Download here

ARML (A Team consists of 15 mathletes.)
Calculators are not allowed in any Round in ARML competition.
Eastern Mass traditionally brings three teams, as well as 5 alternates.
Individual Rounds (150 possible points)
(Currently, 5 pairs of Rounds each with 2 questions – 10 minutes/round)
Sample Individual Questions
(1) ARML 1995 (I-1) Compute the largest prime factor of:

 
  
 
 
3  3  3  3 3 3 3 3 3  3  3  1  1  1  1  1  1  1  1   1   1   1
  
  
(2) ARML 1998 (I-2) Coefficients a, b and c of ax2 + bx + c = 0 are selected without
replacement from N = {4, 3, 2, 1, 1, 2, 3, 4}. Compute the probability that x = 1 is a
solution of the equation.
(3) ARML 2000 (I-6) Let S be a sphere whose equation is x 2  y 2  z 2  25 .
Points P(3, 4, 0) and Q(3, -4, 0) lie on S. Compute the number of planes containing P
and Q that intersect S in a circle whose area is an integer.
(4) ARML 2003 (I-8) The graph of f ( x)  x 4  4 x3  16 x 2  6 x  5 has a common tangent
line at x = p and x = q. Compute pq.
Warning: This was the last individual problem in the contest and as always that last
problem is always very tough!
Team Round (50 points possible)
A team round consists of 10 questions. Time limit is 20 minutes.
All 15 team members work on the questions together and discuss different strategies.
Organization and communication are paramount.
Hurried consultations are necessary, if there is disagreement with respect to the answers.
Five points is awarded each correct answer, zero points for a wrong answer.
Sample Team Round questions
(1) ARML 2001 (T-2) If each distinct letter in ARML  AL represents a distinct
positive digit in base 10, compute the sum A + R + M + L.
(2) ARML 2007 (T-7) The pages of a book are numbered consecutively from 1 to n.
The book is divided into 12 chapters such that the number of digits used to number the
pages of each chapter is the same. Compute the least possible value of n.
Relay Rounds (50 possible points)
Relay – Background information
TNYWR means the number you will receive. A team of 15 is broken down into five relay groups
of three students each. There are three questions A, B and C in a relay round which are passed out
to persons 1, 2 and 3 respectively in each group. Team members have no knowledge of anyone
else’s question and they start working on their problems simultaneously. Person #1 has all the
information required to solve his/her problem. Person #2 and #3 need input from the person
directly in front of them to finish their problem. No collaboration is allowed except person #1
passes back a value to person #2 and person #2 passes back a value to person #3. Only Person
#3’s answer counts. Considerable work needs to be done by person #2 and #3 while they are
waiting. There are two relay rounds and the second is usually much more difficult. A correct
answer in 3 minutes is worth 5 points and a correct answer in 6 minutes is worth 3 points. Wrong
answers are worth nada! Obviously, it’s advantageous to turn in an answer at 6 minutes even if it
is a guess. Equally obvious, turning in the same answer at 6 minutes and at 3 minutes is a very
BAD idea!
A super relay is not counted in the Team results, but is a bonus feature of the competition each
year. All 15 team members are involved. 15 questions are used. Answers are passed from both
ends to the middle of the row. Person #8 is in the hot seat. (S)he receives two inputs, one from
person #7 and one from person #9. Only person #8’s answer counts. First team to finish with a
correct answer has bragging right until they have to defend their title the next year!
Power Round (50 points possible) – A sequential series of in-depth questions on a single
topic, where explanation/justification of answers/assertions is required
Administered after the team round, all 15 mathletes are involved for 60 minutes.
Problems and Solutions to all past contest questions are available in book form.
If you don’t have your own copies, consider buying them. They are a real good investment!
Relay Round #1 – ARML 2005
R1-1
If the product 55  60  65 is written as the product of five distinct positive integers,
compute the least possible value for the largest of the five integers.
R1-2
Let T = TNYWR. Let x and y be real numbers satisfying  x  yi  2  i   T  2Ti .
Compute the value of x.
F
A
R1-3
Let T = TNYWR. In square ABCD, AD = T and DE = T – 4.
M is the midpoint of AD and F is chosen so that AE  MF .
Compute the length of FA .
Answers and solutions will be posted on the MML and GBML
websites next Friday.
ENJOY!
B
P
M
D
E
C