On a circular track, there are several gas stations. The total amount of gas in all these stations are just enough for a car to complete one lap. Prove that, starting with an empty car, one can choose an initial position such that he can complete the lap by subsequently filling gas in these stations.

## Wednesday, September 30, 2009

## Tuesday, September 29, 2009

### Black and white painted sphere

The surface of a sphere is painted with black and white paints such that the area of black paint is less than 1/8 of the area of the sphere (the rest of the sphere is painted with white). Prove that one can inscribe a rectangular box in the sphere such that all eight corners land on white points.

More formally, if each point on the sphere is assigned a color black or white such that there is an injective but not surjective one-to-eight unordered mapping from the black points to the white points, prove that there are eight white points on the sphere that form a rectangular box.

More formally, if each point on the sphere is assigned a color black or white such that there is an injective but not surjective one-to-eight unordered mapping from the black points to the white points, prove that there are eight white points on the sphere that form a rectangular box.

## Thursday, September 24, 2009

### a b c sides of a triangle, perimeter less than 2

If $a,b,c$ are sides of a triangle whose perimeter is less than 2, prove that:

$\displaystyle \frac{(1+a)(1+b)(1+c)}{(1-a)(1-b)(1-c)} < \frac{(2+a+b+c)^2}{(2-(a+b+c))^2}$

$\displaystyle \frac{(1+a)(1+b)(1+c)}{(1-a)(1-b)(1-c)} < \frac{(2+a+b+c)^2}{(2-(a+b+c))^2}$

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## Wednesday, September 23, 2009

### Polynomial not sum of squares

Let $P(x,y) = x^2y^4 + x^4y^2 + 1 - 3x^2y^2$

a.) Prove that $P(x,y)$ is non-negative

b.) Prove that $P(x,y)$ cannot be expressed as a sum of squares of polynomials.

a.) Prove that $P(x,y)$ is non-negative

b.) Prove that $P(x,y)$ cannot be expressed as a sum of squares of polynomials.

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### n-expressible integers

A positive integer is called $n$-expressible if it can be written as a sum of $n$ or less squares.

Prove that the product of two $n$-expressible integers is $\frac{n^2-n+2}{2}$-expressible

Prove that the product of two $n$-expressible integers is $\frac{n^2-n+2}{2}$-expressible

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## Tuesday, September 22, 2009

### Non-negative polynomials as sums of squares

Prove that any non-negative real polynomials can be expressed as a sum of squares.

More formally, if $P(x)$ is a polynomial with real coefficients such that $P(x) \geq 0 \forall x \in \mathbb{R}$, show that there exist $Q_1,Q_2,\cdots, Q_k$ polynomials with real coefficients such that $P(x) \equiv Q_1^2(x) + Q_2^2(x) + \cdots + Q_k^2(x)$.

More formally, if $P(x)$ is a polynomial with real coefficients such that $P(x) \geq 0 \forall x \in \mathbb{R}$, show that there exist $Q_1,Q_2,\cdots, Q_k$ polynomials with real coefficients such that $P(x) \equiv Q_1^2(x) + Q_2^2(x) + \cdots + Q_k^2(x)$.

## Monday, September 21, 2009

### Variant of KBB3 Problem 4

This is a harder version of KBB3 Problem 4.

Suppose $n$ is an odd positive integer. Given an $n \times n$ chessboard where each cell is colored black or white. At any step, we are allowed to pick any cell and flip the colors of the cells that are in the same row or the same column as that cell. Prove or disprove: no matter what the initial coloring is, we can always get an all-white board?

Suppose $n$ is an odd positive integer. Given an $n \times n$ chessboard where each cell is colored black or white. At any step, we are allowed to pick any cell and flip the colors of the cells that are in the same row or the same column as that cell. Prove or disprove: no matter what the initial coloring is, we can always get an all-white board?

