SW-ARML
9-25-10
Problem (I) 1. Compute the number of positive integers less than 25 that cannot be written as the
difference of two squares of integers.
Problem (I) 2. For digits A, B, and C, (AB)2 + (AC)2 = 1313. Compute A + B + C.
Problem (I) 3. For real numbers , B, and C, the zeros of T(x) = x3 +x2 + Bx + C are sin2,
cos2, and –csc2. Compute T(5).
Problem (T) 4. Compute all ordered pairs of real numbers (x, y) that satisfy both of the equations:
x2 + y2 = 6y – 4x + 12 and 4y = x2 + 4x + 12.
Problem (T) 5. Define log*(n) to be the smallest number of times the log function must be
iteratively applied to n to get a result less than or equal to 1. For example, log*(1000) = 2 since
log 1000 = 3 and log(log 1000) = log 3 = 0.477…  1. Let a be the smallest integer such that
log*(a) = 3. Compute the number of zeros in the base 10 representation of a.
Problem (T) 6. The six sides of convex hexagon A1A2A3A4A5A6 are colored red. Each of the
diagonals of the hexagon is colored either red or blue. Compute the number of colorings such
that every triangle AiAjAk has at least one red side.
Problem (T) 7. Given an arbitrary finite sequence of letters (represented as a word), a
subsequence is a sequence of one or more letters that appear in the same order as in the original
sequence. For example, N, CT, OTT, and CONTEST are subsequences of the word CONTEST,
but NOT, ONSET, and TESS are not. Assuming the standard English alphabet {A, B, …, Z},
compute the number of distinct four-letter “words” for which EE is a subsequence.
Problem (R) 8. If A, R, M, and L are positive integers such that A2 +R2 = 20 and M2 +L2 = 10,
compute the product A · R ·M · L.
Problem (R) 9. Compute the last digit, in base 10, of the sum
T2 + (2T)2 + (3T)2 + … + (T2)2 and write the answer in terms of T.
Problem (R) 10. A fair coin is flipped n times. Compute the smallest positive integer n for which
the probability that the coin has the same result every time is less than 10%.
Problem (R) 11. Let T = TNYWR. Compute the smallest positive integer n such that there are at
least T positive integers in the domain of f(x) =
 x 2  2 x  n and write answer in terms of T.
Problem (R) 12. Let T = TNYWR which is positive. Compute the smallest positive real number
 x
x such that    T .
x   x 
1