## Posts

Showing posts from 2020

### Triangle inside a Triangle

In triangle ABC, D, E, and F are on AB, AC, and BC respectively. AD:BD = 2:1; AE:EC = 1:1; BF:FC = 3:1.  AF and BE intersects at G; BE and CD intersects at H; AF and CD intersects at I. If the area of triangle ABC is 504, what is the area of triangle GHI?

### Sharing Mooncakes

It’s mid-autumn day, a festival in China when family members get together and eat mooncakes. Dr. Hong takes out 3 mooncakes of the same flavor and cuts each mooncake into 4 equal pieces. He then asks Leo to allocate them to the Hong family: Dr. Hong, Dr. Wang, Leo, Angela, and Elon.

1. How many different ways can Leo allocate the mooncake pieces?

2. If Leo has to give at least one piece to each family member, how many different ways can Leo allocate the mooncake pieces?

3. If Leo has to give at least one piece to each family member, what’s the likelihood that Leo  gets more pieces than the number of pieces any other family member gets?

4. If Leo has to give at least one piece to each family member, and more pieces to each parent (Dr. Hong and Dr. Wang) than any kid (Leo, Angela, and Elon), how many different ways can Leo allocate the mooncake pieces?

5. (Optional) If Dr. Hong takes out 4 mooncakes instead of 3, under the same settings, what are the answers to the previous 4 questions?

### Mind and Matter

Leo is reading Mind and Matter, a book written by a former NFL player now studying math at MIT. The book has 238 pages. If the pages are numbered consecutively starting from 1, what’s the sum of the digits of all page numbers?

### 1+2+4=7

In the expression I + II + IV = VII, I and V represent different single digit numbers. What’s I?

### Hiding Red Paint

Leo puts 27 small 1x1x1 cubes together to build a 3x3x3 cube. He then paints all six faces of the big cube with red paint. After that, he rearranges the small cubes to a new 3x3x3 cube by minimizing the area of red paint on the surface. What is the red surface area of the rearranged 3x3x3 cube?

### Pulling Weeds

Dr. Hong is taking Leo, Angela, and Elon to pull the weeds out of their front yard. If Dr. Hong does all the work on his own, it will take him 10 hours. If Leo and Angela work together, it will take them 15 hours. If Leo and Elon work together, it will take them 20 hours. If Angela and Elon work together, it will take them 30 hours. How many hours does it take if all four of them work together? Please keep two digits after the decimal point for the final answer. (We can assume that the weeds don’t grow when they work on the lawn.)

### 777 in Base k

The three digit number 777 in base k is equivalent to 4145 in base 8. What is k?

### Playing Dice

The Hong family are playing dice tonight. Dr. Hong serves as the judge. Leo, Angela, Elon, and Dr. Wang are the contestants. Each of them comes to Dr. Hong separately, rolls the dice, memorizes the number on its upper surface, and keeps the number secret to the other three. (Dr. Hong knows everyone’s number.) After all four of them roll the dice, they are trying to guess who gets the smallest or largest number based on the information during the conversation:

• Leo says: I don’t know if my number is the smallest or largest.
• Angela says: I don’t know if my number is the smallest or largest.
• Elon says: I don’t know if my number is the smallest or largest.
• Dr. Wang says: I don’t know if my number is the smallest or largest.
• Then Angela says: I have the largest number!

What’s Angela’s number?

(A dice is a cube with each of its six faces marked with a different number of dots (pips) from one to six.)

### Angela’s Favorite Book

Angela’s favorite book is “The Secret of Cacklefur Castle”. The book has 109 pages. If the pages are numbered consecutively starting from 1. How many times the digit 0 shows up in the page numbers?

### Starting with a Random Number

Pick any number between 1 and 10. Subtract 2. Add 3. Subtract 5. Add 7. Subtract the original number. What’s the largest possible value of the resulting number?

Compute: 2 - 4 + 6 - 8 + 10 - 12 + 14 - 16 + 18.

### Five Squares in a Circle

Five identical squares are placed inside a circle of radius 10. If these squares do not overlap each other, what’s the largest side length for each square?

