MathDB

1

2015 CHMMC Individual - Caltech Harvey Mudd Mathematics Competition

p1. The following number is the product of the divisors of

n

.

2^63^3

What is

n

?
p2. Let a right triangle have the sides

AB =\sqrt3

,

BC =\sqrt2

, and

CA = 1

. Let

D

be a point such that

AD = BD = 1

. Let

E

be the point on line

BD

that is equidistant from

D

and

A

. Find the angle

\angle AEB

.
p3. There are twelve indistinguishable blackboards that are distributed to eight different schools. There must be at least one board for each school. How many ways are there of distributing the boards?
p4. A Nishop is a chess piece that moves like a knight on its first turn, like a bishop on its second turn, and in general like a knight on odd-numbered turns and like a bishop on even-numbered turns. A Nishop starts in the bottom-left square of a

3\times 3

-chessboard. How many ways can it travel to touch each square of the chessboard exactly once?
p5. Let a Fibonacci Spiral be a spiral constructed by the addition of quarter-circles of radius

n

, where each

n

is a term of the Fibonacci series:

1, 1, 2, 3, 5, 8,...

(Each term in this series is the sum of the two terms that precede it.) What is the arclength of the maximum Fibonacci spiral that can be enclosed in a rectangle of area

714

, whose side lengths are terms in the Fibonacci series?
p6. Suppose that

a_1 = 1

and

a_{n+1} = a_n -\frac{2}{n + 2}+\frac{4}{n + 1}-\frac{2}{n}

What is

a_{15}

?
p7. Consider

5

points in the plane, no three of which are collinear. Let

n

be the number of circles that can be drawn through at least three of the points. What are the possible values of

n

?
p8. Find the number of positive integers

n

satisfying

\lfloor n /2014 \rfloor =\lfloor n/2016 \rfloor

.
p9. Let

f

be a function taking real numbers to real numbers such that for all reals

x \ne 0, 1

, we have

f(x) + f \left( \frac{1}{1 - x}\right)= (2x - 1)^2 + f\left( 1 -\frac{1}{ x}\right)

Compute

f(3)

.
p10. Alice and Bob split

5

beans into piles. They take turns removing a positive number of beans from a pile of their choice. The player to take the last bean loses. Alice plays first. How many ways are there to split the piles such that Alice has a winning strategy?
p11. Triangle

ABC

is an equilateral triangle of side length

1

. Let point

M

be the midpoint of side

AC

. Another equilateral triangle

DEF

, also of side length

1

, is drawn such that the circumcenter of

DEF

is

M

, point

D

rests on side

AB

. The length of

AD

is of the form

\frac{a+\sqrt{b}}{c}

, where

b

is square free. What is

a + b + c

?
p12. Consider the function

f(x) = \max \{-11x- 37, x - 1, 9x + 3\}

defined for all real

x

. Let

p(x)

be a quadratic polynomial tangent to the graph of

f

at three distinct points with x values

t_1

,

t_2

and

t_3

Compute the maximum value of

t_1 + t_2 + t_3

over all possible

p

.
p13. Circle

J_1

of radius

77

is centered at point

X

and circle

J_2

of radius

39

is centered at point

Y

. Point

A

lies on

J1

and on line

XY

, such that A and Y are on opposite sides of

X

.

\Omega

is the unique circle simultaneously tangent to the tangent segments from point

A

to

J_2

and internally tangent to

J_1

. If

XY = 157

, what is the radius of

\Omega

?
p14. Find the smallest positive integer

n

so that for any integers

a_1, a_2,..., a_{527}

,the number

\left( \prod^{527}_{j=1} a_j\right) \cdot\left( \sum^{527}_{j=1} a^n_j\right)

is divisible by

527

.
p15. A circle

\Omega

of unit radius is inscribed in the quadrilateral

ABCD

. Let circle

\omega_A

be the unique circle of radius

r_A

externally tangent to

\Omega

, and also tangent to segments

AB

and

DA

. Similarly define circles

\omega_B

,

\omega_C

, and

\omega_D

and radii

r_B

,

r_C

, and

r_D

. Compute the smallest positive real

\lambda

so that

r_C < \lambda

over all such configurations with

r_A > r_B > r_C > r_D

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

10

1

2015 Fall Team #10

Let

P

be the parabola in the plane determined by the equation

y = x^2

. Suppose a circle

C

in the plane intersects

P

at four distinct points. If three of these points are

(-28, 784)

,

(-2, 4)

, and

(13, 169)

, find the sum of the distances from the focus of

P

to all four of the intersection points

9

1

2015 Fall Team #9

Let

T

be a

2015 \times 2015

array containing the integers

1, 2, 3, ... , 2015^2

satisfying the property that

T_{i,a }> T_{i,b}

for all

a > b

and

T_{c,j} > T_{d,j}

for all

c > d

where

1 \le a, b, c, d \le 2015

and

T_{i,j}

represents the entry in the

i

-th row and

j

-th column of

T

. How many possible values are there for the entry at

T_{5,5}

?

8

1

2015 Fall Team #8

Let

f(n) = \sum^n_{d=1} \left\lfloor \frac{n}{d} \right\rfloor

and

g(n) = f(n) -f(n - 1)

. For how many

n

from

1

to

100

inclusive is

g(n)

even?

