## 20 Math Problems for 2020

Happy New Year!

• It’s a leap year — yay!
• It’s the year of the rat — squeak!
• It’s an election year — okay, maybe it’s not that great of a year after all.

To get your mind thinking about something other than the associated realities of that last bullet, here are 20 problems to prepare you for the next 366 days.

1. What’s the difference between the number of positive integer factors of 2020 and the number of positive integer factors of 2020?
2. If all of the numbers from 1 to 2,020 were written down, how many digits would be used?
3. If all of the numbers from 1 to 2,020 were spelled out, how many letters would be used?
4. The numbers 1 through 2,020 are written on a whiteboard. At every stage, two numbers (say, a and b) are erased from the whiteboard and replaced with the sum a + b + 1. For example, if 187 and 2,013 were erased, then 187 + 2,013 + 1 = 2,201 would be written on the whiteboard. This process is repeated until a single number remains on the whiteboard. What is the number?
5. How many positive integers are less than the square root of 2,020?
6. How many different rectangles with integer side lengths and an area of 2,020 square units are possible?
7. A lighthouse is perched on the cliff of a rocky beach. Standing in the lantern room of the lighthouse, you are approximately 2,020 feet above the surface of the ocean below. How far can you see to the horizon?
8. A rectangular prism with integer edge lengths has a volume of 2,020 cubic centimeters. What is the maximum possible surface area of this solid?
9. Mike made two initial bank deposits on Monday and Tuesday. Then every day for the rest of the week, he deposited an amount equal to the sum of the previous two days’ deposits. If he deposited \$2,020 on Saturday, and his deposit on Monday was less than \$10, what was the amounts of his deposit on Tuesday?
10. What’s the sum of 101 + 202 + 303 + … + 2,020?
11. How many postive, four-digit numbers can be formed with the digits 2, 0, 2, and 0?
12. Saying that someone has 20/20 vision means they have normal visual acuity. That is, if you have 20/20 vision, then from 20 feet away, you can see what a normal person would see clearly from 20 feet away. In general, if you have 20/n vision, then from 20 feet away, you can see what normal people would see from n feet away. A person with 20/5 vision is looking at a billboard that is a mile away, and she can clearly see the letters on the billboard. How much closer would a person with 20/80 vision need to stand to clearly see the letters on the billboard? (Assume both people are standing at the boundary of where they can read the sign clearly.)
13. A triangle has two sides of length 20. What is the maximum possible area of this triangle?
14. Using only the digit 2 and the addition symbol, you can create expressions with many different values. For instance, you could use three 2’s and one addition symbol to make 22 + 2 = 24, or you could use  eight 2’s and three additional symbols to make 222 + 22 + 22 + 2 = 268. Using this process, what is the minimum number of 2’s needed to make an expression with a value of 2,020?
15. In how many zeroes does the number 2020 end?
16. Find the smallest possible string of consecutive positive integers that have a sum of 2,020.
17. One integer is removed from the set {1, 2, 3, …, n} so that the sum of the remaining numbers is 2,020. What integer was removed?
18. What is the product of the prime divisors of 2,020?
19. What is the maximum possible product for a set of positive integers that have a sum of 2,020?
20. Which of the following chains — each consisting of regular polygons with side length 1 unit — could be extended to have a perimeter of exactly 2,020 units? 1. 849
2. 6,973 digits
3. 47,123 letters
4. 2,043,229
5. 44 integers
6. 6 rectangles
7. approximately 52.5 miles
8. 8,082 square centimeters
9. \$401 on Tuesday (and \$5 on Monday)
10. 21,210
11. 3 numbers
12. the person would need to be 1/16 mile = 330 feet from the billboard, which is 4,950 feet closer
13. 200 square units
14. 29 2’s
15. 20 zeroes
16. 402, 403, 404, 405, 406
17. 60
18. 1,010
19. 3672 × 22
20. only the squares; for n squares, the perimeter is 2n + 2, and n = 1,009 yields a perimeter of 2 × 1,009 + 2 = 2,020

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