MathDB
Problems
Contests
National and Regional Contests
USA Contests
USA - College-Hosted Events
Harvard-MIT Mathematics Tournament
2000 Harvard-MIT Mathematics Tournament
2000 Harvard-MIT Mathematics Tournament
Part of
Harvard-MIT Mathematics Tournament
Subcontests
(47)
47
1
Hide problems
2000 Guts #47:
Find an
n
<
100
n<100
n
<
100
such that
n
⋅
2
n
−
1
n\cdot 2^n-1
n
⋅
2
n
−
1
is prime. Score will be
n
−
5
n-5
n
−
5
for correct
n
n
n
,
5
−
n
5-n
5
−
n
for incorrect
n
n
n
(
0
0
0
points for answer
<
5
<5
<
5
)
46
1
Hide problems
2000 Guts #46: Sum equals an integer
For what integer values of
n
n
n
is
1
+
n
+
n
2
2
+
⋯
+
n
n
n
!
1+n+\frac{n^2}{2}+\cdots +\frac{n^n}{n!}
1
+
n
+
2
n
2
+
⋯
+
n
!
n
n
an integer?
45
1
Hide problems
2000 Guts #45: Binomial coefficient
Find all positive integers
x
x
x
for which there exists a positive integer
y
y
y
such that
(
x
y
)
=
1999000
\dbinom{x}{y}=1999000
(
y
x
)
=
1999000
44
1
Hide problems
2000 Guts #44: Function over integers
A function
f
:
Z
⟹
Z
f:\mathbb{Z}\implies\mathbb{Z}
f
:
Z
⟹
Z
satisfies
f
(
x
+
4
)
−
f
(
x
)
=
8
x
+
20
f(x+4)-f(x)=8x+20
f
(
x
+
4
)
−
f
(
x
)
=
8
x
+
20
f
(
x
2
−
1
)
=
(
f
(
x
)
−
x
)
2
+
x
2
−
2
f(x^2-1)=(f(x)-x)^2+x^2-2
f
(
x
2
−
1
)
=
(
f
(
x
)
−
x
)
2
+
x
2
−
2
Find
f
(
0
)
f(0)
f
(
0
)
and
f
(
1
)
f(1)
f
(
1
)
.
43
1
Hide problems
2000 Guts #43: Picking marbles from a box
Box A contains
3
3
3
black and
4
4
4
blue marbles. Box B has
7
7
7
black and
1
1
1
blue, whereas Box C has
2
2
2
black,
3
3
3
blue and
1
1
1
green marble. I close my eyes and pick two marbles from
2
2
2
different boxes. If it turns out that I get
1
1
1
black and
1
1
1
blue marble, what is the probability that the black marble is from box A and the blue one is from C?
42
1
Hide problems
2000 Guts #42: Magic square
A
n
n
n
by
n
n
n
magic square contains numbers from
1
1
1
to
n
2
n^2
n
2
such that the sum of every row and every column is the same. What is this sum?
41
1
Hide problems
2000 Guts #41: Angles of inclination
A person observes a building of height
h
h
h
at an angle of inclination
α
\alpha
α
from a point on the ground. After walking a distance
a
a
a
towards it, the angle is now
2
α
2\alpha
2
α
, and walking a further distance
b
b
b
causes it to increase to
3
α
3\alpha
3
α
. Find
h
h
h
in terms of
a
a
a
and
b
b
b
.
40
1
Hide problems
2000 Guts #40: Phi function
Let
ϕ
(
n
)
\phi(n)
ϕ
(
n
)
denote the number of positive integers less than or equal to
n
n
n
and relatively prime to
n
n
n
. Find all natural numbers
n
n
n
and primes
p
p
p
such that
ϕ
(
n
)
=
ϕ
(
n
p
)
\phi(n)=\phi(np)
ϕ
(
n
)
=
ϕ
(
n
p
)
.
39
1
Hide problems
2000 Guts #39: Hypergeometric series
If
x
=
1
3
x=\frac{1}{3}
x
=
3
1
, what is the value, rounded to
100
100
100
decimal digits, of
∑
n
=
0
7
2
n
1
+
x
2
n
\sum_{n=0}^{7}\frac{2^n}{1+x^{2^n}}
∑
n
=
0
7
1
+
x
2
n
2
n
?
38
1
Hide problems
2000 Guts #38: Largest number using basic operations
What is the largest number you can write with three
3
3
3
’s and three
8
8
8
’s, using only symbols
+
,
−
,
/
,
×
+,-,/,\times
+
,
−
,
/
,
×
and exponentiation?
37
1
Hide problems
2000 Guts #37: Water spills from a cone
A cone with semivertical angle
3
0
∘
30^{\circ}
3
0
∘
is half filled with water. What is the angle it must be tilted by so that water starts spilling?
