4th Annual Harvard-MIT November Tournament Guts Round Problems

4th Annual Harvard-MIT November Tournament
Saturday 12 November 2011
Guts Round
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011 — GUTS ROUND
Round 1
1. [5] Determine the remainder when 1 + 2 + · · · + 2014 is divided by 2012.
2. [5] Let ABCD be a rectangle with AB = 6 and BC = 4. Let E be the point on BC with BE = 3,
and let F be the point on segment AE such that F lies halfway between the segments AB and CD. If
G is the point of intersection of DF and BC, find BG.
3. [5] Let x be a real number such that 2x = 3. Determine the value of 43x+2.
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011 — GUTS ROUND
Round 2
4. [6] Determine which of the following numbers is smallest in value: 54
p
3, 144, 108
p
6 − 108
p
2.
5. [6] Charlie folds an 17
2 -inch by 11-inch piece of paper in half twice, each time along a straight line
parallel to one of the paper’s edges. What is the smallest possible perimeter of the piece after two such
folds?
6. [6] To survive the coming Cambridge winter, Chim Tu doesn’t wear one T-shirt, but instead wears up
to FOUR T-shirts, all in different colors. An outfit consists of three or more T-shirts, put on one on
top of the other in some order, such that two outfits are distinct if the sets of T-shirts used are different
or the sets of T-shirts used are the same but the order in which they are worn is different. Given that
Chim Tu changes his outfit every three days, and otherwise never wears the same outfit twice, how
many days of winter can Chim Tu survive? (Needless to say, he only has four t-shirts.)
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011 — GUTS ROUND
Round 3
7. [7] How many ordered triples of positive integers (a, b, c) are there for which a4b2c = 54000?
8. [7] Let a, b, c be not necessarily distinct integers between 1 and 2011, inclusive. Find the smallest
possible value of ab + c
a + b + c
.
9. [7] Unit circle ­ has points X, Y,Z on its circumference so that XY Z is an equilateral triangle. Let
W be a point other than X in the plane such that triangle WY Z is also equilateral. Determine the
area of the region inside triangle WY Z that lies outside circle ­.
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011 — GUTS ROUND
Round 4
10. [8] Determine the number of integers D such that whenever a and b are both real numbers with
−1/4 < a, b < 1/4, then |a2 − Db2| < 1.
11. [8] For positive integers m, n, let gcd(m, n) denote the largest positive integer that is a factor of both
m and n. Compute
X91
n=1
gcd(n, 91).
12. [8] Joe has written 5 questions of different difficulties for a test with problems numbered 1 though 5.
He wants to make sure that problem i is harder than problem j whenever i−j ¸ 3. In how many ways
can he order the problems for his test?
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011 — GUTS ROUND
Round 5
13. [8] Tac is dressing his cat to go outside. He has four indistinguishable socks, four indistinguishable
shoes, and 4 indistinguishable show-shoes. In a hurry, Tac randomly pulls pieces of clothing out of a
door and tries to put them on a random one of his cat’s legs; however, Tac never tries to put more
than one of each type of clothing on each leg of his cat. What is the probability that, after Tac is done,
the snow-shoe on each of his cat’s legs is on top of the shoe, which is on top of the sock?
14. [8] Let AMOL be a quadrilateral with AM = 10, MO = 11, and OL = 12. Given that the perpendicular
bisectors of sides AM and OL intersect at the midpoint of segment AO, find the length of side
LA.
15. [8] For positive integers n, let L(n) be the largest factor of n other than n itself. Determine the number
of ordered pairs of composite positive integers (m, n) for which L(m)L(n) = 80.
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011 — GUTS ROUND
Round 6
16. [10] A small fish is holding 17 cards, labeled 1 through 17, which he shuffles into a random order.
Then, he notices that although the cards are not currently sorted in ascending order, he can sort them
into ascending order by removing one card and putting it back in a different position (at the beginning,
between some two cards, or at the end). In how many possible orders could his cards currently be?
17. [10] For a positive integer n, let p(n) denote the product of the positive integer factors of n. Determine
the number of factors n of 2310 for which p(n) is a perfect square.
18. [10] Consider a cube ABCDEFGH, where ABCD and EFGH are faces, and segments AE,BF,CG,DH
are edges of the cube. Let P be the center of face EFGH, and let O be the center of the cube. Given
that AG = 1, determine the area of triangle AOP.
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011 — GUTS ROUND
Round 7
19. [10] Let ABCD be a rectangle with AB = 3 and BC = 7. Let W be a point on segment AB such that
AW = 1. Let X, Y,Z be points on segments BC,CD,DA, respectively, so that quadrilateral WXY Z
is a rectangle, and BX < XC. Determine the length of segment BX.
20. [10] The UEFA Champions League playoffs is a 16-team soccer tournament in which Spanish teams
always win against non-Spanish teams. In each of 4 rounds, each remaining team is randomly paired
against one other team; the winner advances to the next round, and the loser is permanently knocked
out of the tournament. If 3 of the 16 teams are Spanish, what is the probability that there are 2
Spanish teams in the final round?
21. [10] Let P(x) = x4 + 2x3 − 13x2 − 14x + 24 be a polynomial with roots r1, r2, r3, r4. Let Q be the
quartic polynomial with roots r2
1, r2
2, r2
3, r2
4, such that the coefficient of the x4 term of Q is 1. Simplify
the quotient Q(x2)/P(x), leaving your answer in terms of x. (You may assume that x is not equal to
any of r1, r2, r3, r4).
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011 — GUTS ROUND
Round 8
22. [12] Let ABC be a triangle with AB = 23, BC = 24, and CA = 27. Let D be the point on segment
AC such that the incircles of triangles BAD and BCD are tangent. Determine the ratio CD/DA.
23. [12] Let N = 5AB37C2, where A,B,C are digits between 0 and 9, inclusive, and N is a 7-digit positive
integer. If N is divisible by 792, determine all possible ordered triples (A,B,C).
24. [12] Three not necessarily distinct positive integers between 1 and 99, inclusive, are written in a row
on a blackboard. Then, the numbers, without including any leading zeros, are concatenated to form a
new integer N. For example, if the integers written, in order, are 25, 6, and 12, then N = 25612 (and
not N = 250612). Determine the number of possible values of N.
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011 — GUTS ROUND
Round 9
The answers to the following three problems are mutually dependent, although your answer to each will
be graded independently. Let A be the answer to problem 25, B the answer to problem 26, and C be
the answer to problem 27.
25. [12] Let XY Z be an equilateral triangle, and let K,L,M be points on sides XY, Y Z,ZX, respectively,
such that XK/KY = B, Y L/LZ = 1/C, and ZM/MX = 1. Determine the ratio of the area of
triangle KLM to the area of triangle XY Z.
26. [12] Determine the positive real value of x for which
p
2 + AC + 2Cx +
p
AC − 2 + 2Ax =
p
2(A + C)x + 2AC.
27. [12] In-Young generates a string of B zeroes and ones using the following method:
• First, she flips a fair coin. If it lands heads, her first digit will be a 0, and if it lands tails, her
first digit will be a 1.
• For each subsequent bit, she flips an unfair coin, which lands heads with probability A. If the
coin lands heads, she writes down the number (zero or one) different from previous digit, while if
the coin lands tails, she writes down the previous digit again.
What is the expected value of the number of zeroes in her string?
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011 — GUTS ROUND
Round 10
28. [14] Determine the value of
2X011
k=1
k − 1
k!(2011 − k)! .
29. [14] Let ABC be a triangle with AB = 4, BC = 8, and CA = 5. Let M be the midpoint of BC, and
let D be the point on the circumcircle of ABC so that segment AD intersects the interior of ABC,
and \BAD = \CAM. Let AD intersect side BC at X. Compute the ratio AX/AD.
