A poll asked a number of people if they liked solving mathematics problems. Exactly 74% answered "yes." What is the fewest possible number of people who could have been asked the question?
The notation n! (read "n factorial") is defined as the product of the first n positive integers. (For example, 3!=1⋅2⋅3=6.) Define the superfactorial of a positive number, denoted by n!, to be the product of the first n factorials. (For example, 3!=1!⋅2!⋅3!=12.) How many factors of 7 appear in the prime factorization of 51!, the superfactorial of 51?
记号 n!（读作"n 阶乘"）定义为前 n 个正整数的乘积（例如 3!=1⋅2⋅3=6）。定义超阶乘 n! 为前 n 个阶乘的乘积（例如 3!=1!⋅2!⋅3!=12）。在 51! 的质因数分解中，因子 7 出现了多少次？
Sekou writes the numbers 15, 16, 17, 18, 19. After he erases one of his numbers, the sum of the remaining four numbers is a multiple of 4. Which number did he erase?
A classroom has a row of 35 coat hooks. Paulina likes coats to be equally spaced, so that there is the same number of empty hooks before the first coat, after the last coat, and between every coat and the next one. Suppose there is at least 1 coat and at least 1 empty hook. How many different numbers of coats can satisfy Paulina's pattern?
When Yunji added all the integers from 1 through 9, she mistakenly left out a number. Her incorrect sum turned out to be a square number. Which number did Yunji leave out?
The numbers from 1 to 49 are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number 7. How many of these four numbers are prime?
Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas would like to cover the package with a large number of stamps. Suppose he has a collection of 5-cent, 10-cent, and 25-cent stamps, with exactly 20 of each type. What is the greatest number of stamps Nicolas can use to make exactly $7.10 in postage?
(Note: The amount $7.10 corresponds to 7 dollars and 10 cents. One dollar is worth 100 cents.)
Greta Grasshopper sits on a long line of lily pads in a pond. From any lily pad, Greta can jump 5 pads to the right or 3 pads to the left. What is the fewest number of jumps Greta must make to reach the lily pad located 2023 pads to the right of her starting position?
In a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth term is 4000. What is the first term?
When three positive integers a, b, and c are multiplied together, their product is $100$. Suppose a < b < c. In how many ways can the numbers be chosen?
How many positive integers can fill the blank in the sentence below? “One positive integer is _____ more than twice another, and the sum of the two numbers is 28 .”
If n is an even positive integer, the double factorial notation n!! represents the product of all the even integers from 2 to n . For example, 8!!=2⋅4⋅6⋅8 . What is the units digit of the following sum? 2!!+4!!+6!!+⋯+2018!!+2020!!+2022!!
如果 n 是一个正整数，那么双阶乘记号 n！！代表从 2 到 n 的所有偶整数的乘积。例如， 8!!=2·4·6·8。问下面和式的个位数字是几？
2！！+4！！+6！！+ … +2018！！+2020！！+2022！！
For a positive integer n , the factorial notation n! represents the product of the integers from n to 1 . What value of N satisfies the following equation?
5!⋅9!=12⋅N!
对一个正整数 n，符号 n！表示从 n 到 1 的所有整数的乘积。例如， 6！=6·5·4·3·2·3.2·1 则 N 为何值时，满足下面的方程？
A number is called flippy if its digits alternate between two distinct digits. For example, 2020 and 37373 are flippy, but 3883 and 123123 are not. How many five-digit flippy numbers are divisible by $15?$
A scientist walking through a forest recorded as integers the heights of 5 trees standing in a row. She observed that each tree was either 2 times as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?
When a positive integer N is fed into a machine, the output is a number calculated according to the rule shown below. For example, starting with an input of N = 7, the machine will output 3⋅7+1=22. Then if the output is repeatedly inserted into the machine five more times, the final output is 26: 7→22→11→34→17→52→26. When the same 6-step process is applied to a different starting value of N, the final output is $1$. What is the sum of all such integers N?
