## Posts

Showing posts from April 19, 2020

### 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?

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.)