A set of 12 tokens --- 3 red, 2 white, 1 blue, and 6 black --- is to be distributed at random to 3 game players, 4 tokens per player. The probability that some player gets all the red tokens, another gets all the white tokens, and the remaining player gets the blue token can be written as nm, where m and n are relatively prime positive integers. What is m+n?
On top of a rectangular card with sides of length 1 and 2+3, an identical card is placed so that two of their diagonals line up, as shown (AC, in this case). Continue the process, adding a third card to the second, and so on, lining up successive diagonals after rotating clockwise. In total, how many cards must be used until a vertex of a new card lands exactly on the vertex labeled B in the figure?
(A)6(B)8(C)10(D)12(E)No new vertex will land on B.
Suppose that a1=2 and the sequence (an) satisfies the recurrence relation n−1an−1=(n−1)+1an−1+1 for all n≥2. What is the greatest integer less than or equal to n=1∑100an2?
A disphenoid is a tetrahedron whose triangular faces are congruent to one another. What is the least total surface area of a disphenoid whose faces are scalene triangles with integer side lengths?
A graph is symmetric about a line if the graph remains unchanged after reflection in that line. For how many quadruples of integers (a,b,c,d), where ∣a∣,∣b∣,∣c∣,∣d∣≤5 and c and d are not both 0, is the graph of y=cx+dax+b symmetric about the line y=x?