A palindrome is a number that has the same value when read from left to right or from right to left. (For example, 12321 is a palindrome.) Let N be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of N ?
回环数是指从左向右读和从右向左读，读数是一样的数（例如，12321 是个回环数）。N 是满足以下条件的最小三位整数：它不是个回环数，并且它是 3 个不同的两位回环数的和。问 N 的各个位上的数字之和是多少？
Isabella has $6$ coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every 10 days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the 6 dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?
Isabella 有 6 张优惠券，可用于在 Pete 甜品店里免费兑换冰激凌。为了最后使用优惠券，她决定每 10 天兑换一次直到所有的优惠券都使用完毕。已知 Pete 甜品店周日是不营业的，当她在日历上圈出兑换冰激凌的这 6 天时，发现没有哪天是周日。问 Isabella 兑换她的第一个优惠券是在周几？
After Euclid High School's last basketball game, it was determined that 41 of the team's points were scored by Alexa and 72 were scored by Brittany. Chelsea scored 15 points. None of the other 7 team members scored more than 2 points. What was the total number of points scored by the other 7 team members?
How many positive three-digit integers have a remainder of 2 when divided by 6, a remainder of 5 when divided by 9, and a remainder of 7 when divided by 11?
Let Z be a 6-digit positive integer, such as 247247, whose first three digits are the same as its last three digits taken in the same order. Which of the following numbers must also be a factor of Z ?
Z 表示一个 6 位的正整数，例如 247247，从左往右前三位数字和后三位的数字一样。下面哪个数一定也是 Z 的一个因子？
Malcolm wants to visit Isabella after school today and knows the street where she lives but doesn't know her house number. She tells him, "My house number has two digits, and exactly three of the following four statements about it are true." (1) It is prime. (2) It is even. (3) It is divisible by 7. (4) One of its digits is 9. This information allows Malcolm to determine Isabella's house number. What is its units digit?
The smallest positive integer greater than 1 that leaves a remainder of 1 when divided by 4, 5, and 6 lies between which of the following pairs of numbers?
For any positive integer M , the notation M! denotes the product of the integers 1 through M . What is the largest integer n for which 5n is a factor of the sum 98!+99!+100! ?
对于任何正整数 M 来说，符号 M！表示从 1 到 M 的所有整数的乘积。使得 ＋100！的一个因子的最大整数 n 是多少？
Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by 5 minutes over the preceding day. Each of the four days, her distance traveled was also an integer number of miles. What was the total number of miles for the four trips?
The number N is a two-digit number. • When N is divided by 9, the remainder is 1. • When N is divided by 10, the remainder is 3. What is the remainder when N is divided by 11?
N 是个两位数。
·当 N 除以 9，余数为 1． ·当 N 除以 10，余数为 3． 当 N 除以 11 时，余数是多少？
The least common multiple of a and b is 12 , and the least common multiple of b and c is 15 . What is the least possible value of the least common multiple of a and c ?
a 和 b 的最小公倍数是 12，b 和 c 的最小公倍数是 15，问 a 和 c 的最小公倍数的最小可能值是多少？
The digits 1 , 2 , 3 , 4 , and 5 are each used once to write a five-digit number PQRST . The three-digit number PQR is divisible by 4 , the three-digit number QRS is divisible by 5 , and the three-digit number RST is divisible by 3 . What is P ?
Ralph went to the store and bought 12 pairs of socks for a total of $24 . Some of the socks he bought cost $1 a pair, some of the socks he bought cost $3 a pair, and some of the socks he bought cost $4 a pair. If he bought at least one pair of each type, how many pairs of $1 socks did Ralph buy?
On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6 students in each row. This process continues through June 12 with a different number of students per row each day. However, on June 13, they cannot find a new way of organizing the students. What is the smallest possible number of students in the group?
A baseball league consists of two four-team divisions. Each team plays every other team in its division N games. Each team plays every team in the other division M games with N>2M and M>4 . Each team plays a 76 game schedule. How many games does a team play within its own division?