7

1

2015 Fall Team #7

Let

I

be the incenter and let

\Gamma

be the incircle of

\vartriangle ABC

, and let

P = \Gamma \cap BC

. Let

Q

denote the intersection of

\Gamma

and the line passing through

P

parallel to

AI

. Let

\ell

be the tangent line to

\Gamma

at

Q

and let

\ell \cap AB = S

,

\ell \cap AC = R

. If

AB = 7

,

BC = 6

,

AC = 5

, what is

RS

?

6

1

2015 Fall Team #6

The icosahedron is a convex, regular polyhedron consisting of

20

equilateral triangle for faces. A particular icosahedron given to you has labels on each of its vertices, edges, and faces. Each minute, you uniformly at random pick one of the labels on the icosahedron. If the label is on a vertex, you remove it. If the label is on an edge, you delete the label on the edge along with any labels still on the vertices of that edge. If the label is on a face, you delete the label on the face along with any labels on the edges and vertices which make up that face. What is the expected number of minutes that pass before you have removed all labels from the icosahedron?

5

1

2015 Fall Team #5

Felix is playing a card-flipping game.

n

face-down cards are randomly colored, each with equal probability of being black or red. Felix starts at the

1

st card. When Felix is at the

k

th card, he guesses its color and then flips it over. For

k < n

, if he guesses correctly, he moves onto the

(k + 1)

-th card. If he guesses incorrectly, he gains

k

penalty points, the cards are replaced with newly randomized face-down cards, and he moves back to card

1

to continue guessing. If Felix guesses the

n

th card correctly, the game ends. What is the expected number of penalty points Felix earns by the end of the game?

4

2

2015 Fall Team #4

Let

P(x) = x^{16}-x^{15}+·...-x+ 1

, and let p be a prime such that

p-1

is divisible by

34

(

p = 103

is an example). How many integers a between

1

and

p-1

inclusive satisfy the property that

P(a)

is divisible by

p

?

2015 CHMMC Tiebreaker 4 , 46, 656, 000, 000 = product of divisors of n

The following number is the product of the divisors of

n

.

46, 656, 000, 000

What is

n

?

3

2

2015 Fall Team #3

A trio of lousy salespeople charge increasing prices on tomatoes as you buy more. The first charges you

x^1_1

dollars for the

x_1

th tomato you buy from him, the second charges

x^2_2

dollars for the

x_2

th tomato, and the third charges

x^3_3

dollars for the

x_3

th tomato. If you want to buy

100

tomatoes for as cheap as possible, how many should you buy from the first salesperson?

2015 CHMMC Tiebreaker 3 - 3-digit pair cycles

Define an

n

-digit pair cycle to be a number with

n^2 + 1

digits between

1

and

n

with every possible pair of consecutive digits. For instance,

11221

is a 2-digit pair cycle since it contains the consecutive digits

11

,

12

,

22

, and

21

. How many

3

-digit pair cycles exist?

2

2015 Fall Team #2

You have

4

game pieces, and you play a game against an intelligent opponent who has

6

. The rules go as follows: you distribute your pieces among two points a and b, and your opponent simultaneously does as well (so neither player sees what the other is doing). You win the round if you have more pieces than them on either

a

or

b

, and you lose the round if you only draw or have fewer pieces on both. You play the optimal strategy, assuming your opponent will play with the strategy that beats your strategy most frequently. What proportion of the time will you win?

2015 CHMMC Tiebreaker 2 - a_{n+1} =1/n a_n + a_{n-1}

Let

a_1 = 1

,

a_2 = 1

, and for

n \ge 2

, let

a_{n+1} =\frac{1}{n} a_n + a_{n-1}.

What is

a_{12}

?

1

2

2015 Fall Team #1

3

players take turns drawing lines that connect vertices of a regular

n

-gon. No player may draw a line that intersects another line at a point other than a vertex of the

n-

gon. The last player able to draw a line wins. For how many

n

in the range

4\le n \le 100

does the first player have a winning strategy?

2015 CHMMC Tiebreaker 1 - 2015- cube-invariant postitive integers

Call a positive integer

x

n

-cube-invariant if the last

n

digits of

x

are equal to the last

n

digits of

x^3

. For example,

1

is

n

-cube invariant for any integer

n

. How many

2015

-cube-invariant numbers

x

are there such that

x < 10^{2015}

?

2015 CHMMC (Fall)

Subcontests

2015 CHMMC Individual - Caltech Harvey Mudd Mathematics Competition

2015 Fall Team #10

2015 Fall Team #9

2015 Fall Team #8

2015 Fall Team #7

2015 Fall Team #6

2015 Fall Team #5

2015 Fall Team #4

2015 CHMMC Tiebreaker 4 , 46, 656, 000, 000 = product of divisors of n

2015 Fall Team #3

2015 CHMMC Tiebreaker 3 - 3-digit pair cycles

2015 Fall Team #2

2015 CHMMC Tiebreaker 2 - a_{n+1} =1/n a_n + a_{n-1}

2015 Fall Team #1

2015 CHMMC Tiebreaker 1 - 2015- cube-invariant postitive integers