Easy Olympiad Questions, High Scores, Difficult Olympiad Questions

Doesn't seem to be the right question...

Is their sum equal to 461?

Dividend = divisor * quotient + remainder.

Dividend = divisor * quotient + remainder.

Divisor * Quotient + Remainder + Divisor + Quotient + Remainder.

Divisor * (quotient + 1) + 2 * remainder + quotient.

Divisor. (sum-quotient-2*remainder) (quotient+1).

Dividend = 49 * 8 + 6 = 398

The area of the circle is 4*4*=16

So the area of the sector is 16 -14 = 2

The sector area is 2 (16) = 1 8 of the circular area, and the central angle of the sector is 1 8 * 360 = 45 degrees.

So the triangle is an isosceles right triangle.

Both right-angled edges are equal to the radius of the circle.

The area of the triangle is: 1 2*4*4=8

The dividend is 398 and the dividend is 49

Think of it this way: 463-8-6 = 449 is the sum of the dividend and the divisor, and the dividend is 8 times the divisor and 6 is left according to the known conditions of the question, and all use (449-6) 9 (where 9 contains 1 times the divisor and 8 times the dividend) = 49 and 2So the divisor is 49, and then 49*8+6=398 gives the dividend.

463-8-6=449 is the sum of the dividend and the divisor.

According to the known conditions of the question, the dividend is 8 times the divisor and there is 6 left, and all use (449-6) 9 (where 9 contains 1 times the divisor and 8 times the dividend) = 49 and 2So the divisor is 49

Then 49*8+6=398 gives the dividend.

The dividend is 398 and the dividend is 49

Solution: Set the original water x cubic meters;

A water intake per hour: 100 10 = 10 cubic meters.

B hourly water intake: 100 15 = 20 3 cubic meters A B 6 hours simultaneous water intake: (10 + 20 3) 6 = 100 cubic meters of drainage pipe per hour drainage: (100 + x) 6

Because the water inlet pipe and the drainage pipe are discharged at the same time, it takes 2 hours to drain the water in the pool;

x+2×10=[(100+x)÷6]×2x+20=[50/3+x/6]×2

x+20=100/3+x/3

x = 20 cubic meters.

A: The original water in the pool is 20 cubic meters.

Only open the faucet and put the water for y hours.

6* (1 y-1 10-1 15)=2*(1 y-1 10) can be moved and then solved.

Here's an analysis for you:

First of all, both faucets are turned on, and the drain is also turned on 1 y -1 10-1 15 is the amount of water discharged per hour more than the amount of water entering *6 is the amount of original water.

Then turn on the faucet and open the drain 1 y-1 10 is the amount of waterproof more than the water inlet per hour *2 is the original amount of water.

are equal to the original amount of waterSo we get 6* (1 y-1 10-1 15) = 2*(1 y-1 10).

Not for the sake of points, just to help you, I'm in the third year of junior high school.

Hint, break it down and multiply y y y for two days at the same time, and you can simplify the shift.

A: Already = 3:2 (efficiency ratio) 3 + 2 = 5 parts 100 divided by 5 = 20 cubic meters 20 times (3-2) = 20 cubic meters.

Set the original water XM3, and drain the pipe to drain YM3 in one hour

A's water injection rate is 100 10=10

B's water injection velocity is 100 15=20 3, and the equation x+6 (10+20 3)=6yx+20=2y is obtained

Solution x=20

y=40 and give points to your own calculations.

Landlord, non-equation method.

Expand the numerator denominator of 8 9 by two times to 16 18, and then subtract the denominator by 1 to get 16 17, and in the second question, expand the numerator and denominator by 3 times to become 24 27, and then subtract the denominator by 2 to get 24 25

Equation method: The numerator of a fraction is added to 1, and this fraction is equal to 1, so the denominator is 1 larger than the numerator

The first question assumes that the denominator is x then the numerator x-1

x-1)/(x+1)=8/9

9(x-1)=8(x+1)

9x-8x=8+9

x = 17 The original score is 16 17

Second question (x-1) (x+2) = 8 9

9(x-1)=8(x+2)

9x-9=8x+16

9x-8x=9+16

x=25 The original fraction is 24 25

If the denominator is added by 1, this fraction will be equal to 8 9

8/9=16/18=24/27=...The original score was 8 8, 16 17, 24 26 ,..

If the numerator is added by 1, this fraction is equal to 1" meaning that the denominator is 1 more than the numerator, and only 16 17 satisfies the condition.

So the original score was 16 17

Let the original score be y x.

y+1)/x=1

y/（x+1)=8/9

Get y+1=x

y=(8x+8)/9

The original fraction of 16 17 is obtained by solving the equation

9 out of 8, of course

Because the numerator adds one to make 9/8 = 1

When the denominator is added to 1, it equals 8/9

Let the score be a b, and the question can be obtained: a+1=b; a/(b+1)=8/9;

Solving the equation yields a=16, b=17

The original fraction solved with the equation is 16 17

In the first question, the numerator and denominator of 7 7 7 are multiplied at the same time, and when the difference between the denominators of the two scores is 2, it is the original fraction.

