1. Analysis
1.1 Rolling ellipse
1.2 A limit involving a geometric mean of roots of factorials
1.3 A Lobachevsky-type integral
1.4 A quadratic series with the tail of ζ(2)
1.5 A trigonometric logarithmic integral
1.6 An interesting series with a product of two central binomial coefficients
1.7 A log-tangent integral
1.8 An arccosine integral
1.9 Two norms involving function moments
1.10 Reciprocal power sums of real roots for a complex function
1.11 A monotonic and convex quadratic spline
1.12 An estimate of a series involving the absolute value of sine
1.13 An integral with many solutions
1.14 An inequality on the Wallis integrals
1.15 A sequence defined by inequalities
2. Identities
2.1 Two identities concerning Fibonacci and Lucas numbers
2.2 Summing Catalan numbers
2.3 Perfect squares from an arithmetic function
2.4 Identity related to the parity constant
2.5 Wrestle with Jacobi’s triple product
2.6 A Gaussian q-binomial identity
2.7 A rational function identity
2.8 An identity associated with the zeros of zn + 1
2.9 An identity struck by the elementary symmetric functions
2.10 A sum of products of truncated binomial expansions
2.11 An identity with alternating weighted binomial coefficients
2.12 An identity with the generalized binomial coefficients
3. Geometry
3.1 A line perpendicular to the Newton line
3.2 About two cyclic quadrilaterals
3.3 Concurrency of three lines
3.4 A triangle inscribed in another triangle
3.5 A surprising bisection
3.6 A tangent to a circle
3.7 Collinearity of three points
3.8 Another property of the Nagel Point
3.9 An inequality involving the Fermat point of a triangle
3.10 Supplementary pairs of Heronian triangles
4. Combinatorics
4.1 An equality for integer partitions
4.2 A recurrence involving integer partitions
4.3 Another equality for integer partitions
4.4 Tiling a board with cuts
4.5 About the tilings of a 2 × n strip with squares, dominoes, and trominoes
4.6 Fault-free tilings of 3-by-n strip
4.7 A sum involving a p-root of unity
4.8 Sparse binary representation of an integer
4.9 Subsets with equal sums of powers
4.10 Equality of two sums of reciprocals
4.11 A monochromatic pentagons with given area
4.12 Coloring a graph
4.13 Arranging coins along a line
4.14 Averaging the number of fixed points of permutations
4.15 A one-sided inverse involving Motzkin numbers
5. Number Theory
5.1 Divisibility of a central binomial sum
5.2 A congruence implied by Wolstenholme’s theorem
5.3 A congruence of a multiple sum involving Fibonacci numbers
5.4 A congruence for a product of quadratic forms
5.5 Powers of a prime dividing a product of binomial coefficients
5.6 Divisibility of coefficients of powers of polynomials
5.7 A Diophantine equation with powers
5.8 Multiples with small digits
5.9 Another property of the taxicab number 1729
5.10 An application of prime density
5.11 The asymptotic behavior of a sum
5.12 A family of sums with logarithmic powers
5.13 Another application of Bertrand’s postulate
5.14 A property of the product of consecutive primes
5.15 A congruence for the integer part of a power of a cosine
5.16 A variant of the Collatz map
6. Potpourri
6.1 Fixed point of the distance to the boundary9
6.2 A variant alternating series with the floor function
6.3 A real analytic function with no zeros
6.4 Finding a special half-line
6.5 Squares of palindromes
6.6 Counting some strange mappings
6.7 Comparing the coefficients of two generating functions
6.8 A vector sum of modulus at least 1
6.9 Counting equilateral triangles in hypercubes
6.10 Counting rectangles with prime area
6.11 Expected number of throws of an n-sided die
6.12 Removing tiles
6.13 Writing a Gaussian integer as sum of powers of 1 + i
6.14 Enumerating the positive rationals
Biography
Hongwei Chen received his Ph.D. from North Carolina State University in 1991. He is currently a professor of mathematics at Christopher Newport University. He has published more than 60 research papers in analysis and partial differential equations. He also authored Monthly Problem Gems and Classical Analysis: An Approach through Problems published by CRC Press and Excursions in Classical Analysis published by the Mathematical Association of America.
Roberto Tauraso holds a PhD in Mathematics from Scuola Normale Superiore (Pisa). He is currently a professor of mathematical analysis at Tor Vergata University of Rome. His research interests include complex analysis, number theory, and combinatorics. He has published over 60 research papers in various mathematical journals.
