High School Problem Archive
* Denotes a problem which has not yet been solved.
× Denotes a partially solved or open-ended problem.
o Denotes a work in progress.
+ Denotes a problem whose solution has recently been posted.
- Counting Intersection Points
- Find the Unknown Area
- Rearranging Digits
- What's the Locus?
- Egg Drop Soup
- The Moat
- The Pane of it All
- A Magic Square
- Reflections
- A Problem from Lewis Carroll
- Drawing a Zigzag through 16 points
- How Big is the Softball?
- What's the Radius of the Circle?
- A Belt Around a Pulley
- What's the Area of the Hexagon
- How Long Will the World Series Last?
- What's the Area of the Wedge?
- Find the Dimensions of the Rectangle
- Move the Last Digit to the Front
- A Pyramid of Cubes
- Reverse the Digits
- Two Circles in Two Triangles
- Dissecting Squares and Cubes
- A Sum of Two Reciprocals
- Travels with a Canoeist
- Those Crazy Mathematicians
- Polygons Inside Circles Inside Polygons
- What are the Kids' Ages?
- Use Each Nonzero Digit Once
- Product of Cosines Equation
- A Stack of Balls
- A Magic Hexagram
- How Many Digits Does It Take?
- A Putnam Problem
- Infinite Nested Radical
- Does Your Cup Runneth Over?
- A Box of Valentine's Candy
- Locker Lottery
- David Hilbert's Birthday
- Height of an Octahedron
- Size of the Block?
- Polygonal Partitions
- Digital Divisors
- Another Nested Radical
- Some Divisibility Tests
- The Difference of Two Squares Being a Cube
(Putnam)
- A Clock Puzzle from "Car Talk"
- The Area of a Heart
- An Area Ratio
- The Initial Digits of N Cubed are the
same as N
- More Fun with Digits
- What's the Ratio of Squares' Areas?
- Find the Length of the Triangle's Side
- Four Similar Rectangles
- An (Almost) Regular Octagon
- A Trapezoid with Perpendicular Diagonals
- The Spider and the Fly
- Alice and Bob and the Three Denominations
- How Many Paths from X to Y?
- Area of a Cyclic Quadrilateral
- Area of a Trapezoid
- Sums of Consecutive Integers in Several
Ways
- Packing Five Spheres in a Box
- A Cryptarithm ("Sam's Idea")
- Finding a Cube Ending in Consecutive Digits
- Five Tangent Circles
- Five Numbers in Geometric Progression
- A Ratio of Lengths in a Triangle
- A Circle in a Sector
- How Much Does the Mozzarella Weigh?
- Solve the Nonlinear System of Equations
- A Circle and a Square in a Circle
- Find the Radius of the Circle
- Use Each Digit Exactly Once
- A Quasi-Magic Triangle
- Compute the Length of a Rod from the
Length of its Shadows
- A Japanese Junior High Entrance Exam
Question
- What has the Same Value as MATH?
- Multiples Whose Digits are All 1's and2's
- Find the Tetrahedron's Volume
- A Tribute to Martin Gardner
- Find the Inner and Outer Radii of the
Annulus
- A Palindromic Square
- Find the Village's Population
- Three Circles, a Triangle, and a
Rectangle
- Three Circles in a Semicircle
- Four Semi-circles in a Square
- Find the Area of the Triangular Pieces
- A Cryptarithm
- Decreasing Numbers
- Find the Radii of the Circles
- How Many Equidistant Planes are There?
- Find the Missing Number
- A Magic Pentagram (Revised)
- Packing Two Circles in a 3-4-5 Right
Triangle
- A 4×4 Array of 1's and −1's
- A Quincunx in a Circle
- Dividing Lines with Side-Trisectors
- A 3-4-5 Triangle in a Square
- Find the Largest Power of 2006 that
Divides 2006 Factorial
- Numbering the Vertices of a Cube with No
Consecutive Numbers Adjacent
- A Diophantine Equation
- A Formula for the Area of a Face of an
Ortho-Simplex
- A Point in a Square
- A System of Equations
- A Broken Pole
- When are You Unable to Make Change?