In the second question, when the original fraction numerator is reduced by 124, only the denominator is reduced by 155 at the same time (124 4 times of 5) to ensure that the original fraction remains unchanged, that is, it is still 4 5, and the original denominator is reduced by 11 at this time to get 4 9, that is, the extra 155-11=144 is twice the numerator after the original numerator minus 124, so the original fraction numerator is 144 + 124 = 268, so the original fraction is 268 335

In the first question, you can think of the numerator and denominator are both added by 1, and the numerator and denominator are both subtracted by 1, if you do not divide the denominator, the numerator and denominator of the two fractions should be 2 different, because the 7 8 and 6 7 numerator denominators are both 1 and 1 7 are 1 different by 1, so that the 14 16 and 12 14 numerator denominators are 2 different, and it is obvious that the required number is 13 15

The second problem is not a good algorithm, you can use the method of Olympiad numbers, you can use the method of estimation + number theory, first of all, the amount of numerator reduction is much greater than the amount of denominator reduction, so that we can estimate that the numerator of the original fraction is probably more than 250, and the original denominator is more than 300Then because the denominator of the original fraction is a multiple of 5, and after -11, it is a multiple of 9, so the original denominator is divided by 9 and the remainder is 2, according to the Chinese remainder theorem, the original denominator can be 20 65 110 155 200 245 290 335 380 425....As long as the denominator is more than 300, you can get the answer is 268 335 This method is not very good, if it is an Olympiad problem, you can use equations in elementary school.

So try to do it with equations.

1.If the denominator and numerator are added with 1, the difference remains unchanged; If the denominator and numerator are reduced by 1, the difference will not change; And the former denominator is better than the latter.

The denominator is more than 2. 8-7=1， 7-6=1；8-7=1, 2 1=2, 8 2=16, 7 2=14,16-1=15, 14-1=13, then the previous score is 13 15.

2.124 4 5 = 155, if the numerator of this fraction is reduced by 124 and the denominator is reduced by 155, this fraction is still 4 5;

The actual denominator is reduced by 11, and 155-11=144 is subtracted, so that the denominator is 9-5=4 more parts, 144 4=36··· Number of servings.

36 4=144··· The new fraction of the numerator, 36 9 = 324 · The denominator of the new score.

144+124=268··· The numerator of the original fraction, 324 + 11 = 335 · The denominator of the original fraction.

The original score was 268 335

1.Think of the numerator denominator plus 1, the numerator denominator is subtracted by 1, if you don't divide the two fractions you should get, the numerator denominator of the two fractions should be 2, because 7 8 and 6 7 are 1 different from the denominator, so the two fractions of the numerator and denominator are expanded by 2 times at the same time, so that 14 16 and 12 14 are 2 different from the numerator denominator, and it is obvious that the required number is 13 15

8-7=1， 7-6=1；8-7=1, 2 1=2, 8 2=16, 7 2=14,16-1=15, 14-1=13, then the previous score is 13 15.

2.124 4 5 = 155, if the numerator of this fraction is reduced by 124 and the denominator is reduced by 155, this fraction is still 4 5;

The actual denominator is reduced by 11, and 155-11=144 is subtracted, so that the denominator is 9-5=4 more parts, 144 4=36··· Number of servings.

36 4=144··· The new fraction of the numerator, 36 9 = 324 · The denominator of the new score.

144+124=268··· The numerator of the original fraction, 324 + 11 = 335 · The denominator of the original fraction.

The original score was 268 335

Observations 7 8=14 16=21 24=28 32=....

Obviously, only the second pair fits the topic, and the numerator and denominator at the end are more different and untitled, so the score is 13 15

The second question is not applicable in this way.

2x+7x=63 x=7 This fraction is 14 49 ...49/14 ..63/(2+7)*2=14 63-14=49 14/49 ..Let the numerator be x and the denominator be y x y=2 7

2. a÷b÷c＝a÷（b×c）

Example: Use the simple method to solve (1)4000 125 8 (2)981+5 9810+49 981

3.What number does δ represent in each of the following equations?

4.The father is 32 years old and the son is 5 years old, and in a few years, the father is exactly 4 times the age of the son.

5.A good horse walks 240 kilometers a day, a bad horse walks 150 kilometers a day, a bad horse walks first for 12 days, how many days can a good horse catch up with a bad horse?

6.There are 570 computers in all primary schools in the district, of which 5 primary schools have 30 computers per school and the rest have 20 computers per school.

7.There are 70 pieces of candy, and if the first child gets twice as much as the second child, and the second child gets twice as much as the third child, there are only 7 pieces of candy left. Q: How many pieces of candy does each child get?

8.Xiao Zhang scored 85 points in Chinese in the final exam, and 5 points more in mathematics than English, so how many points do you need to score in English, and the average score of the 3 subjects can reach 90 points?

9.In the 4th grade math test, 1 person scored 98 points, 3 people scored 92 points, 4 people scored 86 points, and 2 people scored 76 points, what is their average score?

10.If the average of A and B is 6, what is the average of A+1 and B+3?

11.The average number of the three numbers of A, B and C is 150, the number A is 48, and the number C is equal to the number B, what is the number B?

Draw a regular 17-sided shape with an ungraduated ruler and a compass. (You certainly can't do it).

1. There is a reservoir equipped with 9 water pipes, of which 1 is the water inlet pipe, and the remaining 8 are the same water outlet pipes The water inlet pipe starts to keep flowing into this reservoir at a uniform speed After injecting some water into the tank, some people want to open the water outlet pipe to make the water in the pool drain all the water If all the 8 outlet pipes are opened, it takes 3 hours to drain the water in the pool; If you only open 3 outlets, it will take 18 hours Q: If you want to drain all the water in the pool within 8 hours, how many outlets do you need to open at least?

2. The clock of a factory is slowing down, so that the standard time coincides with the hour hand every 70 minutes Master Li works according to this slow clock for 8 hours, and the factory stipulates that the overtime wage is 3 5 times more than the original wage, and the original salary of Master Li is 3 yuan per hour, and the factory should pay Master Li an overtime salary of yuan on this day

You can go to the Olympiad online to find it, there are many free ones.

Column equation: Let this number be x

Solution: x=