Want to know some interesting math problems

Cups, Mathematical Induction (2).

Now, let me show you that all positive integers are equal.

To prove this, it is enough to state that for any two positive integers a and b, there is a = b.

To prove this, it is only necessary to state that for all positive integers n, if max(a, b) = n, then a = b.

We generalize to n Shi. When n = 1, since a and b are both positive integers, a and b must both be equal to 1, so a = b. If the proposition is also true when n = k, now assume max(a, b) = k + 1 .

then max(a - 1, b - 1) = k , from the inductive assumption a - 1 = b - 1 , i.e. a = b.

The problem is that a - 1 or b - 1 may not be a positive integer, so the inductive assumption cannot be applied.

For math probability problems, there are 30 students in a class, so what is the probability of two people born on the same day and month in the same year, and the answer is greater than 99%.

If the integers a, b, c, d, m are such that am 3+bm 2+cm+d is divisible by 5, and the number d is not divisible by 5. Try to explain: you can always find such an integer n, so that dn 3 + cn 2 + bn + a can also be divisible by 5.

A method of proving the Pythagorean theorem.

A lot

is a cyclic decimal, then the 80th digit is that number.

Subtract 4,6 two digits from the 80th digit, there are 78 digits, a total of three digits loop 3 26 = 78, three digits cycle 26 times to complete the 80th digit, so the 80th digit is 8

1.There are 3 people to stay, 30 yuan a night.

The three of them each paid 10 yuan, and 30 yuan was handed over to the boss.

Later, the boss said that today's discount was only 25 yuan, and he took out 5 yuan and ordered the waiter to return it to them, and the waiter secretly hid 2 yuan, and then distributed the remaining 3 yuan to the three people, each of whom received 1 yuan.

In this way, at the beginning, each person paid 10 yuan, and now it is returned 1 yuan, that is, 10-1 = 9, and only 9 yuan per person, 9 yuan per person for 3 people, 3 x 9 = 27 yuan + 2 yuan hidden by the waiter = 29 yuan, and one yuan went to **???

2.There was a guy who went to buy green onions.

Ask how much is a pound of green onions.

The person who sells green onions says, "1 yuan and 1 catty, this is 100 catties." After 100 yuan, the person who bought green onions asked again, "Do you sell green onions separately?"

The person who sells green onions says that the green onion is white 7 cents and the green onion is 3 cents.

The people who bought the green onions bought it.

Weigh 50 catties of green onion white and 50 catties of green onion.

The last calculation of green onion white 50 * 7 is equal to 35 yuan.

Green 50*3 is equal to 15 yuan.

35+15 equals 50 yuan.

The person who bought the green onion gave 50 yuan to the person who sold the green onion and left.

And the man who sold the green onion wondered.

Why do you want to sell green onions for 100 yuan?

And why did the person who bought the green onion buy it for 50 yuan?

Why do you say that?

3.A dime a peach.

Three peach for one peach.

How many peaches can you eat for 1 dollar?

4.A sold the clothes for 19 yuan to B 21, and B gave A 100, but A went to C to change money and came back to find money for B, and after B left, C found that the 100 yuan was fake money and went to A, and A gave C another 100, and asked how much A lost.

5.There is a well about 7 meters deep.

There was a snail climbing up from the bottom of the well.

Climb 3 meters during the day and drop 2 meters at night.

Ask how many days the snail can crawl out of the well.

6.Several identical boxes were lined up, and Xiao Ming divided more than 40 identical pieces into these boxes, and only one of them had no pieces in it, and then he went out. After a while, Xiao Guang entered the house, Xiao Guang took one from each box with chess pieces and put it in the empty box, and then rearranged the boxes, Xiao Ming came back and took a closer look, and did not find that anyone had touched the chess pieces in these boxes, but just moved the box position.

Q: How many boxes are there?

Clue 1: The actual transaction amount is 27 yuan, that is, 9 * 3 = 27 yuan paid by the three people, of which the boss gets 25 yuan and the waiter gets 2 yuan. 30 yuan is just a list price, and it is reflected in the final transaction process, which is an image interference factor.

Clue 2: The assumption in the first transaction 30 = 25 from the boss - 2 from the waiter - 2 from the tenant should be 27-2 = 25It shouldn't be 27+2, note that the amount of the actual transaction is different between the two times.

Question 1: Each person spent 9 yuan, and a total of 27 yuan includes the 2 yuan taken by the waiter, which should be 27 yuan plus 1 yuan returned per person (27 yuan + 3 yuan 30 yuan. ）

Question 2: Because green onions are sold by the number of catties, the white onions and green onions are sold separately for 1 yuan and 1 catty.

Question 3: Can eat 15 peaches.

Question 4: Lost 98 yuan.

Question 5: Can you climb out of a well in 5 days.

Question 6: 10 boxes, 45 pieces, press to place.

Add addition, subtraction, multiplication, and division to equals 24

The actual weight of 10 catties of eggs weighed by the boss is x catties.

According to the title, x 10=

Solution x=9 (catty).

10-1=9 (catty).

The boss weighed about 1 pound of eggs less.

Solution 2: Let the boss weigh the weight of the egg as x catty, and the actual weight of the egg as y catty. It is not difficult to find that the actual weight of the egg is a proportional function of the weight x catty weighed by the boss. The actual weight of the basket is catty, and the eggs are weighed together after being put into the basket, the weight is catty, and the increment is catty.

y= * x

When x=10, y= * 10=9 (jin).

10-9 = 1 (jin).

The boss weighed about 1 pound of eggs less.

