Mathematics Olympiad questions for junior high school, Mathematics Olympiad questions for junior hig

S ae*af*sin angle BAF 2

Angle BAF = 90° - Angle BAE - Angle DAF = 90° - 30° - 15° = 45° AE = AB cos30°

af=ad/cos15°

These can then be substituted into the area formula.

The area of the square minus the area of the three small triangles.

3-(3*tan(15)/2+sqrt(

2+(sqrt(

1-tan())sqrt(

Let's find the area of the three triangles outside.

Then subtract the area of the square.

daf, dac are proportional to the edge df, edge dc = 3, so 15° 45° = df dc gives df = 3 3

In the same way, eab, bac, side be, side bc correspond to proportional, 30° 45° = be bc to get be = 2 3 3

The area of AEF = square ABCD- ADF- EFC- ABE- The area of ABCD in a square row is 3, ADF=AD*DF 2=1 2, EFC=EC*CF 2=1 3, ABE=AB*BE 2=1

So aef=3-1 2-1 3-1=7 6

Extend the extension line of AE intersection BC at HSquare abcd, ad=dc,ad bc, dae= h.E is the DC midpoint, de=ec

aed=∠ceh,∴△ade≌△ceh,∴ad=ch,∴dc=ch.dc+cf=af, af=ch+cf=hf, fae= h, fae= dae, ae bisected daf

2.Knowing a*a-a=3, b*b-b=3, and a is not equal to b, try to find the value of a-b.

3.Knowing that a,b satisfies the equation x=a*a+b*b+20, y=4(2b-a), then what is the magnitude relationship between x,y?

4.Knowing that x,y satisfies x*x+y*y+5 4=2x+y, find the value of the algebraic formula xy x+y.

1.If the number of stone pillars on the ground floor is x, then the middle six layers are 2x on each floor, for a total of 12x.

12x+x+12<=250,x<,, assuming x=15, then 12x+x+12=207, which is in line with the topic. So 207.

2.Young couples 22,17, middle-aged couples,31,34, and older couples 98,97 can be solved as x 2-y 2 = 195, 195 = 5 * 39 = 3 * 65 = 1 * 195

When x-y = 5, 3, 1, x + y = 39, 65, 195 solve the equation. 3.Probably, it should be on top of a ten-meter-high building. The height must be greater than 10 meters after being thrown into the sky, so the cup has not yet hit the ground when it falls 10 meters, and it is still falling.

Intercept dg=be on the edge DC

Triangle ABE and triangle ADG congruence.

So angular gad=angular bae=30

And because the angle daf=15

So angular gaf=15

So the angular triangle daf and the triangle haf are congruent.

So the angle AFD = angle AFH = 75

So the angular EFC 30

Because ab root number 3

So be 1

EC (root number 3) 1

fc 3 root number 3

df (2 times root number 3) 3

So the area of the triangle abe (root number 3) 2

The area of the triangular ECF (2 times the root number 3) 3

The area of the triangle FAD (6 3 times the root number 3) 2 i.e.: the area of the triangle AEF The area of the square The area of the triangle ABE The area of the triangle ECF The area of the triangle FAD.

3 (root number 3) 2 (2 times root number 3) 3 (6 2 times root number 3) 23 root number 3

Obviously, a is not equal to 0, so a-3+1 a=0, the original formula =2a —5a—2+1 a=2(a —3a+1)+(a-3+1 a)-1=-1

If you fold the two points of bc along the de, df respectively, then b

c The two points will coincide. Other self-analysis, hint: (the answer is related to the size of the angle a) after b is bf parallel to de and ac is f, then afb=90°+ c, bfc=90°- c, fbc=90°+ c- c= bfc

So bc=cf=4, so af=3, so ae=, so ce=

Intercept dg=be on the edge DC

Triangle ABE and triangle ADG congruence.

So angular gad=angular bae=30

And because the angle daf=15

So angular gaf=15

So the angular triangle daf and the triangle haf are congruent.

So the angle AFD = angle AFH = 75

So the angular EFC 30

Because ab root number 3

So be 1

EC (root number 3) 1

fc 3 root number 3

df (2 times root number 3) 3

So the area of the triangle abe (root number 3) 2

The area of the triangular ECF (2 times the root number 3) 3

The area of the triangle FAD (6 3 times the root number 3) 2 i.e.: the area of the triangle AEF The area of the square The area of the triangle ABE The area of the triangle ECF The area of the triangle FAD.

3 (root number 3) 2 (2 times root number 3) 3 (6 2 times root number 3) 23 root number 3

Let the bottom stone pillar be x times that of 5, then.

The total number of stone pillars = 5x + 60x + 12 = 65x + 12 and the total number of stone pillars is between 200 250, i.e. 200 65x + 12 250

188≤65x≤238

And x is an integer, so x = 3, so the exact number of stone pillars in the leaning tower = 65 3 + 12 = 195 + 12 = 207 (roots) Answer: The exact number of stone pillars in the leaning tower is 207.

If the bottom layer is 5m, then each layer in the middle has 10m roots (m is the natural number) 200 5m+6*10m+12 250

188＜65m≤238

188/65＜m≤238/65

m = 3, the exact number of stone pillars in the leaning tower = 65m + 12 = 207.

Let the bottom stone pillar tree be x, then 200<12x+x+12<250, solution 188 13 total number of stone pillars = 13 * 15 + 12 = 207