Found problems: 54
2007 iTest Tournament of Champions, 4
Black and white coins are placed on some of the squares of a $418\times 418$ grid. All black coins that are in the same row as any white coin(s) are removed. After that, all white coins that are in the same column as any black coin(s) are removed. If $b$ is the number of black coins remaining and $w$ is the number of remaining white coins, find the remainder when the maximum possible value of $bw$ gets divided by $2007$.
2008 iTest Tournament of Champions, 2
Let $N$ be the smallest natural number that, when written to its left, forms an integer with twice as many digits that is a perfect square. Find the remainder when $N$ is divided by $1000$.
2007 iTest Tournament of Champions, 1
Let $a$ and $b$ be perfect squares whose product exceeds their sum by $4844$. Compute the value of \[\left(\sqrt a + 1\right)\left(\sqrt b + 1\right)\left(\sqrt a - 1\right)\left(\sqrt b - 1\right) - \left(\sqrt{68} + 1\right)\left(\sqrt{63} + 1\right)\left(\sqrt{68} - 1\right)\left(\sqrt{63} - 1\right).\]
2008 iTest Tournament of Champions, 3
A regular $2008$-gon is located in the Cartesian plane such that $(x_1,y_1)=(p,0)$ and $(x_{1005},y_{1005})=(p+2,0)$, where $p$ is prime and the vertices, \[(x_1,y_1),(x_2,y_2),(x_3,y_3),\cdots,(x_{2008},y_{2008}),\]
are arranged in counterclockwise order. Let \begin{align*}S&=(x_1+y_1i)(x_3+y_3i)(x_5+y_5i)\cdots(x_{2007}+y_{2007}i),\\T&=(y_2+x_2i)(y_4+x_4i)(y_6+x_6i)\cdots(y_{2008}+x_{2008}i).\end{align*} Find the minimum possible value of $|S-T|$.