一个棒球联盟有两组组成，每组 4 支队伍。每组里面的每支队伍和自己组内的其他每支队伍都打 N 场比赛。每支队伍和另一组的每支队伍都打 M 场比赛，且 N>2M，M>4，每支队伍共需要打 76 场比赛，那么每支队伍和自己组内的队伍共需要打多少场比赛？
Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid their guest speaker $1A2 . What is the missing digit A of this 3 -digit number?
中学数学俱乐部的 11 个成员为了邀请一个客座演讲人在他们的数学俱乐部会议上讲授解题技巧，每个人都支付了同样的美元金额数且这个数是个整数。他们总共支付给客座演讲人，这个三位数中的 A 代表什么数字？
Three members of the Euclid Middle School girls' softball team had the following conversation. Ashley: I just realized that our uniform numbers are all 2 -digit primes. Bethany : And the sum of your two uniform numbers is the date of my birthday earlier this month. Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month. Ashley: And the sum of your two uniform numbers is today's date. What number does Caitlin wear?
Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the smallest number of additional cars she must buy in order to be able to arrange all her cars this way?
When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one?
当 Clara 把她的所有的考试分数相加时，她不小心把某个分数的十位数字和个位数字调换了。那么她不正确的总分和正确总分可能相差多少？
Jamar bought some pencils costing more than a penny each at the school bookstore and paid $1.43 . Sharona bought some of the same pencils and paid $1.87 . How many more pencils did Sharona buy than Jamar?
Students guess that Norb's age is 24, 28, 30, 32, 36, 38, 41, 44, 47 , and 49 . Norb says, "At least half of you guessed too low, two of you are off by one, and my age is a prime number." How old is Norb?
The hundreds digit of a three-digit number is 2 more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result?
The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $1.43. Some of the 30 sixth graders each bought a pencil, and they paid a total of $1.95. How many more sixth graders than seventh graders bought a pencil?
The positive integers x and y are the two smallest positive integers for which the product of 360 and x is a square and the product of 360 and y is a cube. What is the sum of x and y ?
x 和 y 是满足 360 和 x 之积为平方数，360 和 y 之积为立方数的最小正整数。那么 x 和 y 的和是多少？
On the last day of school, Mrs. Wonderful gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought 400 jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class?
In Theresa's first 8 basketball games, she scored 7, 4, 3, 6, 8, 3, 1 and 5 points. In her ninth game, she scored fewer than 10 points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than 10 points and her points-per-game average for the 10 games was also an integer. What is the product of the number of points she scored in the ninth and tenth games?
The product of the two 99 -digit numbers 303,030,303,...,030,303 and 505,050,505,...,050,505 has thousands digit A and units digit B . What is the sum of A and B ?
两个 99 位数字 303,030,303,…,030,303 和 505,050,505,…,050,505 的乘积的千位数字为 A，个位数字是 B。那么 A 与 B 的和是多少？
Pick two consecutive positive integers whose sum is less than 100 . Square both of those integers and then find the difference of the squares. Which of the following could be the difference?
A box contains gold coins. If the coins are equally divided among six people, four coins are left over. If the coins are equally divided among five people, three coins are left over. If the box holds the smallest number of coins that meets these two conditions, how many coins are left when equally divided among seven people?
Barry wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table, as shown. The sums of the two numbers on each of the three cards are equal. The three numbers on the hidden sides are prime numbers. What is the average of the hidden prime numbers?
Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise. The game ends when they stop on the same point. How many turns will this take?
Alice 和 Bob 玩了一个游戏，游戏中有一个圆的周长被 12 个等距点分成 12 等份。这些点按顺时针方向从 1 到 12 编号。两个人都从第 12 个点开始。Alice 顺时针移动，而 Bob 逆时针移动。
在游戏的每个回合，Alice 顺时针移动 5 个点，Bob 逆时针移动 9 个点。当他们停在同一点上时，比赛结束。则到游戏结束一共需要经过多少个回合？
A whole number larger than 2 leaves a remainder of 2 when divided by each of the numbers 3, 4, 5, and 6 . The smallest such number lies between which two numbers?
