Counting numbers by sum of digits

 Challenging Topics: combinatorics, enumeration

Harmonic series diverges

 Perplexing Prerequisites: algebra
Topics: series, infinite, divergent, convergent

Difference of cubes, solving equations

 Challenging Prerequisites: advanced algebra
Topics: polynomials, symmetry

Choosing team captains

 Easy Topics: combinatorics, permutations

Lottery combinations

 Moderate Topics: combinatorics, combinations

Harmonic series with prime divisors restricted

 Perplexing Prerequisites: algebra
Topics: series, infinite, generalization

Fibonacci generating function

 Challenging Prerequisites: calculus
Topics: combinatorics, generating function, calculus, recursion, fibonacci

Difference of fourth powers, solving equations

 Challenging Prerequisites: advanced algebra
Topics: polynomials, symmetry

Counting subsets - dinner party model

 Moderate Topics: combinatorics, enumeration

Solving recursion for Fibonacci-like sequence

 Perplexing Prerequisites: calculus
Topics: combinatorics, generating function, calculus

Polynomials, sum of fifth powers

 Challenging Prerequisites: advanced algebra
Topics: polynomials, symmetry

Counting subsets - staircase model

 Challenging Topics: combinatorics, enumeration

Sum of reciprocals of squares of odd numbers

 Moderate Prerequisites: algebra
Topics: series, infinite

Arbelos radii

 Perplexing Prerequisites: geometry, algebra
Topics: analytic geometry, see heath's "archimedes" for synthetic geometry solution.