KBB2 Problem 10

Let $P$ be a permutation of the number $(1,2,\cdots,n)$. We apply a process as follows:

Start with the leftmost number and move to the right. Everytime we encounter a number that's bigger than the leftmost number, we swap that number and the leftmost number. Continue until we arrive at the rightmost number.

Define the rank of $P$ as the number of swaps that happened. For example, the permutation$(2,3,1,4)$ has rank 2 because:

$\displaystyle (2,3,1,4) \to (3,2,1,4) \to (4,2,1,3)$

Let$S_n$ be the sum of the ranks of all possible permutations of $(1,2,\cdots,n)$. Is there an $n$ for which$S_n > 2009n!$?

KBB2 Problem 9

Find the maximum $k$ such that

$\displaystyle (a^4 + b^4+c^4+d^4) + 4kabcd \geq \frac{k+1}{4}(a^2+b^2+c^2+d^2)^2$

always holds for $a,b,c,d$ positive reals.

KBB2 Problem 8

We are given a circle centered on $O$ with radius $r$, and given lengths $a,b$ with $a > b$. We are also given a point $P$ on the circle.

For any point $B$ on the circle, construct a point $A$ such that $AP = a, BP = b$. Also construct rhombus $ABCD$ such that $D,B,P$ lie on the same line.

Suppose $B$ moves around the circle and we repeat the construction process above, as long as there is a point $A$ that satisfies the conditions. Show that as $B$ moves around the circle, the resulting point $D$ will trace a straight line. Determine the angle formed by that line and $PO$.

KBB2 Problem 7

Prove that for any $n > 1$, the polynomial

$P(x) = 2008x^n + 7x + 56$

cannot be factored into two non-trivial integer polynomials.

(A polynomial is non-trivial if its highest degree is greater than one)

KBB2 Problem 6

An integer is called homogenous if its decimal representation consists of 1 number. For example, 11 and 222 are homogenous. Prove that there are infinitely many homogeneous numbers that are divisible by 2008.

KBB2 Problem 5

In a triangle ABC where AB = 13, BC = 14, CA = 15, a point P is such that $\angle APB = \angle BPC = \angle CPA = 120^0$. Calculate $AP+BP+CP$

KBB2 Problem 4

For $a,b,c$ positive reals, show that:

$\displaystyle \frac{6a}{9a^2+5(a+b+c)^2} + \frac{6b}{9b^2+5(a+b+c)^2} + \frac{6c}{9c^2+5(a+b+c)^2} \leq \frac{1}{a+b+c}$

KBB2 Problem 3

Suppose the Cartesian plane is tiled with infinitely many square tiles such that:

1. The square tiles must have the same size and form a chessboard-like formation.

2. Each tile corner must lie on a point with integer x-coordinate and y-coordinate

3. The tile sides need not be parallel to the X or Y axis.

Given that the points $(0,0)$ and $(3,1)$ are both tile corners (not necessarily of the same tile), determine all the possible tile sizes.

KBB2 Problem 2

The country of Sikinia uses gold, silver, and bronze coins as its currency. One gold coin is worth 334 silver coins, and one silver coin is worth 208 bronze coins. One day, Ali went to the store and bought an item priced at 3 gold coins. How many ways can Ali pay the item in, assuming that he has unlimited supply of gold, silver, and bronze coins?

KBB2 Problem 1

If $S_n = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$ for $n$ natural number, prove that:

$S_1 + S_2 + \cdots + S_n = (n+1)(S_{n+1} - 1)$