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### OSN Proposal: Inequality

*I proposed this problem to OSN 2009 but it was not selected.*

For $a,b,c$ positive numbers such that $a+b+c=3$, prove that:

$2(a^{11} + b^{11} + c^{11}) + 3a^3b^3c^3(ab+bc+ca) \geq 5(a^4b^4 + b^4c^4 + c^4a^4)$

### Inequality of The Day

For $a,b,x,y$ real numbers, prove that:

$(3a^2+2ab+5b^2)(3x^2 + 2xy + 5y^2) \geq (3ax + ay + bx + 5by)^2$

Generalization: for $a,b,x,y$ real numbers, and $P,Q,R$ real numbers such that $Q^2 < PR$, prove that:

$(Pa^2+2Qab+Rb^2)(Px^2 + 2Qxy + Ry^2) \geq (Pax + Qay + Qbx + Rby)^2$

$(3a^2+2ab+5b^2)(3x^2 + 2xy + 5y^2) \geq (3ax + ay + bx + 5by)^2$

Generalization: for $a,b,x,y$ real numbers, and $P,Q,R$ real numbers such that $Q^2 < PR$, prove that:

$(Pa^2+2Qab+Rb^2)(Px^2 + 2Qxy + Ry^2) \geq (Pax + Qay + Qbx + Rby)^2$

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## Thursday, September 17, 2009

### KBB3 Problem 4

Given a 3x3 chessboard where each cell is colored black or white. At any step, we are allowed to pick any column or any row and flip all the cell colors in that column or row. Prove or disprove: no matter what the initial coloring is, we can always get an all-white board?

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## Thursday, September 10, 2009

### KBB3 Problem 3

In a triangle $ABC$, $D$ is the midpoint of $BC$. $O_1$ is the circumcenter of $ABD$ and $O_2$ is the circumcenter of $ACD$.

$M$ is the midpoint of arc $BD$ opposite from $A$.

$N$ is the midpoint of arc $CD$ opposite from $A$.

$P$ is the midpoint of arc $AB$ opposite from $D$.

$Q$ is the midpoint of arc $AC$ opposite from $D$.

$R$ on the circumcircle of $ABD$ such that $O_1R \perp AC$.

$O_1$ and $R$ are on different sides of $AC$ or its extension.

$S$ on the circumcircle of $ACD$ such that $O_2S \perp AB$.

$O_2$ and $S$ are on different sides of $AB$ or its extension.

Prove that $MN, PS, $ and $QR$ are concurrent.

$M$ is the midpoint of arc $BD$ opposite from $A$.

$N$ is the midpoint of arc $CD$ opposite from $A$.

$P$ is the midpoint of arc $AB$ opposite from $D$.

$Q$ is the midpoint of arc $AC$ opposite from $D$.

$R$ on the circumcircle of $ABD$ such that $O_1R \perp AC$.

$O_1$ and $R$ are on different sides of $AC$ or its extension.

$S$ on the circumcircle of $ACD$ such that $O_2S \perp AB$.

$O_2$ and $S$ are on different sides of $AB$ or its extension.

Prove that $MN, PS, $ and $QR$ are concurrent.

## Wednesday, September 9, 2009

### KBB3 Problem 2

Given $x_1, \cdots, x_n, y_1, \cdots y_n$ non-negative real numbers such that:

$x_1 + \cdots + x_n = 1$

$x_1y_1 + \cdots + x_ny_n = n^2$

Find the minimum value of

$\displaystyle \frac{x_1(y_1^2 + n^2)}{y_1 + 1} + \cdots + \frac{x_n(y_n^2 + n^2)}{y_n + 1}$

$x_1 + \cdots + x_n = 1$

$x_1y_1 + \cdots + x_ny_n = n^2$

Find the minimum value of

$\displaystyle \frac{x_1(y_1^2 + n^2)}{y_1 + 1} + \cdots + \frac{x_n(y_n^2 + n^2)}{y_n + 1}$

## Tuesday, September 8, 2009

### KBB3 Problem 1

Find the smallest prime number that divides $54^{54} + 55^{55} + 56^{56}$

Labels:
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