Advanced 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. A Bitangent to a Quartic Curve
  2. Minimizing the Area of a Parabolic Sector
  3. Finding the Height of a Mountain
  4. A Circle Bitangent to a Parabola
  5. The Area and Perimeter of a Certain Region
  6. An Incorrect Quotient Rule for Derivatives
  7. Which Value of a Gives a Unique Solution?
  8. Finding the Area (resp. Volume) of Overlapping Circles (resp. Spheres)
  9. Expected Area of an Inscribed Triangle
  10. The Limit of a Harmonic Sum
  11. Maximize the Area Between a Circle and a Polygon
  12. Maximize the Total Surface Area of an Inscribed Cylinder and Cone
  13. Two Infinite Series
  14. The Number of Solutions of a Trigonometric Equation
  15. Drilling a Cube
  16. Overlapping Tangent Parabolas
  17. Surface Area and Volume of a Pendant
  18. Minimize the Arclength
  19. When Does the Sum Get Big?
  20. Two Leibnizian Infinite Series
  21. A Product of Cosines Equation
  22. Minimizing the Perimeter of a Parabolic Region
  23. Half Full or Half Empty?
  24. A Putnam Problem
  25. Intersecting Cylinders, Part I
  26. Intersecting Cylinders, Part II
  27. A Heart-Shaped Box
  28. Probability of Making a Triangle
  29. Maximize the People-Days
  30. Making a Planter
  31. What's the Average Width?
  32. Turning the Corner with a Rod
  33. Putting a Sphere in a Cone
  34. Average Size of a Piece of a Broken Stick
  35. Rotating Reuleaux
  36. Find the Limit of the Sum (Putnam)
  37. The Average Distance Between Points in the Cantor Set
  38. Put the Heart in a Box
  39. Where Does the Spiral End?
  40. What Shape Does the Sliding Rod Trace Out?
  41. The Expected Area of a Triangle
  42. Another Limit Problem
  43. Find the Area of the Deltoid
  44. Two Cyclists
  45. Three Circles in a Parabola
  46. A Circle Perpendicular to a Parabola
  47. The Smallest Triangle Containing Two Unit Circles
  48. Area of a Lissajous Figure
  49. Another Polygonal Spiral
  50. What's the Area of the Bishop's Hat?
  51. What's the Probability that Alice Makes Fewer Tosses than Bob?
  52. Sum the Series
  53. Buffon's Bent Needle
  54. A Weight on a Rope
  55. Spreading Rumors
  56. Yet Another Infinite Series
  57. The Smallest Circle Containing Two Ellipses
  58. A Tetrahelix (from H.S.M. Coxeter) *
  59. The Area Common to Two Roses
  60. Find the Parabola Given Three Tangent Lines
  61. Again with the Infinite Series?
  62. Area Equals Arclength
  63. Buffon's Dartboard
  64. Light Reflecting in an Ellipse
  65. Maximize the Tetrahedron's Volume
  66. Compute the Area of a Triangle from the Area of its Shadows
  67. An Infinite Series involving the Inverse Tangent
  68. Find the Volume of the Intersecting Cones
  69. A Differential Equation
  70. Find the Volume and Surface Area of this Solid
  71. A Tribute to Martin Gardner
  72. Find the Determinant of the Matrix
  73. How Much Green is There in the Wachovia Logo?
  74. The Locus of Certain Bitangent Points
  75. How Tall Can the Stack of Hemispheres Be?
  76. Evaluate the Integral
  77. An Infinite Series with Factorials
  78. Find the Ellipse with Least Area Containing Two Unit Circles, etc.
  79. A Solid with a Square Shadow, a Circular Shadow, and a Triangular Shadow
  80. What's the Area the Particle Can Reach? o
  81. What's the Probability of Forming a Triangle? ×
  82. Find the Volume of the Great Rhombicuboctahedron
  83. A Limit Involving Partial Sums of the Exponential Function
  84. A Generalization of the Hyperbolic Trig Functions
  85. A Trigonometric Problem
  86. Find the Polynomial
  87. Find the Plane Tangent to the Three Paraboloids
  88. Find the Area of the Hexagon Formed by Side-Trisectors
  89. If You Divide a Cube into Eight Pieces with Three Perpendicular Planes, How Big Can the Largest Piece Be? *
  90. Integrating Powers of the Cantor Function
  91. Three Mutually Perpendicular Lines That Pass Through a Circle
  92. A Problem from Group Theory *
  93. Minimize the Sum of the Reciprocals of the Distances Between Four Points on a Sphere o
  94. Two Infinite Series
  95. Find the Volume of a Region Near a Cone o
  96. What's the Expected Length of a Slice of a Rectangle? o
  97. How Far (on Average) to the Circle's Rim? o
  98. An Integral Involving the Fractional Part Function
  99. Find the Area Inside the Curve
  100. Two Infinite Chains of Circles
  101. A Limit Involving Two Trigonometric Integrals o
  102. Minimize the Product of the Lengths of Two Perpendicular Diameters of an Ellipse o
  103. A Limit Involving the Solution of a Trigonometric Equation o
  104. A Circle Rolling Down a Ramp o
  105. Find the Area Inside the Parametrized Curve o
  106. Find the Volume Inside the Two Ellipsoids o
