Double Under Warm-up

Leo practices jump rope as part of his warm-up routine for basketball. There are three different sets he practices. Set A is 15 seconds long, when he jumps at the rate of 21 double unders per 9 seconds. Set B is 30 seconds long, when he jumps at the rate of 22 double unders per 10 seconds. Set C is 1 minute long, when he jumps at the rate of 23 double unders per 12 seconds. The warm-up period is 10 minutes, starting from the beginning of the first set. Leo would add a set to the remaining seconds if and only if completing the set doesn't make him go beyond the 10-minute period. The warm-up follows the rules below:

1. The adjacent sets must be different;
2. Leo has to practice all three sets during the warm-up;
3. The resting time between two adjacent sets is the sum of duration of the two adjacent sets. 

Previously, Dr. Hong arranged the warm-up plan to minimize the number of sets. This week, Dr. Hong changed the plan to maximize the number of double unders Leo jumps. How many extra double unders in a warm-up does Leo have to jump this week?

Angles of a Star

ABCDE is a convex pentagon. What is the sum of the following five angles?

Making Perfect Squares

The sum n² + 2020 is a perfect square. What's the sum of all positive values of n?

Coolest Father and Son

In the equation TAOHONG + LEOHONG = COOLEST, different letters represent different single-digit non-negative integers. What's the value of COOLEST?

Eight Coins

A group of CMM students decide to buy a math workbook priced at $8.41. Looking at their piggy banks. they find that each student can contribute the same amount using the same set of eight coins. After each of them brings the eight-coin set from their piggy banks, they put the coins in a jar together. How many nickels are in the jar?

Consecutive Even Numbers

The sum of 20 consecutive even numbers is 2020. What's the average of the first five even numbers of the same set?

Maximize the Expression

Fill in the blanks of the expression 1 _ 2 _ 3 _ 2 _ 1 with all four basic arithmetic operators, and then add one pair of parentheses. What's the maximum value of the resulting expression?

Coin Problems

Leo has a money bus to collect his savings. After buying his favorite Lego set, Leo has a few coins left, including 4 pennies, 3 nickels, 2 dimes and 1 quarter. For the following problems, we only consider four types of coins: pennies, nickels, dimes and quarters.

1. How many coins are there? 
2. What's the difference (in cent) between the largest and smallest face values of the coins in Leo's money bus?
3. What's the total value (in dollar) of Leo's coins?
4. What's the average value (in cent) of Leo's coins?
5. What's the median value (in cent) of Leo's coins?
6. How many dollar amounts can Leo make using one or more of his coins?
7. If he can use no more than one coin of each face value, how many positive dollar amounts can Leo make?
8. To make $0.36, what's the least number of coins shall Leo pick from his money bus?
9. What's the least number of coins that sum to the same value as the coins in Leo's money bus? 
10. How many different combinations of coins can Leo make using the coins in his money bus? (A combination has at least one coin.)
11. How many different sets of coins can sum to the same value as the total value of coins in Leo's money bus?
12. If Leo redesigned the face values of coins, what's the least number of face values shall he use to minimize the total number of coins that can cover all values from 1 cent to the total value of the coins in Leo's money bus?