In the pattern below, the cat moves clockwise through the four squares, and the mouse moves counterclockwise through the eight exterior segments of the four squares.
If the pattern is continued, where would the cat and mouse be after the 247th move?
The year 2002 is a palindrome (a number that reads the same from left to right as it does from right to left). What is the product of the digits of the next year after 2002 that is a palindrome?
On a twenty-question test, each correct answer is worth 5 points, each unanswered question is worth 1 point and each incorrect answer is worth 0 points. Which of the following scores is NOT possible?
There are 24 four-digit whole numbers that use each of the four digits 2, 4, 5 and 7 exactly once. Only one of these four-digit numbers is a multiple of another one. Which of the following is it?
You have nine coins: a collection of pennies, nickels, dimes, and quarters having a total value of $1.02 , with at least one coin of each type. How many dimes must you have?
Each of the letters W , X , Y , and Z represents a different integer in the set {1,2,3,4} , but not necessarily in that order. If XW−ZY=1 , then the sum of W and Y is
字母 W、X、Y 和 Z 各代表集合 {1,2,3,4} 中不同的整数，但不一定按此顺序。若 XW−ZY=1，则 W 与 Y 之和为
Terri produces a sequence of positive integers by following three rules. She starts with a positive integer, then applies the appropriate rule to the result, and continues in this fashion. Rule 1: If the integer is less than 10, multiply it by 9. Rule 2: If the integer is even and greater than 9, divide it by 2. Rule 3: If the integer is odd and greater than 9, subtract 5 from it. A sample sequence: 23,18,9,81,76,…. Find the 98th term of the sequence that begins 98,49,….
A rectangular board of 8 columns has squares numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 8, row two is 9 through 16, and so on. A student shades square 1, then skips one square and shades square 3, skips two squares and shades square 6, skips 3 squares and shades square 10, and continues in this way until there is at least one shaded square in each column. What is the number of the shaded square that first achieves this result?
There are positive integers that have these properties: The sum of the squares of their digits is equal to 50 Each digit is larger than the one on it's left The product of the digits of the largest integer with both properties is
All of the even numbers from 2 to 98 inclusive, excluding those ending in 0, are multiplied together. What is the rightmost digit (the units digit) of the product?
are placed in the squares in the figure shown so that the sum of the entries in the vertical column is 23 and the sum of the entries in the horizontal row is 12. The sum of the six digits used is
A lucky year is one in which at least one date, when written in the form month/day/year, has the following property: The product of the month times the day equals the last two digits of the year. For example, 1956 is a lucky year because it has the date and 7×8=56 . Which of the following is NOT a lucky year?
The numbers on the faces of this cube are consecutive whole numbers. The sum of the two numbers on each of the three pairs of opposite faces are equal. The sum of the six numbers on this cube is
Several students are seated at a large circular table. They pass around a bag containing 100 pieces of candy. Each person receives the bag, takes one piece of candy and then passes the bag to the next person. If Chris takes the first and last piece of candy, then the number of students at the table could be
几名学生围坐在一张大圆桌旁。他们传一袋装有 100 颗糖果的袋子。每人接过袋子，取一颗糖果，然后将袋子传给下一个人。如果 Chris 拿走第一颗也是最后一颗糖果，则桌旁的学生人数可能是
A multiple choice examination consists of 20 questions. The scoring is +5 for each correct answer, -2 for each incorrect answer, and 0 for each unanswered question. John's score on the examination is 48 . What is the maximum number of questions he could have answered correctly?
Nine copies of a certain pamphlet cost less than $10.00 while ten copies of the same pamphlet (at the same price) cost more than $11.00. How much does one copy of this pamphlet cost?