trường thcs hoàng xuân hãn

<span class='text_page_counter'>(1)</span><div class='page_container' data-page=1>

<b>2017 WMTC</b>

青年组个人赛第一轮

Advanced Level Individual Round 1

1.Known<i>n</i> is a positive integer number, and 2 is between <i>_nn</i>_₂4
and <i>n_n</i>_₁3. Find<i>n</i>.

2.Known function ( ) 1 1 1



e e







,
2

<i>x</i> <i>x</i>

<i>f x</i> _{    }<i>x</i> <i>x</i> _  <i>x</i>_<b>_R</b>

find
minimum of <i>f</i>(<i>x</i>).

3.The vertex <i>A,B</i> of a square are also the vertex of an ellipse <i>M</i>, and
the vertex <i>C,D</i>of this square are the focus of<i>M</i>. Find eccentricity of <i>M</i>.

4.Known series {<i>an</i>} satisfy 1

1
2 , 0 ;

2
1

2 1, 1.
2

<i>n</i> <i>n</i>
<i>n</i>
<i>n</i> <i>n</i>
<i>a</i> <i>a</i>
<i>a</i>
<i>a</i> <i>a</i>

 _ _

 
 _ _ _

If
1
2
7

<i>a</i>  _,then <i>a</i>₂₀₁₇_=?

</div>
<span class='text_page_counter'>(2)</span><div class='page_container' data-page=2>

5.The area of the right triangle <i>ABC</i> is

2

1 _{. Find the minimum value}

of the triangle’s perimeter.

6.Suppose that [<i>x</i>] is the largest integer not greater than <i>x</i> .Find the
value of

     

lg1  lg 2  lg3  



lg 2017



.

7.Find the value of _{tan 25 2 tan 25 tan 40}2 _  _.

8.Known<i>ABCD</i>﹣<i>A</i>₁<i>B</i>₁<i>C</i>₁<i>D</i>₁is a cuboid, and<i>AB</i>=<i>BC</i>=2,<i>AA</i>₁=1, is

</div>
<span class='text_page_counter'>(3)</span><div class='page_container' data-page=3>

<b>2017 WMTC</b>

青年组个人赛第二轮

Advanced Level Individual Round 2

9.Suppose <i>xn</i>(<i>n</i><b>N</b>) is a positive integer, if<i>xn</i>+2=<i>xn</i>+<i>xn+</i>1, <i>x</i>6=61, <i>x</i>1is

a prime number. Find the maximum value of<i>x</i>1.

10.The point <i>M</i>on the ellipse 2 2 1
8 4

<i>x</i> _ <i>y</i> _ _{,and the point} <i>_F</i> _{is right}

focus of this ellipse, point <i>P</i>(2,1). Find the minimum value of
2 |<i>MF</i>|+|<i>MP</i>|.

11.Know<i>ABCD</i>﹣<i>A</i>₁<i>B</i>₁<i>C</i>₁<i>D</i>₁is a cuboid, if<i>AB</i><i>AD</i>=12, <i>AB</i><i>A A</i>1=36,

<i>AD</i><i>AA</i>1=48.If points <i>A</i>,<i>B</i>,<i>C</i>,<i>D</i>,<i>A</i>1,<i>B</i>1,<i>C</i>1,<i>D</i>1 are on the sphere <i>O</i>,find the

surface area of sphere<i>O</i>.

12.If <i>x</i> and <i>y</i> are integer, and <i>x</i>3 6<i>x</i>2 5<i>x y</i> 3  <i>y</i> 2 .Find the
number of (<i>x</i>,<i>y</i>).

</div>
<span class='text_page_counter'>(4)</span><div class='page_container' data-page=4>

<b>2017 WMTC</b>

青年组个人赛第三轮

Advanced Level Individual Round 3

13.Known the point <i>P</i> is on the unit circle <i>O</i>, <i>A</i>1<i>A</i>2  <i>A</i>2017 is

inscribed regular 2017 polygon of circle <i>O</i>. Find the value of

2 2 2

1 2 2017

<i>PA</i> <i>PA</i>  _ <i>PA</i> _.

14.Known <i>x</i>1=2, and ₁
2

1  _


<i>n</i>

<i>n</i>
<i>n</i> <i>_xx</i>

<i>x</i> ₍<i>_n</i>_=1,2,3, __{). Find integer part of}

2017

1 1

<i>n</i>
<i>n</i> <i>n</i>

<i>x</i>
<i>x</i>

 



.

