MathDB

1

2012 Fall CHMMC Individual - Caltech Harvey Mudd Mathematics Competition

p1. How many nonzero digits are in the number

(5^{94} + 5^{92})(2^{94} + 2^{92})

?
p2. Suppose

A

is a set of

2013

distinct positive integers such that the arithmetic mean of any subset of

A

is also an integer. Find an example of

A

.
p3. How many minutes until the smaller angle formed by the minute and hour hands on the face of a clock is congruent to the smaller angle between the hands at

5:15

pm? Round your answer to the nearest minute.
p4. Suppose

a

and

b

are positive real numbers,

a + b = 1

, and

1 +\frac{a^2 + 3b^2}{2ab}=\sqrt{4 +\frac{a}{b}+\frac{3b}{a}}.

Find

a

.
p5. Suppose

f(x) = \frac{e^x- 12e^{-x}}{ 2}

. Find all

x

such that

f(x) = 2

.
p6. Let

P_1

,

P_2

,

...

,

P_n

be points equally spaced on a unit circle. For how many integer

n \in \{2, 3, ... , 2013\}

is the product of all pairwise distances:

\prod_{1\le i<j\le n} P_iP_j

a rational number? Note that

\prod

means the product. For example,

\prod_{1\le i\le 3} i = 1\cdot 2 \cdot 3 = 6

.
p7. Determine the value

a

such that the following sum converges if and only if

r \in (-\infty, a)

:

\sum^{\infty}_{n=1}(\sqrt{n^4 + n^r} - n^2).

Note that

\sum^{\infty}_{n=1}\frac{1}{n^s}

converges if and only if

s > 1

.
p8. Find two pairs of positive integers

(a, b)

with

a > b

such that

a^2 + b^2 = 40501

.
p9. Consider a simplified memory-knowledge model. Suppose your total knowledge level the night before you went to a college was

100

units. Each day, when you woke up in the morning you forgot

1\%

of what you had learned. Then, by going to lectures, working on the homework, preparing for presentations, you had learned more and so your knowledge level went up by

10

units at the end of the day. According to this model, how long do you need to stay in college until you reach the knowledge level of exactly

1000

?
p10. Suppose

P(x) = 2x^8 + x^6 - x^4 +1

, and that

P

has roots

a_1

,

a_2

,

...

,

a_8

(a complex number

z

is a root of the polynomial

P(x)

if

P(z) = 0

). Find the value of

(a^2_1-2)(a^2_2-2)(a^2_3-2)...(a^2_8-2).

p11. Find all values of

x

satisfying

(x^2 + 2x-5)^2 = -2x^2 - 3x + 15

.
p12. Suppose

x, y

and

z

are positive real numbers such that

x^2 + y^2 + xy = 9,

y^2 + z^2 + yz = 16,

x^2 + z^2 + xz = 25.

Find

xy + yz + xz

(the answer is unique).
p13. Suppose that

P(x)

is a monic polynomial (i.e, the leading coefficient is

1

) with

20

roots, each distinct and of the form

\frac{1}{3^k}

for

k = 0,1,2,..., 19

. Find the coefficient of

x^{18}

in

P(x)

.
p14. Find the sum of the reciprocals of all perfect squares whose prime factorization contains only powers of

3

,

5

,

7

(i.e.

\frac{1}{1} + \frac{1}{9} + \frac{1}{25} + \frac{1}{419} + \frac{1}{811} + \frac{1}{215} + \frac{1}{441} + \frac{1}{625} + ...

).
p15. Find the number of integer quadruples

(a, b, c, d)

which also satisfy the following system of equations:

1+b + c^2 + d^3 =0,

a + b^2 + c^3 + d^4 =0,

a^2 + b^3 + c^4 + d^5 =0,

a^3+b^4+c^5+d^6 =0.

PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Mixer

1

2012 Fall CHMMC Mixer Round - Caltech Harvey Mudd Mathematics Competition

p1. Prove that

x = 2

is the only real number satisfying

3^x + 4^x = 5^x

.
p2. Show that

\sqrt{9 + 4\sqrt5} -\sqrt{9 - 4\sqrt5}

is an integer.
p3. Two players

A

and

B

play a game on a round table. Each time they take turn placing a round coin on the table. The coin has a uniform size, and this size is at least

10

times smaller than the table size. They cannot place the coin on top of any part of other coins, and the whole coin must be on the table. If a player cannot place a coin, he loses. Suppose

A

starts first. If both of them plan their moves wisely, there will be one person who will always win. Determine whether

A

or

B

will win, and then determine his winning strategy.
p4. Suppose you are given

4

pegs arranged in a square on a board. A “move” consists of picking up a peg, reflecting it through any other peg, and placing it down on the board. For how many integers

1 \le n \le 2013

is it possible to arrange the

4

pegs into a larger square using exactly

n

moves? Justify your answers.
p5. Find smallest positive integer that has a remainder of

1

when divided by

2

, a remainder of

2

when divided by

3

, a remainder of

3

when divided by

5

, and a remainder of

5

when divided by

7

.
p6. Find the value of

\sum_{m|496,m>0} \frac{1}{m},

where

m|496

means

496

is divisible by

m

.
p7. What is the value of

{100 \choose 0}+{100 \choose 4}+{100 \choose 8}+ ... +{100 \choose 100}?

p8. An

n

-term sequence

a_0, a_1, ...,a_n

will be called sweet if, for each

0 \le i \le n -1

,

a_i

is the number of times that the number

i

appears in the sequence. For example,

1, 2, 1,0

is a sweet sequence with

4

terms. Given that

a_0

,

a_1

,

...

,

a_{2013}

is a sweet sequence, find the value of

a^2_0+ a^2_1+ ... + a^2_{2013}.

PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

10

1

2012 Fall Team #10

Let

N = {2^{2012} \choose 0} {2^{2012} \choose 1} {2^{2012} \choose 2} {2^{2012} \choose 3}... {2^{2012} \choose 2^{2012}}.

Let M be the number of

0

’s when

N

is written in binary. How many

0

’s does

M

have when written in binary? (Warning: this question is very hard.)

9

1

2012 Fall Team #9

For a positive integer

n

, let

f(n)

be equal to

n

if there is an integer

x

such that

x^2-n

is divisible by

2^{12}

, and let

f(n)

be

0

otherwise. Determine the remainder when

\sum^{2^{12}-1}_{n=0}f(n)

is divided by

2^{12}

.

7

1

2012 Fall Team #7

Consider a

1

by

2

by

3

rectangular prism. Find the length of the shortest path between opposite corners

A

and

B

that does not leave the surface of the prism. https://cdn.artofproblemsolving.com/attachments/7/1/899b79b0f938e63fe7b093cecff63ed1255acb.png

8

1

2012 Fall Team #8

Find the sum of all positive

30

-digit palindromes. The leading digit is not allowed to be

0

.

6

1

2012 Fall Team #6

Suppose you have ten pairs of red socks, ten pairs of blue socks, and ten pairs of green socks in your drawer. You need to go to a party soon, but the power is currently off in your room. It is completely dark, so you cannot see any colors and unfortunately the socks are identically shaped. What is the minimum number of socks you need to take from the drawer in order to guarantee that you have at least one pair of socks whose colors match?

5

1

2012 Fall Team #5

At each step, a rectangular tile of length

1, 2

, or,

3

is chosen at random, what is the probability that the total length is

10

after

5

steps?

4

2

2012 Fall Team #4

Consider the figure below, not drawn to scale. In this figure, assume that

AB \perp BE

and

AD \perp DE

. Also, let

AB = \sqrt6

and

\angle BED =\frac{\pi}{6}

. Find

AC

. https://cdn.artofproblemsolving.com/attachments/2/d/f87ac9f111f02e261a0b5376c766a615e8d1d8.png

2012 Fall CHMMC Tiebreaker 4 lattice points in 3D sum x&#039;^2 <sum x^2

A lattice point

(x, y, z) \in Z^3

can be seen from the origin if the line from the origin does not contain any other lattice point

(x', y', z')

with

(x')^2 + (y')^2 + (z')^2 < x^2 + y^2 + z^2.

