## Posts

Showing posts from April 26, 2020

### Regular Hexagon

ABCDEF is a regular hexagon. G, H, I, J, K, and L are midpoints on AB, BC, CD, DE, EF, and FA respectively. M is a point inside ABCDEF. Connecting M to the 6 midpoints dissects the hexagon into 6 small quadrilaterals. The vertices of each small quadrilateral includes a shared vertex with ABCDEF, two midpoints adjacent to the shared vertex, and the interior point M. The areas of the AGML, DJMI, and EKMJ are 41, 53, and 67, respectively. What’s the area of BHMG?

### Unit Fractions

1/A+1/B=1/18, where A and B are whole numbers, and A ≥ B. How many distinct solutions of (A, B) are there?

### Consecutive Primes

A, B, C, D, E, F, and G are consecutive prime numbers arranged in the increasing order. If the sum of them has a prime factor 2, what is the product of them?

### Fraction to Decimal

Find the 2020th digit after decimal point of 2020/7.

### Counting Perfect Squares

The perfect squares are the squares of the integers. How many perfect squares are there between 20 and 2020?

### Define #

If we define A#B = AB+A+B, what is 1#9#9#9#9#1?

### Handshakes

When two people shake hands with each another, that counts as one handshake. After the COVID-19 quarantine, all students from Dr. Hong's Charlotte Math Meetup have a big party in person. Everyone shakes hands with each other exactly once. There are 1225 handshakes! How many students are in the party?

### Terminal Zeros

Terminal zeros of a number are the zeros to the right of the last nonzero digit. For instance, 506,400,000 has 5 terminal zeros. During a math meetup, students play a game. They form a queue. The first student yells 1, the second student yells 2, the third student yells 6, the fourth student yells 24, the nth student yells the product of 1, 2, 3, ..., n. Leo is the last student to yell his number. He forgets to bring the calculator, but he says there are 7 terminal zeros in his number. If Leo's statement is true, what's the largest possible number of students in the queue?

### Counting Game

Leo, Kyle, Issaic, and several other Charlotte Math Meetup students form a circle to play a game of counting numbers clockwise. Each round starts from Leo, with each student on the circle adds the same prime number continuously. In Round 1, Leo counts 2. When it's Leo's turn to begin Round 6, he counts 107. Then three more students join the circle to continue the same game. What is the number Leo counts in Round 10?

### DDD

A, B, C, D represent four different non-zero digits. AB × CD = DDD. What's the largest possible value of D?

### Hexagon

ABCDEF is a hexagon. G, H, I, J, K, and L are midpoints on AB, BC, CD, DE, EF, and FA respectively. M is a point inside ABCDEF. Connecting M to the 6 midpoints dissects the hexagon into 6 small quadrilaterals. The vertices of each small quadrilateral includes a shared vertex with ABCDEF, two midpoints adjacent to the shared vertex, and the interior point M. Starting from the quadrilateral with vertex A, going clockwise, the areas of the first five quadrilateral are 81, 64, 49, 36, and 25, respectively. What’s the area of the remaining quadrilateral (the one with vertex F)?

### Pokémon cards

Leo collects Pokémon cards. Kyle, Issaic, and Dr. Hong look at his card box and then have the following conversation with Leo --

• Kyle: Leo has at most 65 cards;
• Issaic: Leo has at least 70 cards;
• Dr. Hong: Leo has at least 58 cards.

If exactly one of them is false, what’s the smallest number of cards Leo has?

### Leo's Jump Rope

Leo’s jump rope is 8ft long. He uses his rope to wrap a rectangle ABCD with no remainder of the rope outside ABCD. Kyle draws a circle with center A and radius AC. Kyle’s circle and the extension of AB intersect at E; Kyle’s circle and the extension of AD intersect at F. If AE = 3. What is the area of  BDFE? (FE is an arc, not a chord.)

### Making a Perfect Square

The sum n2 + 2000n is a perfect square. Find the largest n.

### Forming Teams

After the first year of meetups, Dr. Hong identifies 10 students who have the best record on their assignments. He decides to run a competition. He randomly divides the students into 5 teams, with each team having at least 1 student. Student(s) of the winning team will be recognized as the spotlight student(s) of the year. How many different ways can he form the 5 teams?

