Prove that the equation $4x^3-3x+1=2y^2$ has at least $31$ solutions in positive integers $x$ and $y$ with $x\leq 2005$.

Click for solution suppose that $x = 2k+1$. ($k \leq 1002$) $(x+1)(2x-1)^2 = 2y^2 \Longrightarrow (k+1)(4k+1)^2 = y^2$. so, if $(k+1)$ is a square, our equation has a solution. now, we easily see that we have 31 perfect squares $\leq 1002$

Prove that the sum: \[ S_n=\binom{n}{1}+\binom{n}{3}\cdot 2005+\binom{n}{5}\cdot 2005^2+...=\sum_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor}\binom{n}{2k+1}\cdot 2005^k \] is divisible by $2^{n-1}$ for any positive integer $n$.

Let $\mathcal{C}$ be a circle with center $O$, and let $A$ be a point outside the circle. Let the two tangents from the point $A$ to the circle $\mathcal{C}$ meet this circle at the points $S$ and $T$, respectively. Given a point $M$ on the circle $\mathcal{C}$ which is different from the points $S$ and $T$, let the line $MA$ meet the perpendicular from the point $S$ to the line $MO$ at $P$. Prove that the reflection of the point $S$ in the point $P$ lies on the line $MT$.

Let $k$ and $n$ be positive integers. Consider an array of $2\left(2^n-1\right)$ rows by $k$ columns. A $2$-coloring of the elements of the array is said to be acceptable if any two columns agree on less than $2^n-1$ entries on the same row. Given $n$, determine the maximum value of $k$ for an acceptable $2$-coloring to exist.