36
1
Hide problems
2000 Guts #36: Magnitude of angle
If, in a triangle of sides
a
,
b
,
c
a, b, c
a
,
b
,
c
, the incircle has radius
b
+
c
−
a
2
\frac{b+c-a}{2}
2
b
+
c
−
a
, what is the magnitude of
∠
A
\angle A
∠
A
?
35
1
Hide problems
2000 Guts #35: Generating function
If
1
+
2
x
+
3
x
2
+
.
.
.
=
9
1+2x+3x^2 +...=9
1
+
2
x
+
3
x
2
+
...
=
9
, find
x
x
x
.
34
1
Hide problems
2000 Guts #34: Perfect square factorial
What is the largest
n
n
n
such that
n
!
+
1
n! + 1
n
!
+
1
is a square?
33
1
Hide problems
2000 Guts #33: Form of numbers
Characterise all numbers that cannot be written as a sum of
1
1
1
or more consecutive odd numbers.
32
1
Hide problems
2000 Guts #32: Nondegenerate tetrahedrons
How many (nondegenerate) tetrahedrons can be formed from the vertices of an
n
n
n
-dimensional hypercube?
31
1
Hide problems
2000 Guts #31: Constructing point on segment
Given collinear points
A
,
B
,
C
A,B,C
A
,
B
,
C
such that
A
B
=
B
C
AB = BC
A
B
=
BC
. How can you construct a point
D
D
D
on
A
B
AB
A
B
such that
A
D
=
2
D
B
AD = 2DB
A
D
=
2
D
B
, using only a straightedge? (You are not allowed to measure distances)
30
1
Hide problems
2000 Guts #30: Unit square
A
B
C
D
ABCD
A
BC
D
is a unit square. If
∠
P
A
C
=
∠
P
C
D
\angle PAC =\angle PCD
∠
P
A
C
=
∠
PC
D
, find the length
B
P
BP
BP
.
29
1
Hide problems
2000 Guts #29: Infinite nested radical
What is the value of
1
+
2
1
+
3
1
+
4
1
+
5
1
+
⋯
{ \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+\cdots }}}}}}
1
+
2
1
+
3
1
+
4
1
+
5
1
+
⋯
?
28
1
Hide problems
2000 Guts #28: Volume to surface area
What is the smallest possible volume to surface ratio of a solid cone with height =
1
1
1
unit?
27
1
Hide problems
2000 Guts #27: Sum of squares
What is the smallest number that can be written as a sum of
2
2
2
squares in
3
3
3
ways?
26
1
Hide problems
2000 Guts #26: Last 3 digits
What are the last
3
3
3
digits of
1
!
+
2
!
+
⋯
+
100
!
1!+2!+\cdots +100!
1
!
+
2
!
+
⋯
+
100
!
25
1
Hide problems
2000 Guts #25: Next term in sequence
Find the next number in the sequence
131
,
111311
,
311321
,
1321131211
,
⋯
131, 111311, 311321, 1321131211,\cdots
131
,
111311
,
311321
,
1321131211
,
⋯
24
1
Hide problems
2000 Guts #24: Knight on a chessboard
At least how many moves must a knight make to get from one corner of a chessboard to the opposite corner?
23
1
Hide problems
2000 Guts #23: 7 digit numbers divisible by 3
How many
7
7
7
-digit numbers with distinct digits can be made that are divisible by
3
3
3
?
22
1
Hide problems
2000 Guts #22: Minimum value
Find the smallest
n
n
n
such that
2
2000
2^{2000}
2
2000
divides
n
!
n!
n
!
.
21
1
Hide problems
2000 Guts #21: Coloring a necklace
How many ways can you color a necklace of
7
7
7
beads with
4
4
4
colors so that no two adjacent beads have the same color?
20
1
Hide problems
2000 Guts #20: Minimum perimeter of triangle
What is the minimum possible perimeter of a triangle two of whose sides are along the x- and y-axes and such that the third contains the point
(
1
,
2
)
(1,2)
(
1
,
2
)
?
19
1
Hide problems
2000 Guts #19: Operation on integers
Define
a
∗
b
=
a
−
b
1
−
a
b
a*b=\frac{a-b}{1-ab}
a
∗
b
=
1
−
ab
a
−
b
. What is
(
1
∗
(
2
∗
(
3
∗
⋯
(
n
∗
(
n
+
1
)
)
⋯
)
)
)
(1*(2*(3*\cdots (n*(n+1))\cdots )))
(
1
∗
(
2
∗
(
3
∗
⋯
(
n
∗
(
n
+
1
))
⋯
)))
?
18
1
Hide problems
2000 Guts #18: Infinite trigonometric sum
What is the value of
∑
n
=
1
∞
(
tan
−
1
n
−
tan
−
1
n
+
1
)
\sum_{n=1}^\infty (\tan^{-1}\sqrt{n}-\tan^{-1}\sqrt{n+1})
∑
n
=
1
∞
(
tan
−
1
n
−
tan
−
1
n
+
1
)
?
