Math Olympiad questions. There is a reward for satisfaction, and there is a reward for math Olympiad

There is a bunch of balls, 5 out of 5 in a box, 2 less, 6 in a box, 3 less, 10 out of 10 in a box, 7 less, how many balls are there in this pile?

This pile of balls - 3, which is a multiple of , and the minimum multiple is 30. There are at least 30 + 3 = 33 balls in this pile.

A bag of small butter sugar, there are 4 pieces left in a pile of 7 pieces, 3 pieces less in each pile, and every 3 pieces in a pile just count, how many pieces of sugar are there in this bag at least?

3 less in each pile"?

March 12 is Arbor Day, Mr. Zhang led the students to line up in two equal columns to plant trees, and it is known that the trees planted by the teacher are the same as the trees planted by each student, and a total of 111 trees were planted. Q: How many people are there? How many trees?

The question is not rigorous, there is a problem.

The numerator of a true fraction is expanded by 3 times, and the denominator is reduced by 2 times to 17/4, what is the original fraction?

The numerator is expanded by 3 times, the denominator is reduced by 2 times, and the original fraction is expanded by 3*2=6 times. It turns out that the true fraction = (17 4) 6 = 17 24

Note: Are they all original questions or did you make them up yourself?

Question 1 Divide 5 Remainder 3 Divide 6 Remainder 3 Divide 10 Remainder 3 Minimum is 33 Question 2 Less conditions.

Question 3 Didn't you say that you planted 111 trees, how many trees did you ask, 37 people, 3 trees per person.

Question 4 16 17

Let the whole book have n pages, and the torn page numbers are x (the title means 2 digits), and the sum of all the page numbers is:

n(n+1)/2

The remaining page numbers add up to 1993

Then n(n+1) 2>1993

n(n+1)>3986

n>62 takes n=63

So the page number that was torn off is 23

If the book has n pages, then the sum of the page numbers should be slightly larger than 1993(1 n)n 2 > 1993

1+n )n > 3986

Since 64 65 = 4160, i.e. 64 65 1 2 = 2080n = 64

That is, the book is 64 pages long.

2080-1993 = 87 = 43 44 So the book has 64 pages, and the torn one is numbered 43 and 44

Analysis: There are 40 more science books than story books.

There are 4 fewer storybooks than ***.

So there are more science and technology books than ***: 40-4 = 36 books.

It is also known that there are 6 more science and technology books than ***.

That is, 2 times the number of *** plus 6 equals the number of *** plus 36*** has: (40-4-6) (2-1) = 30 storybooks: 30-4 = 26.

Science and technology books: 26 + 40 = 66 books.

The number of storybooks is 4 less than ***, in other words.

There are 4 more books than storybooks.

There are 40 more books of science books than storybooks, so.

There are 36 more science and technology books than ***.

It is also known that the number of science and technology books is more than 6 times that of ***.

So 36 is 1 times more than 6 copies.

of the book = 36-6 = 30

So there are 66 science and technology books.

26 storybooks.

40—4=36

36—6=30（***）

30 + 36 = 66 (science and technology book).

66—40=26 (storybook).

Science and technology books x, then story books x-40, *** x-40 + 4 = x-36 books, and x=2 * (x-36) + 6, the solution is x=66

So there are 66 science and technology books, 26 story books, and 30 books.

*** (40-4-6)/(2-1)=30

Storybook 30-4=26

Science and Technology Book 26+40=66

Grade? Is it okay to use equations?

:54000=2:3

When the car travels 36000*3 5 = 21600 km, the front and rear wheels are replaced, and the front wheels are worn 3 5 and the rear wheels are worn 2 5

It has a driving range of 21,600 km

The maximum distance traveled by the car is 43200 km

minutes for three people to work alone at the same time, the total time of 7 cars stopped remains the same, which is still 123 minutes.

123*11=1353 yuan, loss 1353 yuan.

3.There are 26 floors, the top floor is 25 floors, indicating that the building has a basement, do not take the elevator on the ground 1 floor, the ground 1 floor to the 25th floor is separated by 24 sections, and the elevator stops at 2 3 places, that is, 1 + 24 * 2 3 = 17 floors.

lz indicates whether it is a primary, a secondary school, or a university ==

When they meet, B travels more than A: 12 2 = 24 km.

