Lakshmi has 5 round coins of diameter 4 centimeters. She arranges the coins in 2 rows on a table top, as shown below, and wraps an elastic band tightly around them. In centimeters, what will be the length of the band?
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？
In an equiangular hexagon, all interior angles measure 120°. An example of such a hexagon with side lengths 2, 3, 1, 3, 2, and 2 is shown below, inscribed in equilateral triangle ABC. Consider all equiangular hexagons with positive integer side lengths that can be inscribed in △ABC, with all six vertices on the sides of the triangle. What is the total number of such hexagons? Hexagons that differ only by a rotation or a reflection are considered the same.
2025 AMC8 #21 · Counting, Probability & Statistics · ★★★★
The Konigsberg School has assigned grades 1 through 7 to pods A through G , one grade per pod. Some of the pods are connected by walkways, as shown in the figure below. The school noticed that each pair of connected pods has been assigned grades differing by 1 or more grade levels. (For example, grades 1 and 2 will not be in pods directly connected by a walkway.) What is the sum of the grade levels assigned to pods C, E , and F ?
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?
In trapezoid ABCD , angles B and C measure 60∘ and AB = DC . The side lengths are all positive integers, and the perimeter of ABCD is 30 units. How many non-congruent trapezoids satisfy all of these conditions?
A roll of tape is 4 inches in diameter and is wrapped around a ring that is 2 inches in diameter. A cross section of the tape is shown in the figure below. The tape is 0.015 inches thick. If the tape is completely unrolled, approximately how long would it be? Round your answer to the nearest 100 inches.
Jean made a piece of stained glass art in the shape of two mountains, as shown in the figure below.
One mountain peak is 8 feet high and the other peak is 12 feet high. Each peak forms a 90° angle, and the straight sides of the mountains form 45° angles with the ground.
The artwork has an area of 183 square feet. The sides of the mountains meet at an intersection point near the center of the artwork, h feet above the ground.
2024 AMC8 #25 · Counting, Probability & Statistics · ★★★★
A small airplane has 4 rows of seats with 3 seats in each row. Eight passengers have boarded the plane and are distributed randomly among the seats. A married couple is next to board. What is the probability there will be 2 adjacent seats in the same row for the couple?
2023 AMC8 #21 · Counting, Probability & Statistics · ★★★★
Alina writes the numbers 1,2,…,9 on separate cards, one number per card. She wishes to divide the cards into 3 groups of 3 cards so that the sum of the numbers in each group will be the same. In how many ways can this be done?
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?
2023 AMC8 #23 · Counting, Probability & Statistics · ★★★★
Each square in a 3×3 grid is randomly filled with one of the 4 gray and white tiles shown below on the right. What is the probability that the tiling will contain a large gray diamond in one of the smaller 2×2 grids? Below is an example of such a tiling.
Isosceles triangle ABC has equal side lengths AB and BC. In the figures below, segments are drawn parallel to AC so that the shaded portions of △ABC have the same area. The heights of the two unshaded portions are 11 and 5 units, respectively. What is the height h of △ABC?
Steph scored 15 baskets out of 20 attempts in the first half of a game, and 10 baskets out of 10 attempts in the second half. Candace took 12 attempts in the first half and 18 attempts in the second. In each half, Steph scored a higher percentage of baskets than Candace. Surprisingly they ended with the same overall percentage of baskets scored. How many more baskets did Candace score in the second half than in the first?
A bus takes 2 minutes to drive from one stop to the next, and waits 1 minute at each stop to let passengers board. Zia takes 5 minutes to walk from one bus stop to the next. As Zia reaches a bus stop, if the bus is at the previous stop or has already left the previous stop, then she will wait for the bus. Otherwise she will start walking toward the next stop. Suppose the bus and Zia start at the same time toward the library, with the bus 3 stops behind. After how many minutes will Zia board the bus?
