(Originais Teachy 2023) - Question Medium of Mathematics

In a school presentation, eight friends, including Carlos, Timóteo, and Joana, line up. Calculate the number of different ways this line of friends can be formed so that Carlos, Timóteo, and Joana always stay together:

João is organizing a birthday party and wants to make a larger quantity of his favorite cake recipe. The original recipe yields a cake with 20 slices and uses 3 cups of flour, 2 cups of sugar, and 1 cup of oil. João wants to make a cake with 40 slices. a) How many cups of flour, sugar, and oil should he use to make a cake that yields 40 slices? b) If he wanted to make a cake with 60 slices, what would be the quantity of cups of each ingredient? c) João noticed that the number of cake slices is always half the number of cups of water used in the recipe. How many cups of water are needed for a cake of 80 slices?

Consider a Cartesian coordinate system where a point P has coordinates (a, b) and a point Q has coordinates (c, d). The absolute value of a number is defined as its distance from the origin of this system. If point P is in the second quadrant and point Q is in the fourth quadrant, with a, b, c, and d being integers, and it is known that the absolute difference between the y coordinates of P and Q is 15, determine the value of a - c, knowing that a < 0 and c > 0.

A store decides to reward its customers. Each customer will receive one of the six possible gifts available, according to their arrival order at the store. The gifts to be distributed are: a ball, a keychain, a pen, a soda, an ice cream, and a CD, in that order. The first customer at the store receives a ball, the second receives a keychain, the third receives a pen, the fourth receives a soda, the fifth receives an ice cream, and the sixth receives a CD. The seventh receives a ball, the eighth receives a keychain, and so on, according to the order of the gifts. The one-thousandth customer will receive as a gift a: