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.
  1. Counting Intersection Points
  2. Find the Unknown Area
  3. Rearranging Digits
  4. What's the Locus?
  5. Egg Drop Soup
  6. The Moat
  7. The Pane of it All
  8. A Magic Square
  9. Reflections
  10. A Problem from Lewis Carroll
  11. Drawing a Zigzag through 16 points
  12. How Big is the Softball?
  13. What's the Radius of the Circle?
  14. A Belt Around a Pulley
  15. What's the Area of the Hexagon
  16. How Long Will the World Series Last?
  17. What's the Area of the Wedge?
  18. Find the Dimensions of the Rectangle
  19. Move the Last Digit to the Front
  20. A Pyramid of Cubes
  21. Reverse the Digits
  22. Two Circles in Two Triangles
  23. Dissecting Squares and Cubes
  24. A Sum of Two Reciprocals
  25. Travels with a Canoeist
  26. Those Crazy Mathematicians
  27. Polygons Inside Circles Inside Polygons
  28. What are the Kids' Ages?
  29. Use Each Nonzero Digit Once
  30. Product of Cosines Equation
  31. A Stack of Balls
  32. A Magic Hexagram
  33. How Many Digits Does It Take?
  34. A Putnam Problem
  35. Infinite Nested Radical
  36. Does Your Cup Runneth Over?
  37. A Box of Valentine's Candy
  38. Locker Lottery
  39. David Hilbert's Birthday
  40. Height of an Octahedron
  41. Size of the Block?
  42. Polygonal Partitions
  43. Digital Divisors
  44. Another Nested Radical
  45. Some Divisibility Tests
  46. The Difference of Two Squares Being a Cube (Putnam)
  47. A Clock Puzzle from "Car Talk"
  48. The Area of a Heart
  49. An Area Ratio
  50. The Initial Digits of N Cubed are the same as N
  51. More Fun with Digits
  52. What's the Ratio of Squares' Areas?
  53. Find the Length of the Triangle's Side
  54. Four Similar Rectangles
  55. An (Almost) Regular Octagon
  56. A Trapezoid with Perpendicular Diagonals
  57. The Spider and the Fly
  58. Alice and Bob and the Three Denominations
  59. How Many Paths from X to Y?
  60. Area of a Cyclic Quadrilateral
  61. Area of a Trapezoid
  62. Sums of Consecutive Integers in Several Ways
  63. Packing Five Spheres in a Box
  64. A Cryptarithm ("Sam's Idea")
  65. Finding a Cube Ending in Consecutive Digits
  66. Five Tangent Circles
  67. Five Numbers in Geometric Progression
  68. A Ratio of Lengths in a Triangle
  69. A Circle in a Sector
  70. How Much Does the Mozzarella Weigh?
  71. Solve the Nonlinear System of Equations
  72. A Circle and a Square in a Circle
  73. Find the Radius of the Circle
  74. Use Each Digit Exactly Once
  75. A Quasi-Magic Triangle
  76. Compute the Length of a Rod from the Length of its Shadows
  77. A Japanese Junior High Entrance Exam Question
  78. What has the Same Value as MATH?
  79. Multiples Whose Digits are All 1's and2's
  80. Find the Tetrahedron's Volume
  81. A Tribute to Martin Gardner
  82. Find the Inner and Outer Radii of the Annulus
  83. A Palindromic Square
  84. Find the Village's Population
  85. Three Circles, a Triangle, and a Rectangle
  86. Three Circles in a Semicircle
  87. Four Semi-circles in a Square
  88. Find the Area of the Triangular Pieces
  89. A Cryptarithm
  90. Decreasing Numbers
  91. Find the Radii of the Circles
  92. How Many Equidistant Planes are There?
  93. Find the Missing Number
  94. A Magic Pentagram (Revised)
  95. Packing Two Circles in a 3-4-5 Right Triangle
  96. A 4×4 Array of 1's and −1's
  97. A Quincunx in a Circle
  98. Dividing Lines with Side-Trisectors
  99. A 3-4-5 Triangle in a Square
  100. Find the Largest Power of 2006 that Divides 2006 Factorial
  101. Numbering the Vertices of a Cube with No Consecutive Numbers Adjacent
  102. A Diophantine Equation
  103. A Formula for the Area of a Face of an Ortho-Simplex
  104. A Point in a Square
  105. A System of Equations
  106. A Broken Pole
  107. When are You Unable to Make Change?
  108. An Exponential Equation
  109. Quadrisecting a Square
  110. Five Similar Rectangles
  111. A Fibonacci Question
  112. Five Circles in a Square
  113. A Circle Inscribed in a Triangle Inscribed in a Circle
  114. Dissecting an Isosceles Triangle into Two Isosceles Triangles
