Triangle inside a Triangle

In triangle ABC, D, E, and F are on AB, AC, and BC respectively. AD:BD = 2:1; AE:EC = 1:1; BF:FC = 3:1.  AF and BE intersects at G; BE and CD intersects at H; AF and CD intersects at I. If the area of triangle ABC is 504, what is the area of triangle GHI?

Sharing Mooncakes

It’s mid-autumn day, a festival in China when family members get together and eat mooncakes. Dr. Hong takes out 3 mooncakes of the same flavor and cuts each mooncake into 4 equal pieces. He then asks Leo to allocate them to the Hong family: Dr. Hong, Dr. Wang, Leo, Angela, and Elon. 

1. How many different ways can Leo allocate the mooncake pieces?

2. If Leo has to give at least one piece to each family member, how many different ways can Leo allocate the mooncake pieces?

3. If Leo has to give at least one piece to each family member, what’s the likelihood that Leo  gets more pieces than the number of pieces any other family member gets?

4. If Leo has to give at least one piece to each family member, and more pieces to each parent (Dr. Hong and Dr. Wang) than any kid (Leo, Angela, and Elon), how many different ways can Leo allocate the mooncake pieces? 

5. (Optional) If Dr. Hong takes out 4 mooncakes instead of 3, under the same settings, what are the answers to the previous 4 questions?

 

Mind and Matter

Leo is reading Mind and Matter, a book written by a former NFL player now studying math at MIT. The book has 238 pages. If the pages are numbered consecutively starting from 1, what’s the sum of the digits of all page numbers? 

1+2+4=7

 In the expression I + II + IV = VII, I and V represent different single digit numbers. What’s I?

Hiding Red Paint

Leo puts 27 small 1x1x1 cubes together to build a 3x3x3 cube. He then paints all six faces of the big cube with red paint. After that, he rearranges the small cubes to a new 3x3x3 cube by minimizing the area of red paint on the surface. What is the red surface area of the rearranged 3x3x3 cube?

Pulling Weeds

Dr. Hong is taking Leo, Angela, and Elon to pull the weeds out of their front yard. If Dr. Hong does all the work on his own, it will take him 10 hours. If Leo and Angela work together, it will take them 15 hours. If Leo and Elon work together, it will take them 20 hours. If Angela and Elon work together, it will take them 30 hours. How many hours does it take if all four of them work together? Please keep two digits after the decimal point for the final answer. (We can assume that the weeds don’t grow when they work on the lawn.)

777 in Base k

 The three digit number 777 in base k is equivalent to 4145 in base 8. What is k?

Playing Dice

The Hong family are playing dice tonight. Dr. Hong serves as the judge. Leo, Angela, Elon, and Dr. Wang are the contestants. Each of them comes to Dr. Hong separately, rolls the dice, memorizes the number on its upper surface, and keeps the number secret to the other three. (Dr. Hong knows everyone’s number.) After all four of them roll the dice, they are trying to guess who gets the smallest or largest number based on the information during the conversation:

• Leo says: I don’t know if my number is the smallest or largest.
• Angela says: I don’t know if my number is the smallest or largest.
• Elon says: I don’t know if my number is the smallest or largest.
• Dr. Wang says: I don’t know if my number is the smallest or largest.
• Then Angela says: I have the largest number!

What’s Angela’s number?

(A dice is a cube with each of its six faces marked with a different number of dots (pips) from one to six.) 

Angela’s Favorite Book

Angela’s favorite book is “The Secret of Cacklefur Castle”. The book has 109 pages. If the pages are numbered consecutively starting from 1. How many times the digit 0 shows up in the page numbers?

Starting with a Random Number

Pick any number between 1 and 10. Subtract 2. Add 3. Subtract 5. Add 7. Subtract the original number. What’s the largest possible value of the resulting number?