### Three Roots

a, b, and c are the three roots of 2x3 + 5x2- 19x - 42 = 0. What is (a + b + c)2?

### Folding a Rectangle

ABCD is a rectangle, where AB = 4, and AD = 3. E and F are on AB and CD respectively. If ABCD is folded along EF, B and C meet. How long is the line segment EF?

### n^n

n is a nonzero integer. If nn = n, what’s the sum of all possible values of n?

### Soccer Ball

The surface of a soccer ball is made of 32 pieces of polygons, which are regular pentagons and regular hexagons. Each pentagon is surrounded by 5 hexagons. How many pentagons are on a soccer ball?

### Trail Blazers vs. Lakers

Portland Trail Blazers is playing against Los Angeles Lakers in NBA playoff. The first team winning 4 games wins the series. If Trail Blazers has 40% chance to win Lakers in each game, what’s the probability that Trail Blazers wins the series?

Leo, Angela and Elon are playing pokemon cards. Each of them starts with a few cards in hand. After Leo gives Angela 6 cards, Leo and Angela have the same number of cards. Then Angela gives 6 cards to Elon, so that Angela and Elon have the same number of cards. Finally, Elon gives 3 cards to Leo. Now Leo has twice as many cards as Elon. How many cards does Leo have at the beginning?

### KD

Kevin Durant won the 2013–14 NBA Most Valuable Player (MVP) award. In the expression KD + 35 = MVP, each letter represents a different single digit number. K and M are not zero. If P = 3, what’s the two-digit number KD?

### Counting Triangles

How many triangles are in the figure below?

### Jump Rope Training

Leo is preparing for a jump rope competition to be held within a year. Leo’s personal record is currently less than the national record 80. He plans to improve his personal record for 30 seconds double under by 1 every month. By doing so, the sum of his monthly records (from the current month to the competition month) would be 465. What is the highest possible personal record does Leo have now?

### Missing Operators

Insert an arithmetic operation (addition, subtraction, multiplication, or division) to each slot between two adjacent numbers of the expression N = 1 __  3 __  5 __  6 __  4 __  2. If each of the four operations has to be used at least once, what’s the smallest possible value of N?

### Angela's Coins

Angela has 4 coins: a penny, a nickel, a dime, and a quarter. How many different face values can she make using one or more of these coins?

Dr. Hong bought five books for Leo and Angela. During the first week, Leo finished reading four of them, while Angela finished three of them. At least how many books were read by both Leo and Angela?

### Dinner Time

It’s dinner time for the Hong family. It takes 30 minutes for Dr. Hong to finish the meal, 30 minutes for Dr. Wang, 80 minutes for Leo, and 60 minutes for Angela. If they start the meal together at 6pm, when will the family finish their meals?

### Counting Rectangles

How many rectangles are in the figure below?

Without using a calculator, calculate: 1 + 2 + 8 + 9 + 11 + 12 + 18 + 19 + 21 + 22 + 28 + 29 + 31 + 32 + 38 + 39 = ?

### Math Kangaroo

Math Kangaroo is a popular math contest. There are 24 questions in the test. Each question has point value. 1/3 of questions are 3 points each; 1/3 are 4 points each, 1/3 are 5 points each. The sum of earned points is the student's final score. Leo and five other kids join the contest. The median and mean of their scores are both 90. If they all have different scores, what's the lowest possible score among them?

After Dr. Hong, Leo and William complete a basketball workout, they open a pack of Oreo cookies to share among themselves. If there are 12 cookies in the pack, and if each person gets at least one cookie, how many different ways are there to distribute the cookies? (They do not split a cookie.)

### Tangent Circles

Points A, B and C are the pairwise points of tangency of three externally tangent circles of radius 1. What's the area surrounded by the arcs AB, BC and CA?

### House Cleaning

Leo's mom asks Dr. Hong, Leo and Angela to clean up the house. If Dr. Hong and Leo work together, they can complete the task in 1.5 hours. If Dr. Hong and Angela work together, it would take them 1.75 hours. If Leo and Angela work together, it would take them 4.2 hours. If all three of them work together, how long does it take to clean up the house?