30. [14] Let S be a set of consecutive positive integers such that for any integer n in S, the sum of the
digits of n is not a multiple of 11. Determine the largest possible number of elements of S.
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011 — GUTS ROUND
Round 11
31. [17] Each square in a 3 × 10 grid is colored black or white. Let N be the number of ways this can be
done in such a way that no five squares in an ‘X’ configuration (as shown by the black squares below)
are all white or all black. Determine
p
N.
32. [17] Find all real numbers x satisfying
x9 +
9
8x6 +
27
64x3 − x +
219
512
= 0.
33. [17] Let ABC be a triangle with AB = 5, BC = 8, and CA = 7. Let ¡ be a circle internally tangent
to the circumcircle of ABC at A which is also tangent to segment BC. ¡ intersects AB and AC at
points D and E, respectively. Determine the length of segment DE.
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011 — GUTS ROUND
Round 12
34. [20] The integer 843301 is prime. The primorial of a prime number p, denoted p#, is defined to be
the product of all prime numbers less than or equal to p. Determine the number of digits in 843301#.
Your score will be
max
½¹
60
µ
1
3 −
¯¯¯¯
ln
µ
A
d
¶¯¯¯¯
¶º
, 0
¾
,
where A is your answer and d is the actual answer.
35. [20] Let G be the number of Google hits of “guts round” at 10:31PM on October 31, 2011. Let B be
the number of Bing hits of “guts round” at the same time. Determine B/G. Your score will be
max
µ
0,
¹
20
µ
1 −
20|a − k|
k
¶º¶
,
where k is the actual answer and a is your answer.
36. [20] Order any subset of the following twentieth century mathematical achievements chronologically,
from earliest to most recent. If you correctly place at least six of the events in order, your score will be
2(n − 5), where n is the number of events in your sequence; otherwise, your score will be zero. Note:
if you order any number of events with one error, your score will be zero.
A). Axioms for Set Theory published by Zermelo
B). Category Theory introduced by Mac Lane and Eilenberg
C). Collatz Conjecture proposed
D). Erdos number defined by Goffman
E). First United States delegation sent to International Mathematical Olympiad
F). Four Color Theorem proven with computer assistance by Appel and Haken
G). Harvard-MIT Math Tournament founded
H). Hierarchy of grammars described by Chomsky
I). Hilbert Problems stated
J). Incompleteness Theorems published by Godel
K). Million dollar prize for Millennium Problems offered by Clay Mathematics Institute
L). Minimum number of shuffles needed to randomize a deck of cards established by Diaconis
M). Nash Equilibrium introduced in doctoral dissertation
N). Proof of Fermat’s Last Theorem completed by Wiles
O). Quicksort algorithm invented by Hoare
Write your answer as a list of letters, without any commas or parentheses.
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4th Annual Harvard-MIT November Tournament
Saturday 12 November 2011
Guts Round
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011— GUTS ROUND
School Team Team ID#
1. [5]
2. [5]
3. [5]
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011— GUTS ROUND
School Team Team ID#
4. [6]
5. [6]
6. [6]
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011— GUTS ROUND
School Team Team ID#
7. [7]
8. [7]
9. [7]
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011— GUTS ROUND
School Team Team ID#
10. [8]
11. [8]
12. [8]
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011— GUTS ROUND
School Team Team ID#
13. [8]
14. [8]
15. [8]
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011— GUTS ROUND
School Team Team ID#
16. [10]
17. [10]
18. [10]
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011— GUTS ROUND
School Team Team ID#
19. [10]
20. [10]
21. [10]
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011— GUTS ROUND
School Team Team ID#
22. [12]
23. [12]
24. [12]
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011— GUTS ROUND
School Team Team ID#
25. [12]
26. [12]
27. [12]
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011— GUTS ROUND
School Team Team ID#
28. [14]
29. [14]
30. [14]
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011— GUTS ROUND
School Team Team ID#
31. [17]
32. [17]
33. [17]
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4TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011— GUTS ROUND
School Team Team ID#
34. [20]
35. [20]
36. [20]
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