Solutions to review problems
This page is going to contain some of the solutions I submit
to mathematical periodicals with problem sections such as Mathematical
Reflections and The American
Mathematical Monthly. Problems are usually rewritten in order
to avoid excessive quoting and often to homogenize the notations
used in a problem and its solution.
I don't put up solutions on this site prior to the discussion of
the problem in the respective magazine, so usually they won't
appear here directly after the deadline is over.
Solution to: Titu Andreescu, Problem
U111, Mathematical Reflections
1/2009. (Link to the solution. For the problem
see 1/2009.)
My solution (also the second
published solution) as a PDF file.
Let n be a positive integer. For every k in {0, 1, ...,
n-1}, let ak = 2 cos(pi
/ 2n-k). Prove that product_{k=0}^{n-1} (1-ak) = (-1)n-1 / (1 +
a0).
Solution to: Cezar Lupu and
Valentin Vornicu, Problem
U112, Mathematical Reflections
1/2009. (Link to the solution. For the problem
see 1/2009.)
My solution (also the first
published solution) as a PDF file.
Let x, y, z be real numbers greater or equal to 1. Prove
that x^{x³+2xyz} y^{y³+2xyz} z^{z³+2xyz} >= (xxyyzz)yz+zx+xy.
Solution to: Titu Andreescu, Problem
O111, Mathematical Reflections
1/2009. (Link to the solution. For the problem
see 1/2009.)
My solution (also the second
published solution) as a PDF file.
Prove that, for each integer n >= 0, the number
(binom(n,0) + 2 binom(n,2) + 22 binom(n,4) +
...)² (binom(n,1) + 2 binom(n,3) + 22
binom(n,5) + ...)² is triangular.
Here, binom(n,m) means the (n,m)-th binomial coefficient
(that is, n(n-1)...(n-m+1) / m! if m >= 0, and 0
otherwise).
Solution to: Cezar Lupu and Pham
Huu Duc, Problem
O112, Mathematical Reflections
1/2009. (Link to the solution. For the problem
see 1/2009.)
My solution (not published) as a
PDF file.
Let a, b, c be positive real numbers. Prove that
(a³+abc) / (b+c)² + (b³+abc) / (c+a)² + (c³+abc) /
(a+b)² >= 3/2 * (a³+b³+c³)/(a²+b²+c²).
Solution to: Gabriel Dospinescu, Problem
O114, Mathematical Reflections
1/2009. (Link to the solution. For the problem
see 1/2009.)
My solution (also the first
published solution) as a PDF file.
Prove that for all real numbers x, y, z, the following
inequality holds:
(y²+yz+z²) (z²+zx+x²) (x²+xy+y²) >=
3(x²y+y²z+z²x) (xy²+yz²+zx²).
The American Mathematical Monthly
Currently solved (and solutions submitted): #11391, #11392,
#11393, #11395, #11397, #11398, #11401, #11402, #11403, #11406,
#11407, #11409, #11417.
I am planning to put up solutions to #11391, #11397, #11403,
#11406, and #11407, as well as all other solutions that don't
make it into the journal here when time comes.
Project PEN (Problems in Elementary Number Theory)
Solution to: Problem
E16, a. k. a.: Mathematics
Magazine, Problem 1392 by George Andrews.
(This links to my solution on the PEN server. A local
version can be found here: Solution
to Project PEN Problem E 16.)
If n is a positive integer, and p is a prime lying in the
interval ]n, 4n/3], then prove that p divides
sum_{j=0}^{n} binom(n,j) 4, where binom(n,j)
means n! / (j! (n-j)!).
My solution (which generalizes the problem three times)
is an edited and extended version of my
posting in MathLinks topic #150539 (which only
generalizes it one time).
Solutions to review problems
Darij Grinberg