2022 AMC8 #23 · Counting, Probability & Statistics · ★★★★
A△ or ◯ is placed in each of the nine squares in a 3 -by- 3 grid. Shown below is a sample configuration with three △s in a line. How many configurations will have three △s in a line and three ◯s in a line?
The figure below shows a polygon ABCDEFGH , consisting of rectangles and right triangles. When cut out and folded on the dotted lines, the polygon forms a triangular prism. Suppose that AH = EF = 8 and GH = 14 . What is the volume of the prism?
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?
2020 AMC8 #23 · Counting, Probability & Statistics · ★★★★
Five different awards are to be given to three students. Each student will receive at least one award. In how many different ways can the awards be distributed?
A large square region is paved with n^2 gray square tiles, each measuring s inches on a side. A border d inches wide surrounds each tile. The figure below shows the case for n=3 . When n=24 , the 576 gray tiles cover 64% of the area of the large square region. What is the ratio sd for this larger value of n?
一个大正方形区域用 n2 块灰色方形瓷砖铺成，每块边长为 s 英寸。每块瓷砖四周都有宽 d 英寸的边框。下图为 n=3 的情形。当 n=24 时，576 块灰色瓷砖覆盖了大正方形面积的 64％。那么对于这个较大的 n 值，sd 的比值是多少？
Rectangles R_1 and R_2, and squares S_1, S_2, and S_3, shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of S_2 in units?
A store increased the original price of a shirt by a certain percent and then lowered the new price by the same amount. Given that the resulting price was 84% of the original price, by what percent was the price increased and decreased?
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?
2018 AMC8 #23 · Counting, Probability & Statistics · ★★★★
From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?
In the cube ABCDEFGH with opposite vertices C and E, J and I are the midpoints of edges FB and HD, respectively. Let R be the ratio of the area of the cross-section EJCI to the area of one of the faces of the cube. What is R2?
在正方体 ABCDEFGH 中，顶点 C 和 E 相对，点 J 和点 I 分别是棱 FB 和 HD 的中点。R 表示横截面 EJCI 的面积和正方体的一个面的面积之比，求 R2。
In the right triangle ABC , AC=12 , BC=5 , and angle C is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?
在直角三角形 ABC 中，AC=12，BC=5，且角 C 是个直角。一个半圆如下图所示内切于三角形中，那么这个半圆的半径是多少？
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?
2017 AMC8 #24 · Counting, Probability & Statistics · ★★★★
Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from any of her grandchildren?
In the figure shown, US and UT are line segments each of length 2, and m∠TUS=60∘. Arcs TR⌢ and SR⌢ are each one-sixth of a circle with radius 2. What is the area of the region shown?
2016 AMC8 #21 · Counting, Probability & Statistics · ★★★★
A top hat contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn?
Two congruent circles centered at points A and B each pass through the other circle's center. The line containing both A and B is extended to intersect the circles at points C and D . The circles intersect at two points, one of which is E . What is the degree measure of ∠CED ?
两个圆心分别在点 A 和点 B 的全等圆各自通过对方的圆心。连接点 A 和点 B 的直线和这 2 个圆交于点 C 和点 D。这 2 个圆交于 2 个点，其中之一是点 E．那么的度数是多少？
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 ?
A semicircle is inscribed in an isosceles triangle with base 16 and height 15 so that the diameter of the semicircle is contained in the base of the triangle as shown. What is the radius of the semicircle?
In the given figure hexagon ABCDEF is equiangular, ABJI and FEHG are squares with areas 18 and 32 respectively, △JBK is equilateral and FE=BC . What is the area of △KBC?
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?
2015 AMC8 #23 · Counting, Probability & Statistics · ★★★★
Tom has twelve slips of paper which he wants to put into five cups labeled A , B , C , D , E . He wants the sum of the numbers on the slips in each cup to be an integer. Furthermore, he wants the five integers to be consecutive and increasing from A to E . The numbers on the papers are 2, 2, 2, 2.5, 2.5, 3, 3, 3, 3, 3.5, 4, and 4.5 . If a slip with 2 goes into cup E and a slip with 3 goes into cup B , then the slip with 3.5 must go into what cup?