It is also known that B travels 4 kilometers more per hour than A.

Encounter time: 24 4 = 6 hours.

At the time of the encounter, A traveled: 60-12 = 48 km.

A per hour: 48 6 = 8 km.

12 2 4 = 6 hours of meeting time.

60-12) 6 = 8 (km) The velocity of A.

Bought A, it cost 1000 yuan.

1000 300 = 3 surplus 100

Get 3 gift vouchers of 50 yuan, total:

3 50 = 150 yuan.

B**: 150 + 280 = 430 yuan.

A, B total**: 1000 + 430 = 1430 yuan.

Actual cost: 1000 + 280 = 1280 yuan.

It is equivalent to enjoying a 9% discount.

Let's calculate how much money he shared: 150 yuan in 1000 yuan to the gift certificate, plus 280 yuan later, a total of 1430 yuan 1000 + 150 + 280 = 1430

Then count his own money: 1000 + 280 = 1280

Finally, the discount enjoyed: 1280 1430 is about to enjoy the discount.

Make a straight line parallel to EF through a cross BC with G

Because of AD BC

So ef=ag

ag/ab=sin∠abc

d=ef=ag=ab*sin37°

Is this an Olympiad problem? Am I mistaken?

Let the center of the sphere be O, the sphere is tangent to the plane and G, and the ob, og, oa are because AG, AE, BG, BF are tangent to the ball, and AD CB, so OA is perpendicular ob. Let the radius of the circle be og=r, ob=r sin(b 2) gb=cot(b 2).

ob^2=bg*ba

Algebra calculations can find out.

Circle center o, even oa, ob

Tangent length theorem ab=bf

》△aob=△bof

>> ob deuce ABC

Diameter = 2oa = 2ob * sin abo = 2 * 26 accurate to 1cmsin ( =

dea is not a straight line, the above is all wrong.

The exact answer to the diameter of the ball is, approximately to 1cm, then 28cm for the following reasons:

First of all, make the auxiliary line spherical center to point A line, let the spherical center be O, then OA=of=OE;

The length of ab is.

abc=37°

Connecting the ob, we can see that the ob is the angular bisector of abc.

By tangent tan oba=oa ab=

The diameter of the solved ball is equal to 2*OA= approximate 28cm

Solution: Make a perpendicular line from point A to BC, and the intersection point is denoted as g, then ag = ef is the diameter of the sphere.

So the diameter of the ball = ag = absin37° = * is about 25 cm (accurate to 1cm).

1、x:y:z=2:(-1);1,x/2=y/(-1)=z/1

So x=2z, y=-z

So the original formula = (2z-2z-z) (4z+z+z)=-z 6z

2. Greater than or equal to 0 under the root number

then 4x-1>=0

1-4x>=0, that is, 4x-1<=0

If it is true at the same time, 4x-1=0

So y=0+0=0

So xy=0

Solution: If x:y:z=2:(-1); 1. Find (x+2y-z) (2x-y+z).

x:y:z=2:(-1)：1

Let x=2 y=-1 z=1

x+2y-z)/(2x-y+z)=(2-2-1)/(4+1+1)=-1/6

It is known to be a real number, and y = root number (4x-1) + root number (1-4x), then xy = ?

Any number in the root number is greater than or equal to 0

Root number (4x-1) + root number (1-4x).

4x-1≥0 1-4x≥0

4x-1=0 x=

The two cars met for the first time and traveled together for a full journey, at which time the car from station A traveled 50 kilometers; The two cars met for the second time and traveled together for three whole journeys, at this time the car from station A should have traveled 3 50 = 150 kilometers, and actually traveled more than 30 kilometers for the whole journey.

The whole journey is: 150-30=120 km.

The two stations A and B are x kilometers apart.

50:(x-50)=(x+30):(2x-30)x^2-120x=0

x=120 or x=0 (rounded).

That is, the two stations A and B are 120 kilometers apart.