Weights and Scale

In Dr. Hong's home gym, there are some unlabeled weights. There is also a simple balancing scale, where he can place the weights in either or both sides of the scale. Dr. Hong comes to a sporting goods shop to buy three labeled weights, so that he can use them together with the simple balancing scale to measure any weights in consecutive integer number of pounds, starting from 1 lb, 2 lb, ..., up to N lb. The labeled weights Dr. Hong can choose from include 1 lb, 2 lb, 3 lb, ..., up to 50 lb. If Dr. Hong cannot use one or more unlabeled weights to measure another unlabeled weight, what is the largest possible value for N? 

Sum of Unit Fractions

An integer A satisfies the following inequality:

.

What is A?

Power of 1000s

Given that a + b + c = 3, and a² + b² + c² = 3, what is a¹⁰⁰⁰ + b²⁰⁰⁰ + c³⁰⁰⁰ ?

Wait in Line

Dr. Hong takes Leo, Isaac, and Kyle to the grocery store to buy some snacks for an upcoming CMM party. Each of them holds a shopping basket waiting at the cashier’s line to check out some party goods. It takes 10, 4, 6, 2 minutes Respectively to check out each of them. A cashier can check out one person at a time, the others who haven’t checked out must wait in line. Each person is counting his own waiting minutes plus check out minutes. Dr. Hong figures out a way to arrange themselves properly to minimize the sum of those minutes. What is the total minutes of that sum?

Sum of Prime Factors

What is the sum of 2491's prime factors?

Divisible by 72

A 10-digit number A20200514B is divisible by 72, what is B - A?

Court Time Allocation

There are four quarters in an NBA game, with 12 minutes each quarter. At any time on the basketball court, the two teams are playing 5 on 5. One day, Toronto Raptors is playing against Golden State Warriors. Among the five players in the Raptors starting lineup, two are all-star players. The Raptors coach also adds four bench players into the rotation. If each all-star player plays 3 times as long as a bench player, and if a bench player plays half as long as each of the other three starting player, how many minutes does an all-star player have to play in this game?