trường thcs hoàng xuân hãn

Advanced Level Relay Round 3. 3-B[r]

2018 WMTC

青年组个人赛第一轮 Advanced Level Individual Round 1

1.SupposeAx x 3,x Z , Bx x A  ,2x2A,then B={ }.

2.If 3a 5,5b 7,and 7c 9, then the value of abc is .

3.Use [x] to represent the largest integer that is not larger than x For example,  2 2, 3.1   4,[ ] 3  ,then

[ 1] [ 2] [ 3]    [ 150]＝

4.In△ABC, if tanAtanC 3 tan tanA C，then B= °

5.Suppose y f x ( ) is an even function and f x( ) f x(  4) for any

xR, then the value of f 2018 is

6.Give a sequence  an such that n 2 n 1

n

a a

a

   ，and a2018 1,then the

value of a2000 is

7.Fig.1 is a 5×4 grid that consists of 20 1×1 squares SupposemAC3AB(R)，and m AB，then the value of λ is

Fig

8.In the Fig.2, supposeAC=8,BC=6 and ∠C=60° in △ABC, point E on

AC, point F on BC If SCEF 13SABC , then the minimum value of EF

is

2018 WMTC

青年组个人赛第二轮 Advanced Level Individual Round 2

9.If  4

0

2

x  x a a x a x    a x ，then

  2 2

0

a a a a a     a a a a   

10.In a rectangular box ABCD A B C D 1 1, AB=BC=2 and AA1  If

the angle of the straight line AD1 and DB1 is , then cos 

11.Given that { }an is an arithmetic (equal difference) sequence If a a a a2  6 7 12 51 and a a3: 121: ,then

2 98 100

a a a    a a = .

12.Suppose 0,

  

 ,then the minimum value of

3

sin  cos  sin

is

2018 WMTC

青年组个人赛第三轮 Advanced Level Individual Round 3

13.SupposeF1 and F2 are both foci for the hyperbolaC: x2 a2 

y2 b2 1,

and that an asymptote of C is perpendicular to line y

3 x2 If a point AonCsatisfies F A1  32F A2 , then cosAF1F2 =

14.Suppose the function f x( ) a a a x( 0,a1)has  0，1 as both its

domain and range, and the ABCD is a regular tetrahedron with edge length of a, then the radius of the circumscribed sphere of the tetrahedron ABCD is

2018 WMTC

青年组接力赛第一轮

Advanced Level Relay Round 1

1-A

If the inequality log (3 e )| |

2 x

a   holds for any real number x, then the value range forais

2018 WMTC

青年组接力赛第一轮

Advanced Level Relay Round 1

1-B

LetTbe the number you will receive

Solve the inequality 2log (a x 1) log log (3a  a x) using value of a fromT

2018 WMTC

青年组接力赛第二轮

Advanced Level Relay Round 2

2-A

If ( ) e e

x x

f x 

 , then f( 6)  f( 4)  f( 2)  f(2) f(4) f(6)

2018 WMTC

青年组接力赛第二轮

Advanced Level Relay Round 2

2-B

LetTbe the number you will receive

Suppose a b T where a and b are positive real numbers, then the

minimum possible value of a25 b23

a b is

2018 WMTC

青年组接力赛第三轮

Advanced Level Relay Round 3

3-A

If a sequence { }an is defined as a13, and n 11 n

n

a a

a 

 

 , then

101

a 

2018 WMTC

青年组接力赛第三轮

Advanced Level Relay Round 3

3-B

LetTbe the number you will receive If tan tan

7



 T , then

5 cos

14 sin

7

 

 

  

 

 

  

 

 

=

2018 WMTC

青年组团体赛

Advanced Level Team Round

1.If ( )

4

x x

f x 

 ,then

1 2018

2019 2019 2019

f  f    f  

      = _ 2.Suppose that real numbers x and y satisfy x2y2 1 , then the

maximum value of x24xy y is _

3.Given two sets A{( , ) |x y xy1,x0} and B{( , ) |x y x4y a } If

A B  ,then the value range forais _

4.Suppose M max ,5  x x3,2x and xR ,then the minimal value ofMis _ (Note:max , ,a b c represents the largest of a b c, , )

5.The root of the equation 3

3 1

3   

 x

x isx = _

6.Suppose the lengths of the three sides that are opposite to the three interior angles A, B, and C of △ABC are a, b, and c, respectively.

Suppose

b c  a, and 2sinB3sinC, then the value of cosAis _.

7.The area of the region on the xy-plane over which P ranges , P( , )x y x  1 y 1,xR,yR , is _

8.The number of roots of equation 4 x2 | ln |x 1|| 0 is _

the formula for an  _

10.Let Gbe the center of gravity of△ABC, if GB⊥GC,andBC 2, then the maximum area of△ABCis _

11.In△ABC, letObe the midpoint ofBC If

10

AB ,AC 7,

2

AM  MB

 

,

3

AN  NC

 

, and MO ON  =0, then

cosA= _

12.There are four points A, B, C, and D on the sphere with radius 4, and they satisfy BAC90，CAD90，and DAB90.Let Srepresents the area of the triangle, then the maximum possible value of S△ABC+S△ACD+S△ADB is _

13.If real numbers x and y satisfy

7

3

3

x y x y x y               ， and

(y x y x)( )

z

xy

 

 ,then the value range forzis _

14.It is known that the trilateral lengths a, b, c of △ABC satisfy

, ,

bc ca ab in an equal difference sequence If a c 4, when the area of

△ABCreaches its maximum, the perimeter of△ABCis _

15.Suppose m n2  20and x2y2 18 where m, n, x and y are real

numbers, then the maximum possible value of mx ny is _.

16.The volume of the triangular prism ABC A B C ' ' ' is 1, and '

17.The sum of all real roots of the equation

  

6 1 1 2 4

x x   x  x x  x is _

18.Suppose AB BC 5, and AC 8 in △ABC, and O is the incenter of △ABC If AO mAB nAC m n  ( , R) , then the value of m

n

is _

19.Let Sn be the sum of the first n terms of arithmetic sequences Suppose and Sm2 3 for any integer m, then

_

20 Suppose the function f x  x22x1 has [a，b] as both its

2018WMTC Advanced Level

Individual Rounds

1 2 3 4 5 6 7

-3,3 1162 60 12 351 1

8 9 10 11 12 13 14

4 144 3913 2850 3227

4

6

Relay Rounds

Team Round

1 2 3 4 5 6 7 8 9 10

1009 (-∞,4) 12 log 43 14 4 6 3

n

3

11 12 13 14 15 16 17 18 19 20

3

8 32

21 7, 10 12  

 

  6 10

2

3 -1

8

5 2015 -3

1-B 2-B 3-B

( 1,1)