</div>
<span class='text_page_counter'>(5)</span><div class='page_container' data-page=5>

<b>2017 WMTC</b>

青年组接力赛第一轮

Advanced Level Relay Round 1

1-A

<i>n</i>

<i>S</i> _{is the sum of first} <i>_n</i> _{number of arithmetic progression {}<i>_a_n</i>_{}, if}

1 1

<i>a</i>  ， the difference between any two adjacent number is 2, and

2 36

<i>m</i> <i>m</i>

<i>S</i>  <i>S</i>  , find <i>m</i>.

</div>
<span class='text_page_counter'>(6)</span><div class='page_container' data-page=6>

<b>2017 WMTC</b>

青年组接力赛第一轮

Advanced Level Relay Round 1

1-B

Let<i><b>T</b></i>be the number you will receive.

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

<i>=</i>∠<i>_{APC =}</i>∠<i>_CPB</i> _{= 90°,area of} △<i>APB</i> is <i><b>T</b></i>. Find area of circumscribed

sphere of<i>P-ABC</i>.

</div>
<span class='text_page_counter'>(7)</span><div class='page_container' data-page=7>

<b>2017 WMTC</b>

青年组接力赛第二轮

Advanced Level Relay Round 2

2-A

Suppose <i>A</i> is the region enclose of equation <i>x</i>   2 <i>x</i> 1 <i>y</i> 3 _.

Find area of<i>A</i>.

</div>
<span class='text_page_counter'>(8)</span><div class='page_container' data-page=8>

<b>2017 WMTC</b>

青年组接力赛第二轮

Advanced Level Relay Round 2

2-B

Let<i><b>T</b></i>be the number you will receive.

The radius of the sphere <i>O</i> is <i><b>T</b></i>, the point <i>P,A,B,C</i> are on the sphere

<i>O</i>, and <i>PA,PB,PC</i> are perpendicular to each other, and the height of the
triangle pyramid <i>P-ABC</i>is <i>h</i>. Find maximum of the<i>h</i>.

</div>
<span class='text_page_counter'>(9)</span><div class='page_container' data-page=9>

<b>2017 WMTC</b>

青年组接力赛第三轮

Advanced Level Relay Round 3

3-A

Suppose <i>ab</i>0，<i>a b ab</i>  ，Find minimum value of

1 1

<i>a</i> <i>b</i>

<i>a</i>  <i>b</i>

  .

</div>
<span class='text_page_counter'>(10)</span><div class='page_container' data-page=10>

<b>2017 WMTC</b>

青年组接力赛第三轮

Advanced Level Relay Round 3

3-B

Let<i><b>T</b></i>be the number you will receive.

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

△<i>OAB</i>is<i><b>T</b></i>. Find the value of<i>p</i>.(<i>O</i> is the origin of coordinate system)

</div>
<span class='text_page_counter'>(11)</span><div class='page_container' data-page=11>

<b>2017 WMTC</b>

青年组团体赛

Advanced Level Team Round

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

1 2 2 3 3 4 4 5 17 18

<i>a a</i> <i>a a a a</i> <i>a a</i> _<i>a a</i> _.

2.Known function ( ) 2 2 2
2 1

<i>x</i>
<i>x</i>
<i>f x</i>   

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

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

tetrahedron<i>BDEF</i>.

4.Known real number <i>a</i>,<i>b</i>,<i>c</i>, and













4 4,
4 4,
4 4.
<i>a</i> <i>b</i>
<i>b</i> <i>c</i>
<i>c</i> <i>a</i>
 

 _{ }

 _ _


find the value of

<i>a</i>+<i>b</i>+<i>c</i>.

5.Known function <i>_{f x}</i>_{( )}_<i>_ax</i>2 _<i>_{bx c}</i>_ _,when <i>_d</i> _<i>_x</i>_<i>_d</i>_₂_,

</div>
<span class='text_page_counter'>(12)</span><div class='page_container' data-page=12>

1
)

(  
 <i>f</i> <i>x</i> <i>e</i>

<i>e</i> _, <i>_a</i>_,<i>_b</i>_,<i>_c</i>_,<i>_d</i>_,<i>_e</i> _{are real numbers. Find the maximum value of}
<i>a</i>.

6.Known <i>AB</i> is diameter of circle <i>O</i>, point <i>C</i> is on the circle <i>O</i>,

<i>CD</i><i>AB</i>_{. Find the possible that}<i>AD</i>_,<i>BD</i>_,<i>CD</i> _{can make an acute triangle.}

7.Suppose<i>x</i> is a real number, continuous function <i>f</i>(<i>x</i>) _satisfy:
(1) <i>_{f x}</i>

 

3 _



<i>_{f x}</i>

 



3_;

(2)If<i>x</i>1≠<i>x</i>2, then <i>f x</i>

 

1  <i>f x</i>

 

2 .
Find the value of _<i>f</i>

 

 1 <i>f</i>

 

1_2<i>f</i>

 

0 _.