  107. What's the Expected Distance Between Two Points on a Sphere? (Putnam)
  108. Happy New Year, 2008
  109. A Nested Radical Question o
  110. Find the Limit of the Recursive Sequence
  111. Minimize the Length of the Normal to a Paraboloid
  112. Evaluate the Infinite Divisorial Expression
  113. What's the Probability that the Product is Greater than the Sum?
  114. How Many Ways Are There to Visit Every Room? o
  115. Evaluate the Integral
  116. Find the Ratio of the Areas (Quadric Curve)
  117. When Are the Roots All Real?
  118. A Limit Problem Involving Binomial Coefficients
  119. Evaluate the Power Series (Putnam)
  120. Alice and Bob Play a Coin-Tossing Game
  121. Two Compass and Straightedge Problems Involving an Ellipse
  122. A Hyperbola Tangent to an Ellipse
  123. Maximize the Volume Inside the Hemisphere and Outside the Sphere
  124. What's the Expected Distance Between the Two Points?
  125. Exploring a Common Error in Calculus
  126. Fencing in a Pen Next to a Parabolic Lake
  127. A Surface Integral Over the Unit Sphere
  128. The Surface Area and Volume of a Certain Solid
  129. Maximize the Area of a Rectangle Inscribed in an Annulus
  130. Find the Shortest Path on the Intersection of Three Cylinders
  131. An Infinite Series Involving Arctangent
  132. A Grazing Zebra
  133. An Integral Involving the Fractional Part Function
  134. The Princess and the Moat
  135. Polynomials on [0,1] Bounded by 1 and −1
  136. An Infinite Series Involving the Tangent Function
  137. The Expected Content of Certain Simplices in Two and Three Dimensional Cubes o
  138. A Cubic Diophantine Equation
  139. Find the Limit of the Recursive Sequence
  140. A Limit Involving Factorials
  141. An Infinite Product Involving the Tangent Function
  142. Find the Volume of the Region Bounded by Three Paraboloids
  143. Minimize the Function
  144. When Does the Transition from Concave to Convex Occur?
  145. An Infinite Sum Involving the Nearest Integer Function
  146. An Integral Involving Logarithms and the Floor and Ceiling Functions
  147. The Locus of Polygonal Centroids o
  148. Maximizing Humidifier Efficiency
  149. An Incorrect Version of the Product Rule for Derivatives
  150. A Tilted Conical Water Glass
  151. Buffon's Needle in Three Dimensions o
  152. An Alternating Infinite Series Involving the Natural Logarithm
  153. An Integral Involving the Natural Logarithm and the Floor Function
  154. Maximizing the Area of a Triangle (and a Generalization to Three-Space)
  155. Find the Area of an Ellipse from Three Cross Sectional Lengths
  156. Find the Surface Area and Volume of a Reuleaux Tetrahedron
  157. Quadrisect a 3-4-5 Triangle with Two Perpendicular Lines +
  158. Three Infinite Alternating Series
  159. Find the Surface Area and Volume of a Rotated Cardioid +
  160. Find the Smallest Circle Containing a A Right Triangle and Squares Erected on its Sides o
  161. What's the Probability Two Points Out of n are within Distance d of Each Other? o
  162. A Rod Balanced on a Circular Table o
  163. Evaluate the Integral o
  164. Find the Areas of the Pieces of the Logo o
  165. A Folded Sheet of Paper o
  166. An Erroneous Version of the Chain Rule o
  167. An Infinite Series Involving the Product of Two Arithmetic Progressions o
  168. Evaluate the Limit of a Product o
  169. Maximize the Ratio of the Volume to the Surface Area o
  170. An Infinite Series Involving the Greatest Odd Divisor o
  171. The Area of the Triangle Formed by a Tangent Line to the Curve and the Coordinate Axes is Constant (and Others) o
  172. A 3D Generalization of Advanced Problem #5 o
  173. A Common Tangent to an Ellipse and a Circle o
  174. A Question About the Polygon (resp. Polyhedron) Determined by Random Points on a Circle (Resp. Sphere) o
  175. An Infinite Series Involving a Recursively Defined Sequence o
  176. Double and Triple Integrals Involving the Logarithm and Floor Functions o
  177. Minimize the Length of a Tangent Segment to an Ellipse o
  178. Minimize the Ratio of the Integrals o
  179. Find the Distribution of Unit's Digits of 2014^2014 o
  180. A Bug Walking on the Edges of a Polyhedron o
  181. The Fraction of Water in a Tilted Glass o
  182. Find the Probability that a Point in a Triangle Divides it into Three Obtuse Triangles o
  183. The Probability that Three Points on a Sphere Form an Acute Triangle o
  184. Find the Probability that the Line Segments Meet o
  185. What's the Area the Llama Can Graze? o
  186. What's the Probability of Seeing Exactly One Side of a Regular Tetrahedron? o
  187. Evaluate the Infinite Product o
  188. Find the Volumes of the Slabs of the Hypercube o
  189. Bugs on a Regular Polygon o

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