Advanced Problem Archive

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