AMC8 2027 - Self-practice Set 5

Three of the following statements are true; the remaining statement is false.
I. Đức was born in 1991.
II. Đức was not born in 1992.
III. Đức was born in 1993.
IV. Đức was not born in 1994.
Which statements must be true?

Jenny has 2 dresses, 1 shirt, 2 pairs of trousers, 1 pair of sandals and 2 pairs of shoes. She wears a dress or shirt and trousers, but never sandals with a dress. How many outfits are possible?

Jacky collects stamps from Brazil, France, Peru, Spain. Each Brazilian or French stamp costs 6 cents, each Peruvian stamp 4 cents, and each Spanish stamp 5 cents. The numbers for decades 50 s, 60 s, 70 s, 80 s are Brazil(4, 7, 12, 8), France(8, 4, 12, 15), Peru(6, 4, 6, 10), Spain(3, 9, 13, 9). Brazil and Peru are South American; France and Spain are European. How much did his South American stamps from before the 70 s cost?

A 30-litre mixture contains 25% red paint,30% yellow paint, and 45% water. What percent of the mixture is yellow paint after adding 5 litres of yellow paint?

Find the remainder when the product of 5 consecutive nonnegative integers is divided by 5.

Evaluate the following expression.
$$\left(1-\frac12\right)\left(1-\frac13\right)\left(1-\frac14\right)\left(1-\frac15\right)\left(1-\frac16\right).$$

The three squares $ABCD$, $EFGH$, and $IJGH$ have equal areas. Points $C$ and $D$ are the midpoints of $HI$ and $HE$, respectively. What fraction of the total area of the three squares is shaded?

Three students stand in a line. The last student can see the first two, the middle student sees the first, and the first sees no one. A teacher selects their hats from five hats: two black and three white. No student sees their own hat or the two unused hats.
The last student says, "I do not know." The middle student also says, "I do not know." Hearing this, the first student says, "I know." All three are excellent reasoners. Which conclusion is correct?

Two fair dice are rolled. What is the probability that the product exceeds 10?

Roll two fair standard dice. What is the probability that their product is divisible by 5?

At Anniville,30% of the math club are in the science club, and 80% of the science club are in the math club. The science club has 15 members. How many are in the math club?

A two-digit number equals the product of its digits plus the sum of its digits. Find its units digit.

For positive integers $p$ and $q$, define
$$p*q=2p+2q-pq.$$
If $8*x=4$, what is $x$?

Isosceles right triangles are constructed outward on the sides of a $3-4-5$ right triangle, as shown. Each capital letter represents the area of its triangle. Which statement is true?

Five coins look identical. Four have equal weights, and one counterfeit coin has a different weight. Using only a two-pan balance, what is the minimum number of weighings needed to identify the counterfeit coin and determine whether it is heavier or lighter?

Tyler chooses 1 of 3 meats, 2 distinct vegetables from 4, and 1 of 4 desserts. Order does not matter. How many meals are possible?

The illustrated gasoline prices have maximum 17 and minimum 10. By what percentage is the maximum higher than the minimum?

Bob sees 141 km on his motorbike odometer. Four hours later it reads 161 km. What is his average speed?

Which number has the largest prime divisor?

Define $a\oplus b=2a+3b$. If $(x\oplus3)\oplus4=12$, find $x$.