Perfect Squares

The whole numbers 8, 10, 15, 17, 21, and 28 are arranged, without repetition, in a horizontal row so that the sum of any two numbers in adjacent positions is a perfect square. Find the sum of the middle two numbers.

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A similar problem was given in Assignment 1 of Charlotte Math Meetup.

The whole numbers 3, 4, 5, 6, 12 and 13 are arranged, without repetition, in a horizontal row so that the sum of any two numbers in adjoining positions is a perfect square. Find the sum of the middle two numbers.

Leo's Homework

Every week, Leo works on the 7 problems from CMM assignments. During the 3rd week, he recorded the time for each problem, and found the following interesting facts:

• problems 1, 2, 3 together took him 6 minutes; 
• problems 2, 3, 4 together took him 7 minutes; 
• problems 3, 4, 5 together took him 8 minutes; 
• problems 4, 5, 6 together took him 9 minutes; 
• problems 5, 6, 7 together took him 10 minutes; 
• problems 1, 6, 7 together took him 11 minutes; 
• problems 1, 2, 7 together took him 12 minutes. 

How many minutes did Leo spend on problem #5?

Dr. Hong's Magic Number

Dr. Hong has a magic number ABACDD that satisfies the following conditions:

1. ABACDD = EF × BE × GD; 
2. BE × GD = EGHF; 
3. EF, BE, and GD are prime numbers;  
4. A, B, C, D, E, F, G, H represent eight different single-digit non-zero integers. 

what is Dr. Hong's magic number ABACDD?

Squares in Fractions

Calculate:

. 

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A similar problem was given in Assignment 3 (Level M) of Charlotte Math Meetup:

Calculate:

.

Running drill

Starting at the same time on opposite baselines of a basketball court, Dr. Hong and Leo cross back and forth for 35 seconds without stopping. Dr. Hong needs 5 second to cross court, while Leo needs 7 seconds. What is the number of times during the 35 seconds that the Dr. Hong passes Leo going in the same or opposite direction?

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A similar problem was given in Assignment 1 of Charlotte Math Meetup:

Starting at the same time on opposite shores of a lake, two boats cross back and forth for 35 minutes without stopping. One boat needs 5 minutes to cross the lake. The other boat needs 7 minutes. What is the number of times during the 35 minutes that the faster boat passes the slower boat going in the same or opposite direction?

1v1 Basketball

Dr. Hong and Leo play a series of 1v1 basketball games until one of them has won two games. No game ends in a tie. In any single game, the probability that Dr. Hong wins is 80%. What is the probability that they play exactly 2 games?

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A similar problem was given in Assignment 1 of Charlotte Math Meetup:

Team A and Team B play a series of games until one of them has won two games. No game ends in a tie. In any single game, the probability that Team A win is 70%. What is the probability that they play exactly 2 games?

Multiplication

Calculate 498 × 498 without using a calculator.
(You may use a paper and pencil if you have to, but please think about the fastest way to do it.)

Up-and-down Number

In an up-and-down counting number, the digits increase to a maximum digit and then decrease. This maximum is not the first or last digit. (A few examples: 1247321 is an up-and-down counting number; 12477321 is not an up-and-down counting number; 13557321 is not an up-and-down counting number.) How many different 4-digit up-and-down numbers are there in which the maximum digit is 6 and at least one of the digits is a 3?

Multiples of 33

What's the total number of different ways that the blanks of "__ 3 __ 9" can be filled in so that the resulting four-digit number is a multiple of 33?

Pick a Number

An integer is chosen at random from the set {21, 22, 23, ..., 81}. What's the probability that the chosen number is one more than a multiple or 4 or one more than a multiple of 5?

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A similar problem was given in Assignment 2 of Charlotte Math Meetup:

An integer is chosen at random from the set {41, 42, 43, ..., 67}. What's the probability that the chosen number is one more than a multiple or 4 or one more than a multiple of 5? (Please express the answer as a decimal number rounded to hundredth.)