Let

p

be the probability that a randomly selected point on the cubic lattice

Z^3

can be seen from the origin. Given that

\frac{1}{p}= \sum^{\infty}_{n=i} \frac{k}{n^s}

for some integers

i, k

, and

s

, find

i, k

and

s

.

3

2

2012 Fall Team #3

A particular graph has

6

vertices,

12

edges, and has the property that it contains no Eulerian path; a Eulerian path is a route from vertex to vertex along edges that traces each edge exactly once. Determine all the possible degrees of its vertices in no particular order. There are two solutions, and you need to get both to get credit for this problem.

2012 Fall CHMMC Tiebreaker 3 \sigma (n) < n+ \sqrt{n}

For a positive integer

n

, let

\sigma (n)

be the sum of the divisors of

n

(for example

\sigma (10) = 1 + 2 + 5 + 10 = 18

). For how many

n \in \{1, 2,. .., 100\}

, do we have

\sigma (n) < n+ \sqrt{n}

?

2

2012 Fall Team #2

Consider a triangle

ABC

with points

D

on

AB

,

E

on

BC

, and let

F

be the intersection of

AE

and

CD

. Suppose

AD = 1

,

DB = 2

,

BE = 1

,

EC = 3

, and

CA = 5

. Find the value of the area of

ECF

minus the area of

ADF

.

2012 Fall CHMMC Tiebreaker 2 continuous f(x + f(y)) = f(x + y) + y

Find all continuous functions

f : R \to R

such that

f(x + f(y)) = f(x + y) + y,

for all

x, y \in R

. No proof is required for this problem.

1

2

2012 Fall Team #1

Find the remainder when

5^{2012}

is divided by

3

.

2012 Fall CHMMC Tiebreaker 1 - |P(P([2013]))|, set of non-empty subsets

Let

[n] = \{1, 2, 3, ... ,n\}

and for any set

S

, let

P(S)

be the set of non-empty subsets of

S

. What is the last digit of

|P(P([2013]))|

?

2012 CHMMC Fall

Subcontests

2012 Fall CHMMC Individual - Caltech Harvey Mudd Mathematics Competition

2012 Fall CHMMC Mixer Round - Caltech Harvey Mudd Mathematics Competition

2012 Fall Team #10

2012 Fall Team #9

2012 Fall Team #7

2012 Fall Team #8

2012 Fall Team #6

2012 Fall Team #5

2012 Fall Team #4

2012 Fall CHMMC Tiebreaker 4 lattice points in 3D sum x&#039;^2 <sum x^2

2012 Fall Team #3

2012 Fall CHMMC Tiebreaker 3 \sigma (n) < n+ \sqrt{n}

2012 Fall Team #2

2012 Fall CHMMC Tiebreaker 2 continuous f(x + f(y)) = f(x + y) + y

2012 Fall Team #1

2012 Fall CHMMC Tiebreaker 1 - |P(P([2013]))|, set of non-empty subsets

2012 CHMMC Fall

Subcontests

2012 Fall CHMMC Individual - Caltech Harvey Mudd Mathematics Competition

2012 Fall CHMMC Mixer Round - Caltech Harvey Mudd Mathematics Competition

2012 Fall Team #10

2012 Fall Team #9

2012 Fall Team #7

2012 Fall Team #8

2012 Fall Team #6

2012 Fall Team #5

2012 Fall Team #4

2012 Fall CHMMC Tiebreaker 4 lattice points in 3D sum x&amp;#039;^2 &lt;sum x^2

2012 Fall Team #3

2012 Fall CHMMC Tiebreaker 3 \sigma (n) &lt; n+ \sqrt{n}

2012 Fall Team #2

2012 Fall CHMMC Tiebreaker 2 continuous f(x + f(y)) = f(x + y) + y

2012 Fall Team #1

2012 Fall CHMMC Tiebreaker 1 - |P(P([2013]))|, set of non-empty subsets

2012 Fall CHMMC Tiebreaker 4 lattice points in 3D sum x'^2 <sum x^2

2012 Fall CHMMC Tiebreaker 3 \sigma (n) < n+ \sqrt{n}