AMC 8 specimen paper

A display panel measures 24 cm by 18 cm. Its first dimension is increased by 6 cm, and its second dimension is reduced so that the area stays unchanged. By how many centimetres must the second dimension decrease?

Five volunteers contribute $12,$18, $24,$30 and $66toasharedfund.Theyredistributeitsoeachhascontributedthesamenetamount.Howmuchmustthepersonwhopaid$66 receive back?

Six robots stand in order Ari, Bela, Cleo, Dara, Evan, Faye around a circle. Starting with Ari at 1, they count upward without resetting. A robot leaves whenever its number is divisible by 3 or contains the digit 3. The next survivor continues with the next number. Which robot remains last?

A survey drone initially faces north. Its camera turns clockwise through 11/6 of a revolution and then counterclockwise through 3/4 of a revolution. What is its final direction, expressed as an angle clockwise from north between 0° and 360°?

The integers from 1 through 12 are written once each. Two distinct entries are erased. Their product is odd, and the sum of the remaining entries is divisible by 6. How many unordered erased pairs are possible?

A delivery truck takes 30 minutes to travel a 2 km access road. It then travels 15 km on a highway at three times its access-road speed. How many minutes does the whole journey take?

Three square display boards have side lengths 6 cm, 8 cm and 10 cm. Respectively, one half, one quarter and three fifths of their areas are covered in gold foil. What percentage of the combined board area is gold?

A collection contains only red, green and blue counters. Initially red: green = 2: 3 and green: blue = 4: 5. One counter of each colour is removed; the remaining red: blue ratio is 1: 2. How many counters were in the initial collection?

A warehouse ships one-quarter of its stock in the morning, then one-fifth of what remains in the afternoon. What percentage of the original stock is still in the warehouse?

How many six-digit multiples of 5 can be formed using exactly the digits 0, 0, 1, 1, 2, 3?

A cube's faces carry different letters A, B, C, D, E and F. Three photographs show top/front/right triples (A,B,C), (A,C,D) and (A,D,E). Which letter is opposite A?

Find the smallest integer greater than 2 that leaves remainder 2 when divided by each of 5, 6 and 8.

A design packet contains one of 4 cover colours, two different inserts chosen from 5 designs, and one of 3 seal shapes. Two particular insert designs may not be used together. The order of the two inserts does not matter. How many different packets are possible?

Nonnegative integers a and b satisfy 2^a + 3² = 137 and 5^b − 2^a = 497. Find a + b.

Use each digit 0 through 7 once to form two four-digit integers with the largest possible sum. Among all such pairs, what is the smallest possible absolute difference between the two integers?

A six-digit equipment code may start with zero and may repeat digits. Codes beginning with 4242 are prohibited. How many codes remain?

Three towns have resident populations 80, 120 and 160. Of the residents of A, 1/5 work in B and 1/4 work in C. Of B residents, 1/4 work in A and all others stay in B. Of C residents, 3/8 work in A and all others stay in C. Everyone works in one of these towns. How many people work in A?

How many six-digit positive integers have the pattern ababab, with different digits a and b, and are divisible by 15?

Two fair six-sided dice each have labels 0, 2, 3, 4, 6 and 9. What is the probability that their sum is odd?

Thirty-one consecutive odd-numbered seats have labels whose sum is 961. What is the largest label?

In an ideal tightly wound coil with no gaps, tape has inner radius 2 cm, outer radius 5 cm and thickness 0.02 cm. The same tape length is laminated to have effective thickness 0.03 cm, then rewound without gaps on a core of radius 3 cm. What is the new outer radius squared, in square centimetres?

Eight tokens have weights 1, 1, 1.5, 2, 2, 2.5, 3 and 5. Put every token in one of four crates A, B, C, D so their total weights are consecutive integers in increasing crate order. Crate A contains the token of weight 3, crate B contains a token of weight 2, and crate D contains the token of weight 5. Which crate must contain the token of weight 1.5?

Let P_n = 2! × 4! × 6! × ... × (2 n)!. What is the least positive integer n for which P_n has at least 40 trailing zeros in decimal notation?

Eight children receive a total of 43 counters, with each child receiving at least two. What is the greatest possible median number of counters received?

A counter starts at 1. Each command either adds 1 to the current number or doubles it. What is the fewest commands that reach exactly 176? Give a route and prove that no shorter route succeeds.