  115. A Circle Internally Tangent to Three Others
  116. How Long Does It Take to Do the Job?
  117. Find the Polynomials (Putnam)
  118. Happy New Year, 2008
  119. Rationalize the Denominator
  120. In How Many Ways Can You Write {1,...,7} as the Union of Two Sets?
  121. A Drawer Full of Socks
  122. What's the Probability of No Two in the Same Row or Column?
  123. Fitting Regular Polygons Around a Vertex
  124. A Geometric Progression That Sums to Seven
  125. Find the Ratio of the Areas of the Octagons
  126. Three Students Play a Game
  127. Alice at the 7-Eleven
  128. A Triangle Problem from the Canadian Mathematics Olympiad
  129. An Infinite Nested Radical (Putnam)
  130. How Many Ways Can You Color a Cube?
  131. A Sphere Inscribed in the Frustum of a Cone
  132. Knights and Knaves
  133. Given the Sum and the Sum of the Squares, Find the Sum of the Cubes
  134. What Curve Does the Mechanism Trace Out?
  135. Dividing a Cube Into Two Pieces
  136. A Problem Involving Sums of Powers
  137. Quotient, Remainder, and Divisor in Geometric Progression
  138. A Fractional Cryptarithm
  139. The Magic Number is 56
  140. Find the Volume of the Tetrahedron
  141. An Avian Cryptarithm
  142. Palindromic Dates
  143. One Number in Each Row, Column, and Main Diagonal
  144. A Trapezoid with Perpendicular Diagonals
  145. A Self-Intersecting Curve
  146. A Non-Linear System of Equations
  147. An Exponential Equation
  148. Find the Volume of the Region Where the Cones Overlap
  149. Math Summer Camp Probability Problem
  150. Find the Area of the Region Where A Square Subtends a Given Angle at Each Point
  151. A System of Equations Whose Solution is a Geometric Sequence
  152. Find the Area Between the Two Circles
  153. Five Consecutive Integers: The Sum of the Middle Three Is a Perfect Square; The Sum of All Five is a Perfect Cube
  154. Find the Remainder
  155. Find the Area Inside the Big Circle but Outside the Small Ones
  156. Fold Up the Polyhedron and Find Its Volume
  157. A Quadrilateral Circumscribed About a Circle
  158. Find the Polynomials with the Given Roots
  159. What's the Smallest Circular Region in Which the Coin Game Can Be Played? *
  160. Given the Changes in the Average, Find the Original Average
  161. N and 2N Together Contain All Ten Digits
  162. Find the Area of the Quadrilateral
  163. A Sangaku Problem
  164. A Digit Switching Problem
  165. Sum the Six-Digit Numbers
  166. How Many Paths Are There In the Grid?
  167. A System of Diophantine Equations
  168. A Cryptarithm
  169. Find the Ratio of Octagonal Areas
  170. Four Regular Tetrahedra on the Faces of a Fifth
  171. N/2 is a Perfect Square, N/3 is a Perfect Cube,...
  172. Fill in the Grid
  173. Dividing Diagonals in a Regular Hexagon +
  174. An Equation Involving the Floor Function +
  175. A Dissection of a Regular 18-gon +
  176. Alice, Bob, and Charlie in a Foot Race o
  177. Three Unit Spheres in a Hemisphere o
  178. To Which Points Can the Kangaroo Hop? o
  179. Still a Perfect Square After Increasing Each Digit by Three o
  180. Find the Area of the Triangle Around the Pythagorean Configuration o
  181. What's the Area the Goat Can Graze? o
  182. Polygonal Rings o
  183. Representing 100 as Sums of Powers of 3 o
  184. Numbered Balls in Boxes o
  185. Slicing a Circle with Two Pairs of Perpendicular Chords o
  186. A Ladder Leaning Against a Wall o
  187. How Many Flags are There? o
  188. Numbers Divisible by 2,3,4,...,or 10 o
  189. What's the Period of 1/2014? o
  190. The Google Play Logo o
  191. Find n Such That n Plus the Sum of Its Digits is 2014 o
  192. Make the Pieces of the Triangle Have Equal Area o
  193. Find the Area of the Spherical Shadow o
  194. A Set Where Factorization into "Primes" is not Unique o
  195. A Large Circle and Four Small Semicircles in a Square o
  196. Solve the System and Find the Ratio o
  197. Find All Functions Satisfying the Relation o
  198. Investigating a Grid of Integers o
  199. Another Japanese Temple Problem o

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