Tom 想把 12 张纸条放进 5 个标有 A，B，C，D，E 的杯子中。他希望每个杯子中的纸条上的数字之和均为整数。并且，他希望这 5 个整数是连续的整数，从 A 到 E 递增。这 12 张纸条上的数字分别是 2，2，2，2.5，2.5，3，3，3，3，3.5，4，4.5．如果写有 2 的那张纸条放进了杯子 E，写有 3 的那张纸条放进了杯子 B，那么写有 3.5 的那张纸一定放进了哪个杯子？
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？
One-inch squares are cut from the corners of this 5 inch square. What is the area in square inches of the largest square that can fit into the remaining space?
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?
2014 AMC8 #24 · Counting, Probability & Statistics · ★★★★
One day the Beverage Barn sold 252 cans of soda to 100 customers, and every customer bought at least one can of soda. What is the maximum possible median number of cans of soda bought per customer on that day?
一天 Beverage Barn 向 100 个顾客卖出了 252 罐苏打水，且每个顾客买了至少一罐苏打水。
那么那天每个顾客所买苏打水的罐数的中位数最大可能是多少？
A straight one-mile stretch of highway, 40 feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at 5 miles per hour, how many hours will it take to cover the one-mile stretch? Note: 1 mile = 5280 feet
Angle ABC of △ABC is a right angle. The sides of △ABC are the diameters of semicircles as shown. The area of the semicircle on AB equals 8π, and the arc of the semicircle on AC has length 8.5π. What is the radius of the semicircle on BC?
Squares ABCD , EFGH , and GHIJ are equal in area. Points C and D are the midpoints of sides IH and HE , respectively. What is the ratio of the area of the shaded pentagon AJICB to the sum of the areas of the three squares?
正方形 ABCD，EFGH 和 GHIJ 的面积都相等。点 C 和点 D 分别是边 IH 和 HE 的中点。那么阴影部分五边形 AJICB 的面积和三个正方形面积总和的比值是多少？
A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are R_1 = 100 inches, R_2 = 60 inches, and R_3 = 80 inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from A to B?
直径为 4 英寸的球从 A 开始沿着图示轨道滚动。轨道由 3 段半圆弧组成，半径分别为 R1=100 英寸，R2=60 英寸，R3=80 英寸。球全程都和轨道紧密接触，并且不会滑动。那么当球从 A 滚到 B，球心所经过的路程是多少英寸？
2012 AMC8 #22 · Counting, Probability & Statistics · ★★★★
Let R be a set of nine distinct integers. Six of the elements are 2 , 3 , 4 , 6 , 9 , and 14 . What is the number of possible values of the median of R ?
R 是由 9 个不同的整数组成的集合。其中 6 个元素是 2,3,4,6,9 和 14．那么 R 的中位数有多少个可能的值？
A circle of radius 2 is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle?
A square with area 4 is inscribed in a square with area 5, with one vertex of the smaller square on each side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length a , and the other of length b . What is the value of ab ?
一个面积为 4 的正方形内接在一个面积为 5 的正方形内，小正方形的每个顶点分别落在大正方形的每条边上。小正方形的某个顶点将大正方形的一条边分成了 2 条线段，长度分别是 a 和 b，那么 ab 的值是多少？
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?
2011 AMC8 #23 · Counting, Probability & Statistics · ★★★★
How many 4-digit positive integers have four different digits, where the leading digit is not zero, the integer is a multiple of 5, and 5 is the largest digit?
A circle with radius 1 is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle's shaded area to the area between the two squares?
Hui is an avid reader. She bought a copy of the best seller Math is Beautiful. On the first day, Hui read 1/5 of the pages plus 12 more, and on the second day she read 1/4 of the remaining pages plus 15 pages. On the third day she read 1/3 of the remaining pages plus 18 pages. She then realized that there were only 62 pages left to read, which she read the next day. How many pages are in this book?
