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: Darij Grinberg, Problem
U275, Mathematical Reflections 4/2013.
My solution as a PDF file.
Sourcecode.
Let μ be the number-theoretical Möbius function. Let a_{f} be a real number for every positive integer f. Prove that
the sum of (μ(g)/g) de a_{de/g} over all d | m, all e | n and all g | gcd(d, e)
= the sum of f a_{f} over all f | mn.
Solution to: Darij Grinberg, Problem
U228, Mathematical Reflections 2/2012.
(Link to the solution. For the problem see 2/2012.)
My solution as a PDF file, including two unpublished alternative solutions and a generalization.
Let L/K be a separable algebraic extension of fields and let V, W and U be L-vector spaces.
Let h : V × W -> U be a K-bilinear map satisfying
h(xa, xb) = x^{2} h(a, b) for every x ∈ L, a ∈ V and b ∈ W.
Prove that h is L-bilinear.
Notes: 1. The solution file contains three solutions, one of which generalizes the problem from separable algebraic field extensions to commutative separable algebras over commutative rings.
2. The problem can be discussed on MathLinks.
Solution to: Darij Grinberg, Problem
O222, Mathematical Reflections 1/2012.
(Link to the solution. For the problem see 1/2012.)
My solution (also the second published
solution) as a PDF file.
Let n be a nonnegative integer. Consider two n-tuples (a_{1}, a_{2}, ..., a_{n}) and (b_{1}, b_{2}, ..., b_{n}) of nonnegative reals, and a permutation σ of the set {1,2,...,n}. For every k in the set {1,2,...,n}, let c_{k} be the maximum of the set
{a_{1}b_{k}, a_{2}b_{k}, ..., a_{k}b_{k}} ∪ {a_{k}b_{1}, a_{k}b_{2}, ..., a_{k}b_{k}}.
Prove that a_{1}b_{σ(1)} + a_{2}b_{σ(2)} + ... + a_{n}b_{σ(n)} ≤ c_{1} + c_{2} + ... + c_{n}.
Note: The solution file contains some background on this problem.
Solution to: Gabriel Dospinescu, Problem
O219, Mathematical Reflections 1/2012.
(Link to the solution. For the problem see 1/2012.)
My solution (also the second published
solution) as a PDF file.
Let a, b, c, d be positive reals such that (1-c)/a + (1-d)/b + (1-a)/c + (1-b)/d ≥ 0. Prove that a(1-b) + b(1-c) + c(1-d) + d(1-a) ≥ 0.
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 a_{k} = 2 cos(π / 2^{n-k}). Prove that
product_{k=0}^{n-1} (1-a_{k}) = (-1)^{n-1}
/ (1 + a_{0}).
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} ≥ (x^{x}y^{y}z^{z})^{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) + 2^{2} binom(n,4) + ...)² (binom(n,1) + 2
binom(n,3) + 2^{2} 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
Solution to: Anon, Problem
A5715, AMM February 1970, in a generalized version.
My solution as a PDF file.
(Generalized statement:) Let R be a commutative ring with unity.
Let a_{1}, a_{2}, ..., a_{n} as well as
b_{1}, b_{2}, ..., b_{n} be elements of
R such that a_{1} b_{1} + a_{2} b_{2}
+ ... + a_{n} b_{n} = 1, and such that any two
indices i and j with i < j satisfy b_{i} b_{j} = 0.
Let T be a free R-module with basis (e_{1}, e_{2},
..., e_{n}). Let S be the R-submodule of T generated by
b_{1} e_{1}, b_{2} e_{2}, ...,
b_{n} e_{n}. Show that T/S is a free R-module of
rank n-1.
Solution to: Marian
Tetiva, Problem
11391, AMM November 2008.
My solution as a PDF file. Longer (more detailed) version.
Let p be some prime, s some positive integer. Also, let k and n be
integers such that n ≥ k ≥ p^{s} - p^{s-1}. Let x_{1}, x_{2},
..., x_{n} be integers. Assume that
(p,s,k) ≠ (2,2,2) and (s,k) ≠ (1,p-1). Prove that both of the sums
sum of ( (-1)^{j} · (binomial coefficient (n-k+j) over
j) ·
(number of all (k-j)-element subsets T of {1,2,...,n} such that the sum
of x_{t} over all t in T is divisible by
p) ) over j = 0, 1, ..., k
and
sum of ( (-1)^{j} · (binomial coefficient (n-k+j) over
j) ·
(number of all (k-j)-element subsets T of {1,2,...,n} such that the sum
of x_{t} over all t in T is not divisible
by p) ) over j = 0, 1, ..., k
are divisible by p^{s}.
Solution to: Omran Kouba, Problem
11392, AMM November 2008.
My solution as a PDF file.
