In each square of an 8 by 8 checkerboard I want to write either 1, 0, or -1. Can I do this in such a way that the sums of numbers along individual rows, columns, and long diagonals are all different? Support your answer. Expand and simplify: (1 + x + x2) (1 - x + x2) (1 - x2 + x4) ... (1 - x2y-1 + x2y). Given an equilateral triangle of side 1, what is the minimal length of a brush that can paint the entire interior of the triangle if one end of the brush must always touch a side of the triangle? The sum of several integers is divisible by 6. Show that the sum of their cubes is also divisible by 6. Some integers can be represented as the difference of two squares (e.g., 21 = 52 - 22). Others cannot. (a) Show that every odd number is the difference of two squares. (b) Which even integers can be represented as the difference of two squares? The graph of the function f(x) = (ax+b)/(x+c) has asymptotes given by the equations x = 3 and y = 2. The graph also passes through the point (1,3) and intersects the coordinate axes at the points P and Q. Find an equation of the line through points P and Q. The lateral surface area of a right circular cone is three times the surface area of its inscribed sphere. Find the angle at the vertex of the cone (see figure). A fair coin is flipped until a head occurs. What is the probability that a head first appears on an even numbered toss? Prove: 4 a b < (a + b)2 < 2 (a + b)2. Under what circumstances does equality hold? Given the cube with base ABCD and top EFGH where M is the midpoint of edge AD as shown in the figure: (a) Find the angle between CH and BM. (b) Find the ratio of the volume of the cube to the volume of the pyramid ABME. For what value(s) of a will the quadratic equation a x2 - 2 (a - 3) x + 4 = 0 have two positive roots? A bag contains 150 black marbles and 75 white marbles. A person draws two marbles from the bag. If the drawn marbles are one black and one white, the person replaces the white marble in the bag and discards the black marble. If the two drawn marbles are both the same color, they are discarded and one black marble is placed in the bag (there is an unlimited supply of black marbles). The process is repeated. Eventually there will be just one marble left in the bag (why?). What is its color? Find the smallest positive integer n such that one-half n is a perfect square, one-third n is a perfect cube and one-fifth n is a perfect fifth power. Given: Sn = (4 n3 - 63 n2 - n)/6 is the sum of the first n terms of a sequence t1, t2, t3, ... , tn, ... . (a) Show: tn = 2 n2 - 23 n + 11 for n = 1, 2, 3, ... . (b) Which terms of the sequence are negative? Which are positive? Find the term with the smallest value. Find all possible solutions in natural numbers of the equation . Find all values of b so that . Talent Search Results for 1998-1999 Previous Years: Talent Search 1998-1999 Talent Search 1999-2000 Talent Search 2000-2001 Talent Search 2001-2002 Talent Search 2002-2003 Talent Search 2003-2004 Talent Search 2004-2005 Talent Search 2005-2006 Talent Search 2006-2007 Talent Search 2007-2008 Talent Search 2008-2009