MathDB

1

Finding S(2006) with Simplified Fraction

n

let

D(n)

denote the set of positive integers that divide

n

and let S(n)\equal{}\Sigma_{k \in D(n)} \frac{1}{k}. What is

S(2006)

? Answer with a fraction reduced to lowest terms.

24

1

Prime Factors of a Huge Repdigit

The number 555,555,555,555 factors into eight distinct prime factors, each with a multiplicity of 1. What are the three largest prime factors of 555,555,555,555?

23

1

Mirrors and Lasers

Consider two mirrors placed at a right angle to each other and two points A at

(x,y)

and B at

(a,b)

. Suppose a person standing at point A shines a laser pointer so that it hits both mirrors and then hits a person standing at point B (as shown in the picture). What is the total distance that the light ray travels, in terms of

a

,

b

,

x

, and

y

? Assume that

x

,

y

,

a

, and

b

are positive. [asy]draw((0,4)--(0,0)--(4,0),linewidth(1)); draw((1,3)--(0,2),MidArrow); draw((0,2)--(2,0),MidArrow); draw((2,0)--(3,1),MidArrow); dot((1,3)); dot((3,1)); label("

A (x,y)

", (1,3),NE); label("

B (a,b)

", (3,1),NE);[/asy]

22

1

Conditional Probability

A certain college student had the night of February 23 to work on a chemistry problem set and a math problem set (both due on February 24, 2006). If the student worked on his problem sets in the math library, the probability of him finishing his math problem set that night is 95% and the probability of him finishing his chemistry problem set that night is 75%. If the student worked on his problem sets in the the chemistry library, the probability of him finishing his chemistry problem set that night is 90% and the probability of him finishing his math problem set that night is 80%. Since he had no bicycle, he could only work in one of the libraries on February 23rd. He works in the math library with a probability of 60%. Given that he finished both problem sets that night, what is the probability that he worked on the problem sets in the math library?

21

1

Relatively Prime

How many positive integers less than 2005 are relatively prime to 1001?

20

1

Bits

Given a random string of 33 bits (0 or 1), how many (they can overlap) occurrences of two consecutive 0's would you expect? (i.e. "100101" has 1 occurrence, "0001" has 2 occurrences)

19

1

Going Beyond Gauss

When the celebrated German mathematician Karl Gauss (1777-1855) was nine years old, he was asked to add all the integers from 1 through 100. He quickly added 1 and 100, 2 and 99, and so on for 50 pairs of numbers each adding in 101. His answer was 50 · 101=5,050. Now find the sum of all the digits in the integers from 1 through 1,000,000 (i.e. all the digits in those numbers, not the numbers themselves).

18

1

Free Throws

Alex and Brian take turns shooting free throws until they each shoot twice. Alex and Brian have 80% and 60% chances of making their free throws, respectively. What is the probability that after each free throw they take, Alex has made at least as many free throws as Brian if Brian shoots first?

17

1

Dumb Drivers and a Dumb Fly

Car A is traveling at 20 miles per hour. Car B is 1 mile behind, following at 30 miles per hour. A fast fly can move at 40 miles per hour. The fly begins on the front bumper of car B, and flies back and forth between the two cars. How many miles will the fly travel before it is crushed in the collision?

16

1

Angles and a Circle

Points

A_1

,

A_2

,

...

are placed on a circle with center

O

such that \angle OA_n A_{n\plus{}1}\equal{}35^\circ and A_n\neq A_{n\plus{}2} for all positive integers

n

. What is the smallest

n>1

for which A_n\equal{}A_1?

15

2

Palindromic Odometer Readings

The odometer of a family car shows 15,951 miles. The driver noticed that this number is palindromic: it reads the same backward as forwards. "Curious," the driver said to himself, "it will be a long time before that happens again." Surprised, he saw his third palindromic odometer reading (not counting 15,951) exactly five hours later. How many miles per hour was the car traveling in those 5 hours (assuming speed was constant)?

2006 SMT Team Round #15 - Computation with Geometric Series

Let

c_i

denote the

i

th composite integer so that

\{c_i\}=4,6,8,9,...

Compute

\prod_{i=1}^{\infty} \dfrac{c^{2}_{i}}{c_{i}^{2}-1}

(Hint:

\textstyle\sum^\infty_{n=1} \tfrac{1}{n^2}=\tfrac{\pi^2}{6}

)

14

2

Area of a Region

Determine the area of the region defined by x²+y²≤π² and y ≥ sin x.

2006 SMT Team Round #14 - Smallest Nonnegative $n$

Find the smallest nonnegative integer

n

for which

\binom{2006}{n}

is divisible by

7^3

.

13

2

Inserting Plus and Minus Signs

123456789=100. Here is the only way to insert 7 pluses and/or minus signs between the digits on the left side to make the equation correct: 1+2+3-4+5+6+78+9=100. Do this with only three plus or minus signs.

2006 SMT Team Round #13 - Compute $\cos\theta$

A ray is drawn from the origin tangent to the graph of the upper part of the hyperbola

y^2=x^2-x+1

in the first quadrant. This ray makes an angle of

\theta

with the positive

x

-axis. Compute

\cos\theta

.

12

2

Largest Prime Factor

What is the largest prime factor of 8091?