### Favorite Math Subjects

30 CMM students get together and discuss their favorite math subjects. 23 of them like geometry; 29 of them like algebra; 19 of them like probability. At most how many students like all three subjects?

### Logo Shot

In the expression LOGO + SHOT = HARD, each letter represents a single digit number from 0 to 9. Different letters represent different numbers. L, S and H cannot be zero. What's the difference between the largest and smallest values of HARD?

### Circle in the Corner

A regular hexagon ABCDEF has the side length of 6. Circle O is inscribed in the hexagon. The tangent points on AB and AF are G and H respectively. Circle P is inscribed in the corner area sounded by HA, AG, and arc GH. What's the radius of circle P? (Do not use calculator. Let the square root of 3 be 1.73. When submitting the result, keep two digits after decimal point.)

### Triangles in a Grid

Nine dots form a 3 by 3 grid. The distance between any two adjacent dots in the left-right or up-down direction is the same. How many triangles can be formed from this grid by choosing three dots as the vertices?

### Meetup in Classroom

Dr. Hong decides to gather the CMM students for a face-to-face meetup in a classroom, where 90 seats are arranged in a 9 by 10 grid. The distance between two adjacent seats in any row or column is 5 feet. Due to the concern of COVID-19, Dr. Hong wants to keep at least 6 feet distance between any two students. What's the maximum number of students Dr. Hong could bring to the classroom?

### Splitting a Pizza

Leo's mom has made a big (round) pizza for a CMM party. It's Dr. Hong's job to split it. If Dr. Hong cuts one line through the circle, he can split the pizza into two pieces. If Dr. Hong cuts two lines, he can split the pizza into four pieces at most. If Dr. Hong cuts 6 lines, what's the greatest number of pieces can he get? (Dr. Hong cannot stack the pieces before making a cut.)

### GRIT

In the expression STARTUP + GRIT = SUCCESS, each letter represents a single digit number from 0 to 9. Different letters represent different numbers. S and G cannot be zero. What's GRIT?

### Three Composite Numbers

The greatest common factor of three different composite numbers is 21. The product of them is 1,018,710. What's the smallest possible sum of these three numbers?

Warm up: how many possible values for each two-digit number in bold?

JH + 13 = MVP
KD + 35 = MVP

Find the unique solution

DREAM + TEAM = WINNER
BLACK + MAMBA = KOBE24
BLACK + MAMBA = CLUTCH
NY × 17 = LIN
KOBE × 24 = MAMBA
CHICAGO + BULLS = CLASSIC (This one is really hard.)

### Alphametic Warmup

Find the range of each letter in bold:

A + B = C
AB = C
A + B = CD
AA + B = CD
AA + B = CDE
A × B = C
A × B = CD
AA × BB = CDE
AA × BB = CDEF

Find the unique solution

SEND + MORE = MONEY
SMART + MOVE = DOCTOR
TRIAL + ERROR = ASSURE
SMART + MIRACLE = SUCCESS

### Three Circles

Three circles of equal radius pairwise meet at tangent points A, B, and C. What's the ratio of the area of triangle ABC to the area of one of the circles?

### Triangles in a Pentagon

Leo draws a convex pentagon, and then connects all its diagonal lines. How many triangles of all sizes are shown in Leo's picture?

### Third Term

In a sequence such that the next term in the sequence is the sum of the previous two terms.The sum of the first ten terms in the sequence is 231. The fifth term in the sequence is 13. What is the third term?

### Two Families

When Dr. White and Dr. Green met each other 5 years ago, each family had only one child. Now they each have two children under 10 years old. These are the facts about their children:
1) James is the oldest among the four;
2) Russell is 4 years older than his brother;
3) Stephen's age is half of one of Dr. White's children's age;
4) Kevin is 2 years older than Dr. Green's younger child.
If each child is a different age, how old is Stephen?

### Timed Exam

Leo plans to take a timed exam. He fills the corresponding bubble in the answer sheet right after completing each problem. If he spends 2 minutes to complete each problem, he would have 6 problems on the answer sheet left empty. If he spends 1 minute on each problem, he would have 4 minutes left before the end of the exam. Let N be the number of problems in the exam. What's the least value for N?