2010 AMC8 #25 · Counting, Probability & Statistics · ★★★★
Everyday at school, Jo climbs a flight of 6 stairs. Jo can take the stairs 1 , 2 , or 3 at a time. For example, Jo could climb 3 , then 1 , then 2 . In how many ways can Jo climb the stairs?
Jo 每天在学校里都要爬 6 节楼梯。Jo 可以一次爬 1 节，2 节或者 3 节。例如，J0 可以先一次爬 3 节，然后 1 节，然后 2 节。则 Jo 爬楼梯一共有多少种可能的方法？
Andy and Bethany have a rectangular array of numbers greater than 0 with 40 rows and 75 columns. Andy adds the numbers in each row. The average of his 40 sums is A . Bethany adds the numbers in each column. The average of her 75 sums is B . What is the value of BA ?
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?
A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is 1/2 foot from the top face. The second cut is 1/3 foot below the first cut, and the third cut is 1/17 foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet?
Jerry cuts a wedge from a 6 -cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters?
2008 AMC8 #24 · Counting, Probability & Statistics · ★★★★
Ten tiles numbered 1 through 10 are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?
Margie's winning art design is shown. The smallest circle has radius 2 inches, with each successive circle's radius increasing by 2 inches. Which of the following is closest to the percent of the design that is black?
2007 AMC8 #24 · Counting, Probability & Statistics · ★★★★
A bag contains four pieces of paper, each labeled with one of the digits 1 , 2 , 3 or 4 , with no repeats. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. What is the probability that the three-digit number is a multiple of 3 ?
2007 AMC8 #25 · Counting, Probability & Statistics · ★★★★
On the dart board shown in the figure, the outer circle has radius 6 and the inner circle has a radius of 3. Three radii divide each circle into three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to the area of the region. When two darts hit this board, the score is the sum of the point values in the regions. What is the probability that the score is odd?
Three different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added and the sum is placed in the cell above them. In the second row, continue the same process to obtain a number in the top cell. What is the difference between the largest and smallest numbers possible in the top cell?
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?
A company sells detergent in three different sized boxes: small (S), medium (M) and large (L). The medium size costs 50% more than the small size and contains 20% less detergent than the large size. The large size contains twice as much detergent as the small size and costs 30% more than the medium size. Rank the three sizes from best to worst buy.
Isosceles right triangle ABC encloses a semicircle of area 2π. The circle has its center O on hypotenuse AB and is tangent to sides AC and BC. What is the area of triangle ABC ?
等腰直角三角形 ABC 内含一个面积为 2π 的半圆。该半圆圆心 O 在斜边 AB 上，且与边 AC 和 BC 相切。那么三角形 ABC 的面积是多少？
2005 AMC8 #24 · Counting, Probability & Statistics · ★★★★
A certain calculator has only two keys [+1] and [x2]. When you press one of the keys, the calculator automatically displays the result. For instance, if the calculator originally displayed "9" and you pressed [+1], it would display "10." If you then pressed [x2], it would display "20." Starting with the display "1," what is the fewest number of keystrokes you would need to reach "200"?
A square with side length 2.0 and a circle share the same center. The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square. What is the radius of the circle?
At a party there are only single women and married men with their wives. The probability that a randomly selected woman is single is 52 . What fraction of the people in the room are married men?
In the figure, ABCD is a rectangle and EFGH is a parallelogram. Using the measurements given in the figure, what is the length d of the segment that is perpendicular to HE and FG ?
如下图所示，ABCD 是个矩形，EFGH 是个平行四边形。使用图中所标线段的长度，则同时垂直于和的线段长度 d 是多少？
Two 4×4 squares intersect at right angles, bisecting their intersecting sides, as shown. The circle's diameter is the segment between the two points of intersection. What is the area of the shaded region created by removing the circle from the squares?
