91Ó°ÊÓ

If \(A=(0,-10)\) and \(B=(2,0)\), find the point(s) \(C\) on the parabola \(y=x^{2}\) which minimizes the area of triangle \(A B C\).

Find all solutions in nonnegative integers to the system of equations $$ 3 x^2-2 y^2-4 z^2+54=0, \quad 5 x^2-3 y^2-7 z^2+74=0 $$

Given three lines in the plane which form a triangle (that is, every pair of the lines intersects, and the three intersection points are distinct), what is the set of points for which the sum of the distances to the three lines is as small as possible? (Be careful not to overlook special cases.)

Find all perfect squares whose base 9 representation consists only of ones.

Find a positive integer \(n\) such that \(2011 n+1\) and \(2012 n+1\) are both perfect squares, or show that no such positive integer \(n\) exists.

Let \(f_1(x)=x^2+4 x+2\), and for \(n \geq 2\), let \(f_n(x)\) be the \(n\)-fold composition of the polynomial \(f_1(x)\) with itself. For example, $$ f_2(x)=f_1\left(f_1(x)\right)=x^4+8 x^3+24 x^2+32 x+14 . $$ Let \(s_n\) be the sum of the coefficients of the terms of even degree in \(f_n(x)\). For example, \(s_2=1+24+14=39\). Find \(s_{2012}\).

Let \(A B C D\) be a convex quadrilateral (a four-sided figure with angles less than \(180^{\circ}\) ). Find a necessary and sufficient condition for a point \(P\) to exist inside \(A B C D\) such that the four triangles \(A B P, B C P, C D P, D A P\) all have the same area.

Let \(k\) be a positive integer. Find the largest power of 3 which divides \(10^k-1\).

There is no analog of the quadratic formula that solves polynomial equations of degree 5 and higher, such as \(x^5-5 x^4+8 x^3-6 x^2+3 x+3=0\). However, this particular polynomial has two roots that sum to 2 . Using this information, find all solutions.

Call a convex pentagon (five-sided figure with angles less than \(180^{\circ}\) ) "parallel" if each diagonal is parallel to the side with which it does not have a vertex in common. That is, \(A B C D E\) is parallel if the diagonal \(A C\) is parallel to the side \(D E\) and similarly for the other four diagonals. It is easy to see that a regular pentagon is parallel, but is a parallel pentagon necessarily regular?