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?

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.

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

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

KBB3 Problem 1

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