If the eggs are 10 catties, it means that there is no problem with the stall owner's weighing device, then it should be catties if they are weighed together with the basket. However, the result is catty, which shows that the weight value weighed by the stall owner's weighing device is larger than the actual weight value, so it can be judged that the stall owner's weighing device is problematic.

The following begins to explain Mr. Zhang's judgment method.

The weighing device is problematic, this has been determined, and now we need to determine the ratio of the weighed weight value to the actual value.

Mr. Zhang's basket weighs in catties, but the weight weighed in the stall owner's weighing device is catties, that is to say, the weight value of the stall owner's weighing device will be about 10% more than the actual weight value, so Mr. Zhang deduced that the weight of the 10 catties of eggs is about 1 catty less.

Note: The 10% value here is only an approximate value, not an exact value.

Let an article with an actual weight of x, and the displayed weight of it be said to be f(x)*x, then there is:

Display weight: f(

Therefore, f( and since the so-called operating characteristic curve is usually approximately regarded as linear, it can be considered that f(x) is a constant, that is, for any x, there is f(x)=

And the 10 catties of the displayed weight can be expressed as f (m) times of m (actual weight), so:

f(m)*m=10 m=10 f(m)= about 9, that is, the actual gain is 9 catties, so the boss gives one pound less.

The boss weighed 1 catty, and the actual weight was x catty.

The weight of the basket as weighed.

The equations can be listed as follows.

x=10 The actual weight of the eggs bought.

So this person said that the boss gave him a pound less.

Maybe the basket wasn't big enough to hold ten pounds at once, so he divided it into three parts. After weighing, the total was a catty, so the boss gave one pound less.

It's still quite a lot.

1. The problem of baking cakes: It takes two minutes for my mother to bake a cake, one minute for the front and back sides, and two cakes are placed in the pot at the same time, so it takes at least a few minutes to bake three cakes?

2.Socks problem, there are 5 pairs of socks of different colors in the drawer, the light is not turned on, to take out a pair of socks of the same color, how many of them need to be found at most?

3.Egg problem: Xiao Zhang sells eggs, a basket of eggs, the first person comes to buy half of it, and then gives him another one.

The second person bought half of it, and Xiao Zhang gave him another egg. The third person bought half of the eggs, and Xiao Zhang gave him another one. The fourth person came to buy half of it, Xiao Zhang gave him another one, and the eggs were just finished!

How many eggs does Xiao Zhang have in total?

4 table questions, a square table, cut off a corner and how many corners?

5.Tofu cutting problem: A piece of tofu can be cut into three knives, and a maximum of how many pieces can be cut into 6 watermelon Question: Three knives cut 7 petals, and there are 8 pieces of skin left after eating, how to cut it?

7.Bamboo pole question: Can a 5-meter-long bamboo pole pass through a one-meter-high door?

8. Carton problem: Can a square box with a side length of one meter put down a wooden stick of rice?

9.Clock Question: 12 Hours, How Many Times Does the Clock and Minute Hand Repeat?

10.Origami Question: How thick is a piece of paper with a thickness of 1 mm, folded in half 1000 times? ......

[Topic].

1.Bridge. There is a b c d today and all four of them are to go from the left side of the bridge to the right side at night. This bridge can only walk for a maximum of two people at a time, and there is only one flashlight, so you must use a flashlight to cross the bridge. The fastest time for four people to cross the bridge is as follows:

a 2 points; b 3 points; c 8 points; d 10 points. Those who walk fast have to wait for those who walk slowly, how can I walk in 21 minutes to let all people cross the bridge?

2.Insert numbers.

This equation is obviously unequal, but if you cleverly insert two numbers 7 into the equation, this equation can be true, do you know where these two 7s should be inserted?

3.Warm all seasons.

Spring-summer Autumn-winter = Spring-summer-autumn-winter.

Spring-Winter Autumn-Summer = Spring-Summer-Autumn-Winter.

where spring, summer, autumn, and winter each represent four different numbers, can you point out what numbers each of them represents?

4.Broken car down the hill.

A broken car has to travel two miles, one mile up and one mile downhill, and the average speed when going up the hill is 15 miles per hourQ How fast does it have to go down the second mile to reach an average speed of 30 miles per hour? Is it 45 miles? You've got to think about it!

5.How many eggs are sold.

Mrs. Wang went to the market to sell eggs, the first person bought half of the eggs in the blue and one again, and the second person bought half and one of the remaining eggs, at this time there was still one egg left in the blue, how many eggs did Mrs. Wang sell?

6.How many people took the exam.

There are 6 multiple-choice questions on the test paper, each question has 3 options, and the teacher found that in all the papers, any 3 answer sheets were selected, and there was a question with different choices from each other, how many people took this exam at most?

Answer] Answer to Question 1: A and B cross the bridge together, then B is left on the other side and A returns alone. After A returns, give the flashlight to C and D, let C and D cross the bridge together, when C and D reach the opposite bank, give the flashlight to B, let B bring the flashlight back, and finally A and B cross the bridge together again. The time required is: 3+2+10+3+3=21 minutes.

Answer to question 2: The formula after inserting the number is: 172543=20700

Answer to question 3: Spring = 2; Summer=1; Autumn = 8; Winter = 7

Answer to Question 4: No matter what, the average speed of a broken car cannot reach 30 mph. Because when the average speed is 30 mph, the total time it takes to get up and down the hill should be 1 15 hours.

It took 1 15 hours to go up the mountain with a broken car. So the average speed of a broken car is less than 30 mph.

Answer to Question 5: Mrs. Wang sold a total of 10 eggs.

Question 6 Answer: A maximum of 13 people take the exam

I think the last one only sold three.