### Memorial Day

Memorial Day, observed on the last Monday of May, is a U.S. federal holiday for honoring and mourning the military personnel who had died while serving in the United States Armed Forces. One year, there were four Saturdays and four Wednesdays in the month of May. On which day of May was Memorial Day?

### Greatest Possible GCF

Sum of five different positive integers is 2020. What is the greatest possible greatest common factor of the five?

### Arithmetic Sequence

In the year-end ceremony, 12 top-performing CMM students form a line. Each of them holds a different number from 1 to 12. Dr. Hong calls their numbers one at a time to give them awards. At first, Dr. Hong tries to call the numbers so that no three of them form an arithmetic sequence. What's the maximum number of students Dr. Hong calls before he sees an arithmetic sequence of at least three numbers.

### Circle in a Right Triangle

A circle is inscribed in a right triangle of leg lengths 8 and 15. What's the area of this circle? (Let Pi = 3.14.)

### Squares and Median

The sum of 3 different positive integers is 55. The sum of any two of them is a perfect square. What is the median of these three positive integers?

### Party at Dr. Hong's House

During the COVID-19 lock down, CMM was conducted online. Many students don't know each other before. After the lock down, Dr. Hong invites a few CMM students to his house for a party. To make sure that the students can have fun with each other, Dr. Hong invites as many students as he can under the condition that among each three students, there are two who know each other and two who don't know each other. How many students were invited by Dr. Hong?

### Biggest V

In the following expression, ONE + TWO + FOUR = SEVEN, different letters represent different nonnegative integers. What's the largest possible value for V?

### Greatest Common Factor

What's the greatest common factor of 25200, 14850 and 8190?

### Sum of Digits

What's the sum of digits of 999,999,999 × 666,666?

### Product of Two Numbers

There are eight balls in a bag, which are labeled as -7, -5, -3, -1, 1, 3, 5, 7 respectively. Leo randomly picks two balls. What's the probability that the product of the two is 35?

### Dissecting a Triangle

ABC is a triangle. D and E are on AB and AC respectively. BE and CD intersects at F. The areas of triangles CEF, BDF, and BCF are 10, 15, and 20 respectively.  What is the area of quadrilateral AEFD?

### Angle LEO

A regular 26-sided polygon has its 26 vertices marked as A, B, C, .... , Z clockwise. What's the angle LEO? (Please keep one digit after decimal point.)

### Sum of 2020 Numbers

N is the sum of 2020 numbers: N = 20 + 2020 + 202020 + 20202020 + ... + 2020...2020. What's the last four digits of N?

### Crossing Bridge

Four people want to cross a dangerous bridge in dark. They have only one flash light. The bridge can hold up to two persons at a time. People walking on the bridge must use the flash light to avoid drop off the bridge. The highest walking speed for each person is different. It takes at least 2, 3, 7, 10 minutes respectively for each person to cross the bridge. What is the minimum time (in minutes) for the group of four to cross the bridge safely?

### After Leo Gets a PhD

In the expression LEO + DOCTOR = DRHONG, different letters represent different non-negative integers. What's the largest value for LEO?

### Most Number of Factors

Several numbers between 1 and 100 have the most number of distinct factors. What's the sum of these numbers?

### Sum and Difference of Prime Numbers

A prime number P is a sum of two prime numbers. Meanwhile, P is also a difference of two prime numbers. What's P?

### 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?

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?

### 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?

### 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?

----------------------

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?

### 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.)

### Leo's Magic Number

Leo has a magic number ABCDE that satisfies the following conditions:
1. ABCDE = FD × HI × GB;
2. HI × GB = DGBE;
3. FD, HI and GB are prime numbers;
4. A, B, C, D, E, F, G, H, and I are nine different single-digit non-negative integers.
What is Leo's magic number ABCDE?

### Counting Trails

48 Charlotte Math Meetup students have a meetup together. Each holds a square tile of 1 m2. They put the tiles together to form a big rectangle of 48 m2. Leo tries to complete a trail from the bottom left tile to the top right tile. (The trails are not along the edges of the tiles, but from one tile to another tile.) If every time he can only go up or right, and if the students try to make the rectangle such that there are as many trails as possible. what’s the most number of trails?