In the figure, the area of square WXYZ is 25 cm2 . The four smaller squares have sides 1 cm long, either parallel to or coinciding with the sides of the large square. In △ABC , AB = AC , and when △ABC is folded over side BC , point A coincides with O , the center of square WXYZ . What is the area of △ABC , in square centimeters?
在下图中，正方形 WXYZ 的面积为 25 平方厘米。四个小正方形边长为 1 厘米，它们的各边与大正方形的边平行或重合。在中，，若将沿着边折叠，则点 A 和正方形的中心 重合。那么的面积是多少平方厘米？
Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What percent of the blend is pear juice?
Loki, Moe, Nick and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, Loki gave Ott one-fourth of his money and Nick gave Ott one-third of his money. Each gave Ott the same amount of money. What fractional part of the group's money does Ott now have?
Loki，Moe，Nick 和 Ott 是好朋友。Ott 没有钱，但其他人有。Moe 给了 Ott 五分之一的钱， Loki 给了 Ott 四分之一的钱，Nick 给了 Ott 三分之一的钱。每个人都给了 Ott 同样的钱。那么现在 Ott 有的钱占了四个人全部的钱的几分之几？
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?
Points R , S and T are vertices of an equilateral triangle, and points X , Y and Z are midpoints of its sides. How many noncongruent triangles can be drawn using any three of these six points as vertices?
点 R，S 和 T 是一个等边三角形的三个顶点，点 X，Y 和 Z 分别是边的中点。使用这六个点中的任意三个作为顶点，可以画出多少个不全等的三角形？
2001 AMC8 #24 · Counting, Probability & Statistics · ★★★★
Each half of this figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. When the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. There are 2 red-white pairs. How many white pairs coincide?
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?
There is a list of seven numbers. The average of the first four numbers is 5 , and the average of the last four numbers is 8 . If the average of all seven numbers is 674 , then the number common to both sets of four numbers is
Points B , D , and J are midpoints of the sides of right triangle ACG . Points K , E , I are midpoints of the sides of triangle JDG , etc. If the dividing and shading process is done 100 times (the first three are shown) and AC=CG=6 , then the total area of the shaded triangles is nearest
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?
Three generous friends, each with some money, redistribute the money as followed: Amy gives enough money to Jan and Toy to double each of their amounts. Jan then gives enough to Amy and Toy to double their amounts. Finally, Toy gives enough to Amy and Jan to double their amounts. If Toy had 36 dollars at the beginning and 36 dollars at the end, what is the total amount that all three friends have?
三位慷慨的朋友各自拥有一些钱，按如下方式重新分配：Amy 给 Jan 和 Toy 足够的钱，使他们各自的金额翻倍。然后 Jan 给 Amy 和 Toy 足够的钱，使他们的金额翻倍。最后，Toy 给 Amy 和 Jan 足够的钱，使他们的金额翻倍。如果 Toy 一开始有 36 美元，最后也有 36 美元，那么三位朋友共有多少钱？
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
Diameter ACE is divided at C in the ratio 2:3 . The two semicircles, ABC and CDE , divide the circular region into an upper (shaded) region and a lower region. The ratio of the area of the upper region to that of the lower region 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?
The manager of a company planned to distribute a $50 bonus to each employee from the company fund, but the fund contained $5 less than what was needed. Instead the manager gave each employee a $45 bonus and kept the remaining $95 in the company fund. The amount of money in the company fund before any bonuses were paid was
1996 AMC8 #25 · Counting, Probability & Statistics · ★★★★
A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region?
A plastic snap-together cube has a protruding snap on one side and receptacle holes on the other five sides as shown. What is the smallest number of these cubes that can be snapped together so that only receptacle holes are showing?
In parallelogram ABCD , DE is the altitude to the base AB and DF is the altitude to the base BC . [Note: Both pictures represent the same parallelogram.] If DC=12 , EB=4 , and DE=6 , then DF=
在平行四边形 ABCD 中，DE 是底边 AB 上的高，DF 是底边 BC 上的高。[注：两幅图表示同一平行四边形。] 若 DC=12，EB=4，DE=6，则 DF=
Buses from Dallas to Houston leave every hour on the hour. Buses from Houston to Dallas leave every hour on the half hour. The trip from one city to the other takes 5 hours. Assuming the buses travel on the same highway, how many Dallas-bound buses does a Houston-bound bus pass in the highway (not in the station)?
