2006 SMT/General Problems

Problem 1

After a cyclist has gone $\frac{2}{3}$ of his route, he gets a flat tire. Finishing on foot, he spends twice as long walking as he did riding. How many times as fast does he ride as walk?

Solution

Problem 2

A customer enters a supermarket. The probability that the customer buys bread is $0.60$ , the probability that the customer buys milk is $0.50$ , and the probability that the customer buys both bread and milk is $0.30$ . What is the probability that the customer would buy either bread or milk or both?

Solution

Problem 3

After a typist has written ten letters and had addressed the ten corresponding envelopes, a careless mailing clerk inserted the letters in the envelopes at random, one letter per envelope. What is the probability that exactly nine letters were inserted in the proper envelopes?

Solution

Problem 4

In a certain tournament bracket, a player must be defeated three times to be eliminated. If $512$ contestants enter the tournament, what is the greatest number of games that could be played?

Solution

Problem 5

A geometric series is one where the ratio between each two consecutive terms is constant (ex. $3, 6, 12, 24,\cdots$ ). The fifth term of a geometric series is $5!$ , and the sixth term is $6!$ . What is the fourth term?

Solution

Problem 6

An alarm clock runs $4$ minutes slow every hour. It was set right $3\frac{1}{2}$ hours ago. Now another clock which is correct shows noon. In how many minutes, to the nearest minute, will the alarm clock show noon?

Solution

Problem 7

An aircraft is equipped with three engines that operate independently. The probability of an engine failure is $.01$ . What is the probability of a successful flight if only one engine is needed for the successful operation of the aircraft?

Solution

Problem 8

Given two $2$ 's, "plus" can be changed to "times" without changing the result: $2+2=2\cdot2$ . The solution with three numbers is easy too: $1+2+3=1\cdot2\cdot3$ . There are three answers for the five-number case. Which five numbers with this property has the largest sum?

Solution

Problem 9

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.

Solution

Problem 10

What is the square root of the sum of the first $2006$ positive odd integers?

Solution

Problem 11

An insurance company believes that people can be divided into $2$ clases: 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?

Solution

Problem 12

What is the largest prime factor of $8091$ ?

Solution

Problem 13

$123456789=100$ . 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.

Solution

Problem 14

Determine the area of the region defined by $x^2+y^2\le\pi^2$ and $y\ge\sin x$ .

Solution

Problem 15

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 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)?

Solution

Problem 16

Points $A_1, A_2, \cdots$ are placed on a circle with center $O$ such that $\angle OA_nA_{n+1}=35^\circ$ and $A_n\not=A_{n+2}$ for all positive integers $n$ . What is the smallest $n>1$ for which $A_n=A_1$ ?

Solution

Problem 17

Car A is traveling $20$ 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?

Solution

Problem 18

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?

Solution

Problem 19

When the celebrated German mathematician Karl Gauss (1777-1855) was nine years old, he was asked to add all the integers from $1$ to $100$ . He quickly added $1$ and $100$ , $2$ and $99$ , and so on for $50$ pairs of numbers each adding in $101$ . His answer was $50\cdot101=5,050$ . 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).

Solution

Problem 20

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

Solution

Problem 21

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

Solution

Problem 22

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?

Solution

Problem 23

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

Solution

Problem 24

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 $555,555,555,555$ ?

Solution

Problem 25

For positive integers $n$ let $D(n)$ denote the set of positive integers that divide $n$ and let $S(n)=\sum_{k\in D(n)}\frac{1}{k}$ . What is $S(2006)$ ? Answer with a fraction reduced to lowest terms.

Solution

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2006 SMT/General Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

See Also