A point M in the plane of a regular n-gon P is orthogonally projected
onto each side of this n-gon. Assume that these projections all lie
inside the respective sides of the n-gon. Then, the n segments joining
M to the vertices of P, as well as the n segments joining M to these
projections, subdivide the n-gon P into 2n triangles. We label these
triangles by 1, 2, ..., 2n in a counterclockwise way. Prove that the
sum of the areas of the triangles with odd labels is half the area of
P.
Solution to: Cosmin Pohoata, Problem
11393, AMM November 2008.
My solution as a PDF file.
In triangle ABC, let M and Q be points on the segment AB, let N and R
be points on the segment CA, and let P and S be points on the segment
BC. Denote m = BM / MA, n = CN / NA, p = CP / PB, q = AQ / QB, r = AR /
RC, s = BS / SC. Prove that the lines MN, PQ and RS are concurrent if
and only if mpr + nqs + mq + nr + ps = 1.
Solution to: Catalin Barboianu, Problem
11402, AMM December 2008.
My solution as a PDF file.
Given a continuous function f from [0, 1] to the set of nonnegative
reals such that f(0) = f(1) = 0 and every 0 < x < 1 satisfies
f(x) > 0. Prove that there exists a square which has two of its
vertices in the interval (0,1) on the x-axis and its other two vertices
on the graph of f.
Solution to: Finbarr Holland, Problem
11415, AMM February 2009.
My solution as a PDF file. Warning: it's
very long and boring (because written down with maximal detail - I
really had to do this to get sure it was right).
Show that the determinant of the sum of n positive-definite Hermitian
complex 2x2 matrices is greater or equal to the sum of their largest
eigenvalues times the sum of their smallest eigenvalues. (Actually, the
problem states this in a slightly different way, but that's the main
point.)
If my solution is correct, it renders the "positive-definite" condition
useless. In order to make sure it is correct, I have written out every
single step of the proof, but a residual uncertainty remains. You
should best scroll down to Lemma 2 (it's on page 14) immediately after
reading Lemma 1, because the proof of Lemma 1 is well-known.
Solution to: Christopher Hillar, Problem
11422, AMM March 2009.
My solution as a PDF file.
Let H be a real symmetric square matrix whose eigenvalues are pairwise
distinct. Let A be a real matrix of the same size. Let H_{0} =
H, let H_{1} = AH_{0} - H_{0}A and let H_{2}
= AH_{1} - H_{1}A. Assume that the matrices H_{1}
and H_{2} are symmetric. Prove that AA^{t} = A^{t}A
(in other words, prove that the matrix A is normal).
One day I might add a solution to #11397
(I am still pondering whether to write up a proof of Karamata...).
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).
Other PEN solutions posted by me on MathLinks. You can find them all on the MathLinks forum in a better readable format; I am just including them here for the sake of completeness.
Octogon
This was a Vietnamese project at creating a journal for mathematical problems. Sadly, it has not kicked off.A modulus and square root inequality.
PDF file. Also, the sourcecode.
For any four reals a, b, c, d, prove that
|a + b - c - d| + |a + c - b - d| + |a + d - b - c|
≥ |sqrt(a^{2} + b^{2}) - sqrt(c^{2} + d^{2})| + |sqrt(a^{2} + c^{2}) - sqrt(b^{2} + d^{2})|
+ |sqrt(a^{2} + d^{2}) - sqrt(b^{2} + d^{2})|.
Problems from the Book (diverse mathematical problems)
Solution
to: Problem
19.9.
PDF file.
Let n be a positive integer. Let w_{1}, w_{2}, ..., w_{n}
be n reals. Prove the inequality
sum_{i=1}^{n} sum_{j=1}^{n} ijw_{i}w_{j} / (i+j-1) ≥ (sum_{i=1}^{n} w_{i})².
Click here for a
longer and more complete version of the solution. (Warning: the
completeness comes at the cost of readability.)
QED-Mathematikolympiade (click here for a description of this context)
Solutions to several problems in German
Other problems and solutions
At least |S|^{k-1} · (|S| - 1) frontiers: a graph theory problem.
PDF file. Also, the sourcecode.
Let S be a finite set, and k be a positive integer. Let A be a subset of S^{k} satisfying |A| = |S|^{k-1}. Let B = S^{k} \ A.
We let F be the set of all pairs (a, b) ∈ A × B such that the k-tuples a and b differ in at most one entry. Prove that |F| ≥ |S|^{k-1} · (|S| - 1).
Some AoPS/MathLinks solutions
a graph
with 2n+1 vertices |
some
inequalities for 2 triangles | a
combinatorial identity from number theory | an
Oddtown variation for acyclic graphs
| a
nonstandard inequality | generalizing
a BMO problem | ARO
2005 10.2 combinatorics | an
unwieldy proof for an unwieldy combinatorial identity | generalizing
the ordered Shapiro inequality | a
double count | rearrangement
inequality redux |
Solutions to review problems
Darij Grinberg