trường thcs hoàng xuân hãn

O , and PA,PB,PC are perpendicular to each other, and the height of the triangle pyramid P-ABC is h.. Find the value of.[r]

2017 WMTC 青年组个人赛第一轮

Advanced Level Individual Round 1

1.Knownn is a positive integer number, and is between nn24 and nn13 Findn

2.Known function ( ) 1 1e e  ,

x x

f x     x x   xR

find minimum of f(x)

3.The vertex A,B of a square are also the vertex of an ellipse M, and the vertex C,Dof this square are the focus ofM Find eccentricity of M

4.Known series {an} satisfy

1 , ;

2

2 1, n n n n n a a a a a             If

a  ,then a2017=?

5.The area of the right triangle ABC is

2

1 Find the minimum value

of the triangle’s perimeter

6.Suppose that [x] is the largest integer not greater than x Find the value of      lg1  lg  lg3  lg 2017

7.Find the value of tan 25 tan 25 tan 402   .

8.KnownABCD﹣A1B1C1D1is a cuboid, andAB=BC=2,AA1=1, is

2017 WMTC 青年组个人赛第二轮

Advanced Level Individual Round 2

9.Suppose xn(nN) is a positive integer, ifxn+2=xn+xn+1, x6=61, x1is

a prime number Find the maximum value ofx1

10.The point Mon the ellipse 2

x  y  ,and the point F is right

focus of this ellipse, point P(2,1) Find the minimum value of |MF|+|MP|

11.KnowABCD﹣A1B1C1D1is a cuboid, ifABAD=12, ABA A1=36,

ADAA1=48.If points A,B,C,D,A1,B1,C1,D1 are on the sphere O,find the

surface area of sphereO

12.If x and y are integer, and x3 6x2 5x y  y Find the number of (x,y)

2017 WMTC 青年组个人赛第三轮

Advanced Level Individual Round 3

13.Known the point P is on the unit circle O, A1A2  A2017 is

inscribed regular 2017 polygon of circle O Find the value of

2 2

1 2017

PA PA   PA .

14.Known x1=2, and 1

1  

 n

n n xx

x (n=1,2,3, ) Find integer part of

2017

1

n n n

x x

 



2017 WMTC 青年组接力赛第一轮

Advanced Level Relay Round 1

1-A n

S is the sum of first n number of arithmetic progression {an}, if

1

a  ， the difference between any two adjacent number is 2, and

2 36

m m

S  S  , find m

2017 WMTC 青年组接力赛第一轮

Advanced Level Relay Round 1

1-B

LetTbe the number you will receive

Known triangle pyramid P-ABC,PA=PB=PC,AB=BC=CA,∠BPA

=∠APC =∠CPB = 90°,area of △APB is T Find area of circumscribed

sphere ofP-ABC

2017 WMTC 青年组接力赛第二轮

Advanced Level Relay Round 2

2-A

Suppose A is the region enclose of equation x   2 x y 3 .

Find area ofA

2017 WMTC 青年组接力赛第二轮

Advanced Level Relay Round 2

2-B

LetTbe the number you will receive

The radius of the sphere O is T, the point P,A,B,C are on the sphere

O, and PA,PB,PC are perpendicular to each other, and the height of the triangle pyramid P-ABCis h Find maximum of theh

2017 WMTC 青年组接力赛第三轮

Advanced Level Relay Round 3

3-A

Suppose ab0，a b ab  ，Find minimum value of

1

a b

a  b

 

2017 WMTC 青年组接力赛第三轮

Advanced Level Relay Round 3

3-B

LetTbe the number you will receive

Point F is the focus of parabola ᖄ썈ɦᖄ ߅ ⁀斦, the line whose angle of inclination is 60° intersect parabola at point A,B, if area of

△OABisT Find the value ofp.(O is the origin of coordinate system)

2017 WMTC 青年组团体赛

Advanced Level Team Round

1.Suppose an 2n1(n=1,2,3, ) Calculate:

1 2 3 4 17 18

a a a a a a a a a a .

2.Known function ( ) 2 2

x x f x   

 ,when -4≤x≤4, M and m is the maximum and minimum of f x( )，respectively Find M+m

3.ABCD A B C D 1 1 is a cuboid, point E is on the edge B1D1,point F is on the AE, and AF=2FE If AB=4, AD=2, AA1=3 Find the volume of

tetrahedronBDEF

4.Known real number a,b,c, and

      4, 4, 4 a b b c c a           

find the value of

a+b+c

5.Known function f x( )ax2 bx c ,when d xd2,

1 )

(    f x e

e , a,b,c,d,e are real numbers Find the maximum value of a

6.Known AB is diameter of circle O, point C is on the circle O,

CDAB Find the possible thatAD,BD,CD can make an acute triangle.

7.Supposex is a real number, continuous function f(x) satisfy: (1) f x 3 f x 3;

(2)Ifx1≠x2, then f x 1  f x 2 Find the value of f  1 f 12f 0 .

8.Known a,b,c,d are distrinct positive integers Find the minimum value of a b c d  abcd

9.Solve the equation:

1

x x

x x

x x

   

 

10.Known arithmetic sequence { }an and { }bn , Sn and Tnis the sum from a1 to an,b1 to bn, respectively, and n 23 14

n

S n

T n

 

6 a b

11.Letab>0, anda+2b=1 Find the maximum value of a ab

12.KnownA(-3,0),B(-1,-2), point Cis on ellipse 2

x  y  .Find the minimum value of the area of△ABC

13.Suppose positive integer an and 190 are coprime, and

a1<a2<a3<<an Finda2017

14.Point A,B,C,D are not in the same plane, and AB=1,BC=2,CD=3,

ACBD Find the length of AD.

15.Suppose function f x( )x2 2ax a 21,if there is only onexcan make f f x( ( )) 0≤ ,find the range ofa.

16.Known the line l x y:   6 and the parabola C y: 2px intersect atA,B, pointFis the focus of C, if FA FB  0, find the value

17.Known the point Pis on the image of the function yex,the image is tangent linel at the pointP , and the linel, linex=1,x=2 and x

axis form a trapezoid Find the maximum of area of the trapezoid

18.The football is made of xpieces of the same regular pentagon and

y pieces of the same regular hexagon, andx+y=32 Find the value ofx

19.Known a,b,c are all positive numbers, and

b a b c b a a    2

≤3

2 Find the maximum value ofc

20.Suppose function   2017

1 ! k x f x k        

 , x is the largest integer less than x If the equation f x n (1n2017and n is an even number) has solution Find out how many ncan make it

（

2017

1 ! 1! 2! 2017! k

x x x x

2017WMTC Advanced Level Individual Rounds

1 2 3 4 5 6 7

3

2

2

7 2 4944

8 9 10 11 12 13 14

1

3 17 507 4034

Relay Rounds

Team Round

1 2 3 4 5 6 7 8 9 10 687 6 2 125 12

11 12 13 14 15 16 17 18 19 20

2

4

 3 5321 6 { 2} 1

e 12 587

1-B 2-B 3-B