Solution: 3 Variable Inequality

Original problem: http://dharmath.thehendrata.com/2009/11/24/3-variable-inequality/

For $a,b,c > 0$, prove that:

$! (ab(a+b) + bc(b+c) + ca(c+a))^2 \geq 4abc(a+b+c)(a^2+b^2+c^2)$

Solution: Integer Inequality

Original problem: http://dharmath.thehendrata.com/2009/11/27/integer-inequality/

For any positive real number $t$, prove that there are integers $a,b,c,d$ such that

$! a^3b+b^3c+c^3d < tabcd$

System Change

Starting with today, I am going to change the system at which problems and solutions are posted, to avoid spoiling solutions for people who do not want to be spoiled.

For each problem post, there will be a corresponding solution post, with category Solution. The two posts will refer to each other.

The problem post will simply post the problem and a link to the solution post.

The solution post will repeat the problem, and then post the solution behind a "More" link.

Integer Inequality

For any positive real number $t$, prove that there are integers $a,b,c,d$ such that

$! a^3b+b^3c+c^3d < tabcd$

Solution: http://dharmath.thehendrata.com/2009/11/30/solution-integer-inequality/

Polynomial Algorithm

Credit for this problem goes to Zeke Chin

Does there exist an algorithm for the decision problem:

Given a polynomial over $n$ variables $x_1,\cdots,x_n$ with positive integer coefficients, and a positive integer $c$, is it true that $cx_1\cdots x_n \leq P(x_1,\cdots,x_n)$ for all assignments of positive integers to $x_1,\cdots,x_n$?

For example, for $P = x_1^2 + x_2^2$ and $c=2$, the algorithm should return "Yes" since $2x_1x_2 \leq x_1^2 + x_2^2$

3 Variable Inequality

For $a,b,c > 0$, prove that:

$! (ab(a+b) + bc(b+c) + ca(c+a))^2 \geq 4abc(a+b+c)(a^2+b^2+c^2)$

Solution: http://dharmath.thehendrata.com/2009/11/30/solution-3-variable-inequality/

Mean of Means

If $a_i,b_i$ are positive numbers for $i=1,\cdots,n$, and

$! M_r (a,b) = \left( \frac{a^r+b^r}{2} \right)^\frac{1}{r}$

Prove that for $0 < r < s$,

$! \displaystyle \sum M_r(a_i,b_i) \sum M_{-r}(a_i,b_i) \leq \sum M_s(a_i,b_i) \sum M_{-s}(a_i,b_i) \leq \sum a_i \sum b_i$

Triangle Inequality

Given $n$ right triangles each with sides $a_i, b_i, c_i$, with $c_i$ being the hypotenuse ($i=1,\cdots, n)$. Let $A = \sum a_i, B = \sum b_i, C= \sum c_i$.

Prove that

$\displaystyle \frac{a_1b_1}{c_1} + \cdots + \frac{a_nb_n}{c_n} \leq \frac{AB}{\sqrt{A^2+B^2}}$

Harder version: prove that

$\displaystyle \frac{a_1b_1}{c_1} + \cdots + \frac{a_nb_n}{c_n} \leq \frac{AB}{C}$

3 Sequence Inequality

If $a_i, b_i, c_i$ are sequences of positive numbers for $i = 1,2,\cdots,n$, prove the following inequality:

$\sum (a_i+b_i+c_i) \sum \frac{a_ib_i + b_ic_i+ c_ia_i}{a_i+b_i+c_i} \sum \frac{a_ib_ic_i}{a_ib_i + b_ic_i+ c_ia_i} \leq \sum a_i \sum b_i \sum c_i$

where all summations are taken from $i=1$ to $i=n$. When does equality happen?

Matching Binary String

We are given two binary strings $A$ and $B$ each of length $2n$. Both strings contain equal number of ones and zeros to each other. We write $A$ on top of $B$, so that each element of $A$ is paired with each element of $B$ at the corresponding position.

We call the two strings a "match" if there are at least $n$ common elements where the element of $A$ at a certain position matches that of $B$ at that position.

Prove that, we can cyclically shift $B$ a number of times (and wrap around at the end/beginning) so that $A$ and $B$ form a match.

Simplifying Series

Let $a_n = \binom{2n}{n}$ and $b_n = \binom{3n}{n}$. Find a more compact expression for:

$A(x) = a_0 + a_1x + a_2x^2 +\cdots + a_nx^n + \cdots$

$B(x) = b_0 + b_1x + b_2x^2 +\cdots + b_nx^n + \cdots$

ABC String

How many strings of A's, B's, and C's are there that satisfy the following conditions:

1. There are $n$ A's and $2n$ total of B's and C's.

2. Every two adjacent letters are different.

Coin Toss

A boy tosses a fair coin repeatedly. Every time he gets a Head, he earns 1 penny, and every time he gets a Tail, he earns 2 pennies. He starts with no money. Prove that the probability of him at one time having exactly $n$ pennies is $\frac{2+(-\frac{1}{2})^n}{3}$

Path and Parallelogram

A path from $A$ to $B$ consists of several piecewise straight segments (where $b \neq A$). A flea starts from point $P$ and for each segment, it performs the following operation:

If the segment starts at $X$ and ends at $Y$, and the flea is currently at $Z$, the flea will jump to the point $Z'$ such that $XZYZ'$ is a parallelogram (in that order).

The flea starts from $P$ and performs the operation on the segments sequentially until it is done with the last segment, where the flea is now in $Q$. Prove that $APBQ$ is a parallelogram.

Integral

Evaluate $ \int e^x \sec x \tan^2x dx$