(This problem is beyond Level E, so it is not counted against spotlight student eligibility.)

### Counting Squares

48 CMM students have a meetup together. Each holds a square tile. They lay the tiles down to form a big 7 by 7 grid, with the center position empty. How many squares are there that are fully covered by tiles.

### Triangle in Parallelogram

ABCD is a parallelogram. E, F, and G are on AB, BC, and CD respectively. AF and DE intersect at H; AF and EG intersect at I; AF and BG intersect at J; DF and BG intersect at K; DF and EG intersect at L. The area of quadrilateral BEIJ is 36; the area of quadrilateral DHIL is 40; the area of triangle GKL is 4; the area of triangle FJK is 5. What’s the area of triangle AEH?

### Rearranging Digits

If we rearrange all digits of a three digit number from largest to smallest, we get ABC. Moreover, the original three digit number equals ABC - CBA. What is the original three digit number?

### Grouping CMM Students

Dr. Hong is trying to group students in his meetup. If he forms groups of 2, one student is left out. If he forms groups of 3, one student is left out. If he forms groups of 4, one student is left out. If he forms groups of 5, one student is left out. If he forms groups of 6, how many students would be left out?

### Page Numbers in Leo's Diary

Leo plans to publish his diary. The pages of his diary book are numbered consecutively starting from 1. The digit 4 is printed 44 times on page numbers. What's the minimum number of pages his diary book can have?

### Denominator

The fraction 360/N in the simplest form is a whole number. N is a positive integer. What is the total number of different values that N can be?

Leo is one of the fastest jump ropers in his age group. Dr. Hong plans to take him and the family to the national jump rope competition in Orlando. They are going to visit a few Southeast cities between Charlotte (NC) and Orlando (FL), such as Atlanta (GA), Charleston (SC), Columbia (SC), Greenville (SC), Jacksonville (FL), Myrtle Beach (SC), Savannah (GA), Spartanburg (SC), and Tallahassee (FL). If Dr. Hong wants to start from Charlotte, visit each city only once, and go back to Charlotte, how many miles does he have to drive on the shortest path? (Please use Google Map to find out the shortest path between two cities.)

### Sum of Digits

A is a two-digit number. The sum of its digits is S. If S3 = A2, what is the sum of all possible values of A?

### Age Problem

One day, Dr. Hong tells Leo: "When I was at your age, you were 9 years old. When you reach my age, I will be 90 years old." When Leo was born, how old was Dr. Hong?

### Sidewalk Running

Leo wants to go to the library with his mom, but he has not complete the running drill of the day yet. Instead of running in the gym, Leo decides to run on the sidewalk. He runs at 4m/s towards the library the moment they leave home. Once he arrives at the library, he turns around and runs at the same speed towards his mom; once he meets his mom, he turns around and runs at the same speed towards the library again; until both Leo and his mom arrive at the library. If the distance between his home and library is 500m, and Leo’s mom is walking at 2m/s, how many meters does Leo run from the moment they leave home until they both arrive at the library?

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A similar problem was given in Assignment 7 of Charlotte Math Meetup.

Kyle is playing Pokémon Go while walking from home to library with his dad. To maximize the distance Pokémon Go records, Kyle decides to run at 4m/s towards the library the moment they leave home; once he arrives at the library, he turns around and runs at the same speed towards his dad; once he meets his dad, he turns around and runs at the same speed towards the library again; until both Kyle and his dad arrive at the library. If the distance between his home and library is 500m, and Kyle’s dad is walking at 2m/s, how many meters does Kyle run from the moment they leave home until they both arrive at the library?

### Double Dutch Rope

Leo has a double dutch rope that is 16 feet long. He is using it to wrap rectangles with each side as a whole number. Each time he may or may not use the entire rope. How many rectangles with different shapes can he create using this double dutch rope?

### Liquid Drops

Leo and his little sister Angela are working on a science project together. They use pipettes to drop liquid into a 0.7 liter container. It takes Angela 24 seconds to fill in a 10 cm3 tube. Leo can do this 3 times as fast as Angela. The how many minutes does it take them to fill in the container?

### Common Factors

N is a natural number. There are 12 common factors between 1,000,000,000,000 and N. What's the smallest possible value for N?