Answer-Guess-Matrix Math

When Alice walks up the hill, her speed is 1 m/s and when she goes down the hilll, her speed is 3 m/s. Then when Alice walks up and down the hill, what is her average speed? 9 days among 10 days, Alice goes out for drink. She only goes to three bars for drink with equal frequency. Bob is trying to find Alice today. However, he fails after going to two bars. What is the probability that Alice shows up in the third bar. You can select one closed door of three. A prize, a car, is behind one of the doors. The other two doors hide goats. After you have made your choice, Monty Hall will open one of the remaining doors and show that it does not contain the prize. He then asks you if you would like to switch your choice to the other unopened door. What is the probability to get the prize if you switch? One peach costs one cent. You can use 3 peach pits to exchange for one peach. If you have 10 cents, then what is the maximal number of peaches you can eat? You finish drinking 20% of a cup of milk. You then fill up water, drink 40% of it and fill up water again. What is the percentage of milk in this cup now? We put 10 g sat into 100 g water. What is the percentage of salt now? We have 1 ton apples which contain 90% water. After a few days, those apples only have 80% water. What is the weight of those apples now? A family has two children. They can be brothers/sisters/big brother and little sister/big sister and little brother with equal probability. If you see one of them is a boy, then what is the probability that the other one is a boy? We roll a dice and count the number of times we roll. If we get an even number, we will reset the count to zero. We will stop if we get one. What is the expected number of final counts? We flip two coins simultaneously. What is the probability that we get one heads and one tails?
We pick four points arbitrarily from a ball. What is the probability that the triangular pyramid formed by those four points contain the ball's center? Please calculate the following formula: In the figure below, all edges of the two pyramids are equal in length. If we put a face of the triangular pyramid on the face of the quadrangular pyramid, how many exposed faces would the resulting solid have? 0.1234567891011121314…is a non-repeating decimal that consists of consecutive natural numbers. What is the position where 201 first show up? We assign fruits to kids in a line. Starting from the first kid, we assign pears to every three kids (1,4,7...). Starting from the last kid, we assign apples to every five kids. Nine kids get both pear and apple. How many kids here? We use a four minutes hourglass and a five minutes hourglass to measure six minutes. How many times we need to flip the hourglass at least? We have a diameter 2 half circle. Point C,D,E divide the diameter AB into four pieces. The angles marked by ○ are all 45°. What is the difference between the blue area and yellow area? There are ininite number of stones. Starting from the first stone, Alice rolls a dice to decide the number steps she will move forward. P(X) denotes the probability that Alice will pass through the Xth stone. When X goes to infinite, what is the limit of P(X)? Nine squares, whose sides have an integer length, form a regtangle without overlapping. What is the minimal area of the regtangle? For a rectangle, we divide one of its sides into 6 pieces equally and another into 4 pieces equally. If the area of the blue triangles is 18, what is the area of "?"?
At most how many distinct natural numbers sum to 100? We have 30 2cents coins and 8 5cents coints. How many values between 1 cent to 100 cents those coins can make? We divide 50 into 4 natural numbers a,b,c,d such that 2a = b/2 and c+2=d-2. How many divisions we can make? When a goat runs 5 steps, a horse runs 4 steps. When the horse runs 4 steps, the goat runs 7 steps. Now the goat has moved forward 30 meters and the horse starts to chase it. How far can the goat run when the horse catches up with it? A cuboid where the sum of the areas of the front and the top is 209 square centimeter, the width, length, height are prime numbers (the unit is centimeter). What is the volume of this cuboid? In a 600 meters circular runway, Alice and Bob run from the same location clockwise. They meet with each other every 12 minitues. If they do not change the speed and run from the same location again while this time Alice runs counterclockwise, then they meet with each other every 4 minites. How long does Alice need to run one circle? We flip a fair coint and among 100 times, 95 times are heads. What is the probability that the 101th time is heads? A family has three children. If we have already known that one of them is a girl, then what is the probability that at least there is a boy there? 20 people get full marks for math. 20 people get full marks for history. 20 people get full marks for physics. 7 people get full marks for both math and physics. 9 for both history and physics. 3 do not get full marks for any of the subjects. How many people here at most? How many proper irreducible fraction whose denominator is 1001?