MathDB

6

Part of 2008 Harvard-MIT Mathematics Tournament

Problems(5)

Roots of Unity and a Quadratic

Source:

3/2/2008
A root of unity is a complex number that is a solution to z^n \equal{} 1 for some positive integer n n. Determine the number of roots of unity that are also roots of z^2 \plus{} az \plus{} b \equal{} 0 for some integers a a and b b.
quadraticsHMMTtrigonometryalgebrapolynomialFTWVieta
Combinatorial Limit

Source: HMMT 2008 Calculus Problem 6

3/2/2008
Determine the value of \lim_{n\rightarrow\infty}\sum_{k \equal{} 0}^n\binom{n}{k}^{ \minus{} 1}.
limitratiocalculuscalculus computations
Sudoku Matrix

Source:

3/3/2008
A Sudoku matrix is defined as a 9×9 9\times9 array with entries from {1,2,…,9} \{1, 2, \ldots , 9\} and with the constraint that each row, each column, and each of the nine 3×3 3 \times 3 boxes that tile the array contains each digit from 1 1 to 9 9 exactly once. A Sudoku matrix is chosen at random (so that every Sudoku matrix has equal probability of being chosen). We know two of the squares in this matrix, as shown. What is the probability that the square marked by ? contains the digit 3 3? \setlength{\unitlength}{6mm} \begin{picture}(9,9)(0,0) \multiput(0,0)(1,0){10}{\line(0,1){9}} \multiput(0,0)(0,1){10}{\line(1,0){9}} \linethickness{1.2pt} \multiput(0,0)(3,0){4}{\line(0,1){9}} \multiput(0,0)(0,3){4}{\line(1,0){9}} \put(0,8){\makebox(1,1){1}} \put(1,7){\makebox(1,1){2}} \put(3,6){\makebox(1,1){?}} \end{picture}
linear algebramatrixprobability
Congruent Circumcircles

Source:

3/6/2008
Let ABC ABC be a triangle with \angle A \equal{} 45^\circ. Let P P be a point on side BC BC with PB \equal{} 3 and PC \equal{} 5. Let O O be the circumcenter of ABC ABC. Determine the length OP OP.
geometrycircumcircletrigonometrytrig identitiesLaw of SinesLaw of Cosines
Counting Rectangles

Source:

3/23/2008
Determine the number of non-degenerate rectangles whose edges lie completely on the grid lines of the following figure. \begin{tabular}{|c|c|c|c|c|c|} \hline & & & & & \\ \hline & & & & & \\ \hline & & \multicolumn{1}{c}{} & & & \\ \cline{1 \minus{} 2}\cline{5 \minus{} 6} & & \multicolumn{1}{c}{} & & & \\ \hline & & & & & \\ \hline & & & & & \\ \hline \end{tabular}
geometryrectangle