**problems 1-10 (problems only)**

**problems 1-10 (problems only)**

*These problems were posted as an ongoing contest on www.dansmath.com from late 1997 to late 2008, divided into 12 seasons.*

### Problem #1** **- Posted Friday, November 21, 1997

*The Most Divisors*

*The Most Divisors*

A **divisor** of a positive whole number n is a whole number that divides evenly into n (with no remainder). What integer from 1 through 1000 has the most divisors? *(**To win you must prove it and explain your method.)*

### Problem #2 - Posted Friday, November 28, 1997

**Trains versus Fly**

**Trains versus Fly**

Two trains are 2 miles apart and are traveling towards each other on the same track, each train going 30 mph. A fly going 60 mph starts at the nose of one train, flies toward the other train, and upon reaching the second train immediately turns around and flies back towards the first train. The fly buzzes back and forth until all three collide. How far did the fly fly?

### Problem #3 - Posted Sunday, December 7, 1997

**Unit Fraction Problems **

**Unit Fraction Problems**

A **unit fraction** is of the form** 1/n** (where n = whole no. ≥ 1.)

a) Starting with 1/1 , then 1/2 , 1/3 , etc., how many unit fractions does it take to add up to more than pi ?

b) Express 3/23 as the sum of two unit fractions; 3/23 = 1/a + 1/b .

c) Write 5/4 as the sum of distinct unit fractions.

*Fine print: **The winner in part c) will be the one using the smallest number of fractions. **In case of a tie, the winner is the one with the "smallest biggest denominator."*

### Problem #4 - Posted Thursday, December 18, 1997

**The Census Taker Problem**

**The Census Taker Problem**

A census-taker rings Mr. Simpson's bell and asks how many children he has.

"Three daughters," he replies.

"And how old are they, in whole numbers?" asks the census-taker.

"Well, I'll tell you this: the product of their ages is 72, and the sum of their ages is my house number."

"But that isn't enough information!" complains the census taker.

"Okay, my oldest daughter (in years) likes chocolate milk," replies Mr. Simpson.

With that, the census-taker nods and writes down the three ages.

**How old are the Simpson girls, and how did the census-taker figure it out?***Include a full explanation!*

### Problem #5 - Posted Monday, December 29, 1997

**A Tree-o of Square Root Problems**

**A Tree-o of Square Root Problems**

Let **Sqrt(x) or -/x** denote the (positive) square root of x, as in Sqrt(100) = -/100 = 10.

Also **x^2** will mean x squared, as in 10 ^ 2 = 10 * 10 = 100.

a ) If: Sqrt(m) + Sqrt(n) = 13 , and m and n differ by 65, what is the largest possible value of m ?

b ) Notice that the equation x^2 - 3 = 0 has a solution x = Sqrt(3). Find a polynomial equation in x, with integer coefficients, having x = Sqrt(3) + Sqrt(5) as a solution.

c ) What is a really good fraction approximation for Sqrt(17), and why? Generalize your answer if possible to Sqrt(n^2 + 1).

### Problem #6 - Posted Wednesday, January 7, 1998

**New Year, More House Numbers!**

**New Year, More House Numbers!**

The people living on Sesame Street all decide to buy new house numbers, so they line up at the store in order of their addresses: 1, 2, 3, . . . .

If the store has 100 of each digit, **what is the first address that won't be able to buy its house numbers?**

### Problem #7 - Posted Friday, January 23, 1998

**Patterns and Sequences **

**Patterns and Sequences**

### a) 2, 3, 5, 8, 13, _?_

### b) 2, 3, 5, 7, 11, _?_

### c) 3, 3, 5, 4, 4, 3, 5, 5, 4, _?_

### d) 1, 3, 7, 15, 31, _?_

### e) 1, 4, 27, 256, 3125, _?_

### f) 1, 2, 6, 24, 120, 720, _?_

### g) 1, 2, 6, 30, 210, _?_

**What number comes next in each sequence? Give reasons!**

### Problem #8 - Posted Friday, January 30, 1998

**Clinking Wine Glasses**** **

**Clinking Wine Glasses**

When I have wine with a few people and we clink glasses and say *salud*, I can always tell if everyone has "clinked" with everyone else, because I know math! Let's assume each person clinks each other person exactly once. If there are 2 people, there is one "clink." If there are 3 people, there are 3 clinks.

a) ** How many clinks** are there for 4, 5, 6, . . . 10 people?

b) ** How many people** were there if I heard 903 clinks?

c) **What is the formula **for the quantity : c(n) = number of clinks for a group of n people ?

### Problem #9 - Posted Sunday, February 15, 1998

**Strange Powers**

**Strange Powers**

The expression **b ^ n** means b to the power n.

**a) If 3 ^ a = 4 and 4 ^ b = 8, what is 9 ^ (a – b) ?**

**b) Can you find numbers a ≠ b such that a ^ b = b ^ a ?**

**c) If a $ b means a ^ b – b ^ a , what is 4 $ 6 ?**

**d) Is 2 $ (3 $ 4) the same as (2 $ 3) $ 4 ?**

### Problem #10 - **Posted Saturday, February 28, 1998**

**Odd and Abundant? **

**Odd and Abundant?**

A natural number is **abundant** if its proper divisors (not including itself) add up to more than the number. (12 is abundant: the divisors of 12 are 1, 2, 3, 4, 6, and 12, and 1 + 2 + 3 + 4 + 6 = 16 > 12.)

Note: 6 = 1 + 2 + 3 ; 6 is called **perfect**; 15 is **deficient** (not abundant or perfect): 1 + 3 + 5 = 9 < 15.

**What is the smallest odd abundant number? ***Prove your answer and spell out your thinking.*

*(Hint: Numbers like 12 = 2 * 2 * 3 . that have lots of small prime factors, tend to be abundant.)*