17
1
Hide problems
2000 Guts #17: Power of 3
Find the highest power of 3 dividing
(
666
333
)
\dbinom{666}{333}
(
333
666
)
.
16
1
Hide problems
2000 Guts #16: System of linear+quintic equations
Solve for real
x
,
y
x,y
x
,
y
:
x
+
y
=
2
x+y=2
x
+
y
=
2
x
5
+
y
5
=
82
x^5+y^5=82
x
5
+
y
5
=
82
15
2
Hide problems
2000 Guts #15: Filling grid with 0 and X
Find the number of ways of filling a
8
8
8
by
8
8
8
grid with
0
0
0
's and
X
X
X
's so that the number of
0
0
0
's in each row and each column is odd.
2000 HMMT RMT Team #15
lim
n
→
∞
n
r
1
−
cos
2
π
n
2
=
?
\lim_{n \to \infty} nr\sqrt[2]{1-\cos \frac{2\pi}{n}}=?
n
→
∞
lim
n
r
2
1
−
cos
n
2
π
=
?
14
2
Hide problems
2000 Guts #14: Cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
is a cyclic quadrilateral inscribed in a circle of radius
5
5
5
, with
A
B
=
6
AB=6
A
B
=
6
,
B
C
=
7
BC=7
BC
=
7
,
C
D
=
8
CD=8
C
D
=
8
. Find
A
D
AD
A
D
.
2000 HMMT RMT Team #14
Define a sequence
<
x
n
>
<x_n>
<
x
n
>
of real numbers by specifying an initial
x
0
x_0
x
0
and by the recurrence
x
n
+
1
=
1
+
x
n
1
−
x
n
x_{n+1}=\frac{1+x_n}{1-x_n}
x
n
+
1
=
1
−
x
n
1
+
x
n
. Find
x
n
x_n
x
n
as a function of
x
0
x_0
x
0
and
n
n
n
, in closed form. There may be multiple cases.
13
2
Hide problems
2000 Guts #13: Polynomial division
Determine the remainder when
(
x
4
−
1
)
(
x
2
−
1
)
(x^4-1)(x^2-1)
(
x
4
−
1
)
(
x
2
−
1
)
is divided by
1
+
x
+
x
2
1+x+x^2
1
+
x
+
x
2
.
2000 HMMT RMT Team #13 max no of intersections inside convex n-gon
Let
P
1
,
P
2
,
.
.
.
,
P
n
P_1, P_2,..., P_n
P
1
,
P
2
,
...
,
P
n
be a convex
n
n
n
-gon. If all lines
P
i
P
j
P_iP_j
P
i
P
j
are joined, what is the maximum possible number of intersections in terms of
n
n
n
obtained from strictly inside the polygon?
12
2
Hide problems
2000 Guts #12: Choosing 2 consecutive numbers
Calculate the number of ways of choosing
4
4
4
numbers from the set
1
,
2
,
⋯
,
11
{1,2,\cdots ,11}
1
,
2
,
⋯
,
11
such that at least
2
2
2
of the numbers are consecutive.
2000 HMMT RMT Team #12
At a dance, Abhinav starts from point
(
a
,
0
)
(a, 0)
(
a
,
0
)
and moves along the negative
x
x
x
direction with speed
v
a
v_a
v
a
, while Pei-Hsin starts from
(
0
,
6
)
(0,6)
(
0
,
6
)
and glides in the negative
y
y
y
-direction with speed
v
b
v_b
v
b
. What is the distance of closest approach between the two?
11
2
Hide problems
2000 Guts #11: Maximum value
Let
M
M
M
be the maximum possible value of
x
1
x
2
+
x
2
x
3
+
⋯
+
x
5
x
1
x_1x_2+x_2x_3+\cdots +x_5x_1
x
1
x
2
+
x
2
x
3
+
⋯
+
x
5
x
1
where
x
1
,
x
2
,
⋯
x
5
x_1, x_2, \cdots x_5
x
1
,
x
2
,
⋯
x
5
is a permutation of
(
1
,
2
,
3
,
4
,
5
)
(1,2,3,4,5)
(
1
,
2
,
3
,
4
,
5
)
and let
N
N
N
be the number of permutations for which this maximum is attained. Evaluate
M
+
N
M+N
M
+
N
.
2000 HMMT RMT Team #11
Find all polynomials
f
(
x
)
f(x)
f
(
x
)
with integer coefficients such that the coefficients of both
f
(
x
)
f(x)
f
(
x
)
and
[
f
(
x
)
]
3
[f(x)]^3
[
f
(
x
)
]
3
lie in the set
{
0
,
1
,
−
1
}
\{0,1, -1\}
{
0
,
1
,
−
1
}
.
10
6
Show problems
9
7
Show problems
8
7
Show problems
7
7
Show problems
6
7
Show problems
5
7
Show problems
4
7
Show problems
3
7
Show problems
2
7
Show problems
1
7
Show problems