- An Exponential Equation
- Quadrisecting a Square
- Five Similar Rectangles
- A Fibonacci Question
- Five Circles in a Square
- A Circle Inscribed in a Triangle Inscribed in
a Circle
- Dissecting an Isosceles Triangle into Two
Isosceles Triangles
- A Circle Internally Tangent to Three Others
- How Long Does It Take to Do the Job?
- Find the Polynomials (Putnam)
- Happy New Year, 2008
- Rationalize the Denominator
- In How Many Ways Can You Write {1,...,7} as the
Union of Two Sets?
- A Drawer Full of Socks
- What's the Probability of No Two in the Same
Row or Column?
- Fitting Regular Polygons Around a Vertex
- A Geometric Progression That Sums to Seven
- Find the Ratio of the Areas of the Octagons
- Three Students Play a Game
- Alice at the 7-Eleven
- A Triangle Problem from the Canadian Mathematics Olympiad
- An Infinite Nested Radical (Putnam)
- How Many Ways Can You Color a Cube?
- A Sphere Inscribed in the Frustum of a Cone
- Knights and Knaves
- Given the Sum and the Sum of the Squares, Find the Sum of the Cubes
- What Curve Does the Mechanism Trace Out?
- Dividing a Cube Into Two Pieces
- A Problem Involving Sums of Powers
- Quotient, Remainder, and Divisor in Geometric Progression
- A Fractional Cryptarithm
- The Magic Number is 56
- Find the Volume of the Tetrahedron
- An Avian Cryptarithm
- Palindromic Dates
- One Number in Each Row, Column, and Main Diagonal
- A Trapezoid with Perpendicular Diagonals
- A Self-Intersecting Curve
- A Non-Linear System of Equations
- An Exponential Equation
- Find the Volume of the Region Where the Cones Overlap
- Math Summer Camp Probability Problem
- Find the Area of the Region Where A Square Subtends a Given Angle at Each Point
- A System of Equations Whose Solution is a Geometric Sequence
- Find the Area Between the Two Circles
- Five Consecutive Integers: The Sum of the Middle Three Is a Perfect Square; The Sum of All Five is a Perfect Cube
- Find the Remainder
- Find the Area Inside the Big Circle but Outside the Small Ones
- Fold Up the Polyhedron and Find Its Volume
- A Quadrilateral Circumscribed About a Circle
- Find the Polynomials with the Given Roots
- What's the Smallest Circular Region in Which the Coin Game Can Be Played? *
- Given the Changes in the Average, Find the Original Average
- N and 2N Together Contain All Ten Digits
- Find the Area of the Quadrilateral
- A Sangaku Problem
- A Digit Switching Problem
- Sum the Six-Digit Numbers
- How Many Paths Are There In the Grid?
- A System of Diophantine Equations
- A Cryptarithm
- Find the Ratio of Octagonal Areas
- Four Regular Tetrahedra on the Faces of a Fifth
- N/2 is a Perfect Square, N/3 is a Perfect Cube,...
- Fill in the Grid
- Dividing Diagonals in a Regular Hexagon +
- An Equation Involving the Floor Function +
- A Dissection of a Regular 18-gon +
- Alice, Bob, and Charlie in a Foot Race o
- Three Unit Spheres in a Hemisphere o
- To Which Points Can the Kangaroo Hop? o
- Still a Perfect Square After Increasing Each Digit by Three o
- Find the Area of the Triangle Around the Pythagorean Configuration o
- What's the Area the Goat Can Graze? o
- Polygonal Rings o
- Representing 100 as Sums of Powers of 3 o
- Numbered Balls in Boxes o
- Slicing a Circle with Two Pairs of Perpendicular Chords o
- A Ladder Leaning Against a Wall o
- How Many Flags are There? o
- Numbers Divisible by 2,3,4,...,or 10 o
- What's the Period of 1/2014? o
- The Google Play Logo o
- Find n Such That n Plus the Sum
of Its Digits is 2014 o
- Make the Pieces of the Triangle Have Equal Area o
- Find the Area of the Spherical Shadow o
- A Set Where Factorization into "Primes" is not Unique o
- A Large Circle and Four Small Semicircles in a Square o
- Solve the System and Find the Ratio o
- Find All Functions Satisfying the Relation o
- Investigating a Grid of Integers o
- Another Japanese Temple Problem o
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