1994 AMC8 #24 · Counting, Probability & Statistics · ★★★★
A 2 by 2 square is divided into four 1 by 1 squares. Each of the small squares is to be painted either green or red. In how many different ways can the painting be accomplished so that no green square shares its top or right side with any red square? There may be as few as zero or as many as four small green squares.
1993 AMC8 #22 · Counting, Probability & Statistics · ★★★★
Pat Peano has a collection of numbers to label the pages of his scrapbook. He has an infinite number of 0's, 1's, 3's, 4's, 5's, 6's, 7's, 8's and 9's, but only twenty-two 2's. How far can he number the pages of his scrapbook with these digits?
Pat Peano 有一批数字用于给他的剪贴簿页码编号。他有无限多个 0、1、3、4、5、6、7、8 和 9，但只有二十二个 2。用这些数字他最多能将剪贴簿页码编到多少？
1993 AMC8 #23 · Counting, Probability & Statistics · ★★★★
Five runners, P , Q , R , S , T , have a race, and P beats Q , P beats R , Q beats S , and T finishes after P and before Q . Who could NOT have finished third in the race?
A checkerboard consists of one-inch squares. A square card, 1.5 inches on a side, is placed on the board so that it covers part or all of the area of each of n squares. The maximum possible value of n is
棋盘由边长为 1 英寸的方格组成。一张边长为 1.5 英寸的正方形卡片放在棋盘上，使其覆盖 n 个方格的部分或全部面积。n 的最大可能值是
Eight 1×1 square tiles are arranged as shown so their outside edges form a polygon with a perimeter of 14 units. Two additional tiles of the same size are added to the figure so that at least one side of each tile is shared with a side of one of the squares in the original figure. Which of the following could be the perimeter of the new figure?
One half of the water is poured out of a full container. Then one third of the remainder is poured out. Continue the process: one fourth of the remainder for the third pouring, one fifth of the remainder for the fourth pouring, etc. After how many pourings does exactly one tenth of the original water remain?
1991 AMC8 #23 · Counting, Probability & Statistics · ★★★★
The Pythagoras High School band has 100 female and 80 male members. The Pythagoras High School orchestra has 80 female and 100 male members. There are 60 females who are members in both band and orchestra. Altogether, there are 230 students who are in either band or orchestra or both. The number of males in the band who are NOT in the orchestra is
A cube of edge 3 cm is cut into N smaller cubes, not all the same size. If the edge of each of the smaller cubes is a whole number of centimeters, then N=
一个棱长为 3 cm 的正方体被切成 N 个小正方体，不全部大小相同。如果每个小正方体的棱长都是整数厘米，则 N=
An equilateral triangle is originally painted black. Each time the triangle is changed, the middle fourth of each black triangle turns white. After five changes, what fractional part of the original area of the black triangle remains black?
A list of 8 numbers is formed by beginning with two given numbers. Each new number in the list is the product of the two previous numbers. Find the first number if the last three are shown:
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
1990 AMC8 #25 · Counting, Probability & Statistics · ★★★★
How many different patterns can be made by shading exactly two of the nine squares? Patterns that can be matched by flips and/or turns are not considered different. For example, the patterns shown below are not considered different.
Suppose a square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed-a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle?
1989 AMC8 #25 · Counting, Probability & Statistics · ★★★★
Every time these two wheels are spun, two numbers are selected by the pointers. What is the probability that the sum of the two selected numbers is even?