2006 SMT Team Round #12 - Number of k-tuples

Find the total number of

k

-tuples

(n_1,n_2,...,n_k)

of positive integers so that

n_{i+1}\ge n_i

for each

i

, and

k

regular polygons with numbers of sides

n_1,n_2,...,n_k

respectively will fit into a tesselation at a point. That is, the sum of one interior angle from each of the polygons is

360^{\circ}

.

11

2

A Black and White View on Insurance

An insurance company believes that people can be divided into 2 classes: those who are accident prone and those who are not. Their statistics show that an accident prone person will have an accident in a yearly period with probability 0.4, whereas this probability is 0.2 for the other kind. Given that 30% of people are accident prone, what is the probability that a new policyholder will have an accident within a year of purchasing a policy?

2006 SMT Team Round #11 - Determine Sum of Roots

Polynomial

P(x)=c_{2006}x^{2006}+c_{2005}x^{2005}+\ldots+c_1x+c_0

has roots

r_1,r_2,\ldots,r_{2006}

. The coefficients satisfy

2i\tfrac{c_i}{c_{2006}-i}=2j\tfrac{c_j}{c_{2006}-j}

for all pairs of integers

0\le i,j\le2006

. Given that

\sum_{i\ne j,i=1,j=1}^{2006} \tfrac{r_i}{r_j}=42

, determine

\sum_{i=1}^{2006} (r_1+r_2+\ldots+r_{2006})

.

10

4

9

3

Numerators and Denominators

If to the numerator and denominator of the fraction

\frac{1}{3}

you add its denominator 3, the fraction will double. Find a fraction which will triple when its denominator is added to its numerator and to its denominator and find one that will quadruple.

2006 SMT Algebra #9 - Figure out the Trick

Principal Skinner is thinking of two integers

m

and

n

and bets Superintendent Chalmers that he will not be able to determine these integers with a single piece of information. Chalmers asks Skinner the numerical value of

mn+13m+13n-m^2-n^2

. From the value of this expression alone, he miraculously determines both

m

and

n

. What is the value of the above expression.

2006 SMT Team Round #9 - Geometry

\triangle ABC

has

AB=AC

. Points

M

and

N

are midpoints of

\overline{AB}

and

\overline{AC}

, respectively. The medians

\overline{MC}

and

\overline{NB}

intersect at a right angle. Find

(\tfrac{AB}{BC})^2

.

8

3

Addition and Multiplication Swapping

Given two 2's, "plus" can be changed to "times" without changing the result: 2+2=2·2. The solution with three numbers is easy too: 1+2+3=1·2·3. There are three answers for the five-number case. Which five numbers with this property has the largest sum?

2006 SMT Algebra #8 - Evaluate

Evaluate:

\sum^{10}_{x=2} \dfrac{2}{x(x^2-1)}

.

2006 SMT Team Round #8 - Evaluate

Evaluate:

\lim_{n\rightarrow\infty}\sum_{k=n^2}^{(n+1)^2} \dfrac{1}{\sqrt{k}}

4

4

4

4

3

Bread and Milk

A customer enters a supermarket. The probability that the customer buys bread is .60, the probability that the customer buys milk is .50, and the probability that the customer buys both bread and milk is .30. What is the probability that the customer would buy either bread or milk or both?

2006 SMT Team Round #2 - Second Term in Sequence

In a given sequence

\{S_1,S_2,...,S_k\}

, for terms

n\ge3

,

S_n=\sum_{i=1}^{n-1} i\cdot S_{n-i}

. For example, if the first two elements are 2 and 3, respectively, the third entry would be

1\cdot3+2\cdot2=7

, and the fourth would be

1\cdot7+2\cdot3+3\cdot2=19

, and so on. Given that a sequence of integers having this form starts with 2, and the 7th element is 68, what is the second element?

Minimum of Expression with x, y, and z

Find the minimum value of 2x^2\plus{}2y^2\plus{}5z^2\minus{}2xy\minus{}4yz\minus{}4x\minus{}2z\plus{}15 for real numbers

x

,

y

,

z

.

1

4

2006 Stanford Mathematics Tournament

Subcontests

Finding S(2006) with Simplified Fraction

Prime Factors of a Huge Repdigit

Mirrors and Lasers

Conditional Probability

Relatively Prime

Bits

Going Beyond Gauss

Free Throws

Dumb Drivers and a Dumb Fly

Angles and a Circle

Palindromic Odometer Readings

2006 SMT Team Round #15 - Computation with Geometric Series

Area of a Region

2006 SMT Team Round #14 - Smallest Nonnegative $n$

Inserting Plus and Minus Signs

2006 SMT Team Round #13 - Compute $\cos\theta$

Largest Prime Factor

2006 SMT Team Round #12 - Number of k-tuples

A Black and White View on Insurance

2006 SMT Team Round #11 - Determine Sum of Roots

Numerators and Denominators

2006 SMT Algebra #9 - Figure out the Trick

2006 SMT Team Round #9 - Geometry

Addition and Multiplication Swapping

2006 SMT Algebra #8 - Evaluate

2006 SMT Team Round #8 - Evaluate

Bread and Milk

2006 SMT Team Round #2 - Second Term in Sequence

Minimum of Expression with x, y, and z