1988 AMC8 #21 · Counting, Probability & Statistics · ★★★★
A fifth number, n , is added to the set {3,6,9,10} to make the mean of the set of five numbers equal to its median. The number of possible values of n is
将第五个数 n 加入集合 {3,6,9,10} 中，使得这五个数的平均数等于其中位数。n 的可能值有多少个？
The square in the first diagram "rolls" clockwise around the fixed regular hexagon until it reaches the bottom. In which position will the solid triangle be in diagram 4 ?
1988 AMC8 #25 · Counting, Probability & Statistics · ★★★★
A palindrome is a whole number that reads the same forwards and backwards. If one neglects the colon, certain times displayed on a digital watch are palindromes. Three examples are: 1:01, 4:44, and 12:21. How many times during a 12-hour period will be palindromes?
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?
1987 AMC8 #25 · Counting, Probability & Statistics · ★★★★
Ten balls numbered 1 to 10 are in a jar. Jack reaches into the jar and randomly removes one of the balls. Then Jill reaches into the jar and randomly removes a different ball. The probability that the sum of the two numbers on the balls removed is even is
Suppose one of the eight lettered identical squares is included with the four squares in the T-shaped figure outlined. How many of the resulting figures can be folded into a topless cubical box?
假设八个标有字母的全等正方形之一与 T 形外框中的四个正方形合并。所得图形中有多少个可以折叠成一个无盖的正方体盒子？
1986 AMC8 #22 · Counting, Probability & Statistics · ★★★★
Alan, Beth, Carlos, and Diana were discussing their possible grades in mathematics class this grading period. Alan said, "If I get an A, then Beth will get an A." Beth said, "If I get an A, then Carlos will get an A." Carlos said, "If I get an A, then Diana will get an A." All of these statements were true, but only two of the students received an A. Which two received A's?
The large circle has diameter AC . The two small circles have their centers on AC and just touch at O , the center of the large circle. If each small circle has radius 1 , what is the value of the ratio of the area of the shaded region to the area of one of the small circles?
大圆的直径为 AC。两个小圆的圆心在 AC 上，且恰好在 O 点（大圆的圆心）相切。如果每个小圆的半径为 1，着色区域与一个小圆的面积之比是多少？
1986 AMC8 #24 · Counting, Probability & Statistics · ★★★★
The 600 students at King Middle School are divided into three groups of equal size for lunch. Each group has lunch at a different time. A computer randomly assigns each student to one of three lunch groups. The probability that three friends, Al, Bob, and Carol, will be assigned to the same lunch group is approximately
King 中学的 600 名学生被平均分成三组用午餐。每组在不同的时间用餐。一台计算机随机将每名学生分配到三个午餐组之一。三个朋友 Al、Bob 和 Carol 被分到同一午餐组的概率大约是
1985 AMC8 #22 · Counting, Probability & Statistics · ★★★★
Assume every 7-digit whole number is a possible telephone number except those that begin with 0 or 1 . What fraction of telephone numbers begin with 9 and end with 0 ?
King Middle School has 1200 students. Each student takes 5 classes a day. Each teacher teaches 4 classes. Each class has 30 students and 1 teacher. How many teachers are there at King Middle School?
1985 AMC8 #24 · Counting, Probability & Statistics · ★★★★
In a magic triangle, each of the six whole numbers 10-15 is placed in one of the circles so that the sum, S , of the three numbers on each side of the triangle is the same. The largest possible value for S is
在一个魔力三角形中，六个整数 10-15 各放入一个圆圈，使得三角形每条边上的三个数之和 S 相同。S 的最大可能值是
1985 AMC8 #25 · Counting, Probability & Statistics · ★★★★
Five cards are lying on a table as shown. Each card has a letter on one side and a whole number on the other side. Jane said, "If a vowel is on one side of any card, then an even number is on the other side." Mary showed Jane was wrong by turning over one card. Which card did Mary turn over?
如图所示，桌上放着五张卡片。每张卡片一面是字母、另一面是整数。Jane 说：「如果任何一张卡片的一面是元音字母，那么它的另一面就是偶数。」Mary 只翻开一张卡片就证明了 Jane 是错的。Mary 翻开的是哪张卡片？