8.Known <i>a</i>,<i>b</i>,<i>c</i>,<i>d</i> are distrinct positive integers. Find the minimum
value of <i>_{a b c d}</i>_{  }<i>abcd</i> .

9.Solve the equation: 2 1

1 1

<i>x</i> <i>x</i>

<i>x</i> <i>x</i>

<i>x</i> <i>x</i>

   

  .

10.Known arithmetic sequence { }<i>an</i> and { }<i>bn</i> , <i>Sn</i> and <i>Tn</i>is the sum
from <i>a</i>1 to <i>an</i>,<i>b</i>1 to <i>bn</i>, respectively, and <i>n</i> 2_{3 1}4

<i>n</i>

<i>S</i> <i>n</i>

<i>T</i> <i>n</i>




</div>
<span class='text_page_counter'>(13)</span><div class='page_container' data-page=13>

6
5
<i>a</i>
<i>b</i> .

11.Let<i>ab</i>>0, and<i>a</i>+2<i>b</i>=1. Find the maximum value of <i>a</i> <i>ab</i>.

12.Known<i>A</i>(-3,0),<i>B</i>(-1,-2), point <i>C</i>is on ellipse 2 2 1
4 3

<i>x</i> _ <i>y</i> _ _.Find
the minimum value of the area of△<i>ABC</i>.

13.Suppose positive integer <i>an</i> and 190 are coprime, and

<i>a</i>1<<i>a</i>2<<i>a</i>3<<<i>an</i>. Find<i>a</i>2017.

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

<i>AC</i><i>BD</i>_{. Find the length of} <i>AD</i>_.

15.Suppose function <i>_{f x}</i>_{( )}_<i>_x</i>2 _₂<i>_{ax a}</i>_ 2_₁_{,if there is only one}<i>_x</i>_can
make <i>f f x</i>( ( )) 0≤ _{,find the range of}<i>_a</i>_.

16.Known the line <i>l x y</i>: 2   6 0 _{and the parabola} <i>_{C y}</i>_: 2 _₂<i>_px</i>
intersect at<i>A</i>,<i>B</i>, point<i>F</i>is the focus of <i>C</i>, if <i>FA FB</i>  0, find the value

</div>
<span class='text_page_counter'>(14)</span><div class='page_container' data-page=14>

17.Known the point <i>P</i>is on the image of the function <i>_y</i>_e<i>x</i>_,the
image is tangent line<i>l</i> at the point<i>P</i> , and the line<i>l</i>, line<i>x</i>=1,<i>x</i>=2 and <i>x</i>

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

18.The football is made of <i>x</i>pieces of the same regular pentagon and

<i>y</i> pieces of the same regular hexagon, and<i>x</i>+<i>y</i>=32. Find the value of<i>x</i>.

19.Known <i>a</i>,<i>b</i>,<i>c</i> are all positive numbers, and

<i>b</i>

<i>a</i>
<i>b</i>
<i>c</i>
<i>b</i>
<i>a</i>
<i>a</i>


 2
2
2

≤3 2

2 .Find the maximum value of<i>c</i>.

20.Suppose function

 

2017

1 !
<i>k</i>
<i>x</i>
<i>f x</i>
<i>k</i>

 
 _{ }
 



,

 

<i>x</i> _{is the largest integer less}
than <i>x</i>. If the equation <i>f x</i>

 

<i>n</i> ₍1<i>n</i>2017and <i>n</i> is an even number)

has solution. Find out how many <i>n</i>can make it.

（

2017

1 ! 1! 2! 2017!
<i>k</i>

<i>x</i> <i>x</i> <i>x</i> <i>x</i>

</div>
<span class='text_page_counter'>(15)</span><div class='page_container' data-page=15>

<b>2017WMTC Advanced Level</b>

<b>Individual Rounds</b>

<b>1</b> <b>2</b> <b>3</b> <b>4</b> <b>5</b> <b>6</b> <b>7</b>

3 3 2

2

2

7 2 2 4944 1

<b>8</b> <b>9</b> <b>10</b> <b>11</b> <b>12</b> <b>13</b> <b>14</b>

1

3 17 2 507 0 4034 1

<b>Relay Rounds</b>

<b>Team Round</b>

<b>1</b> <b>2</b> <b>3</b> <b>4</b> <b>5</b> <b>6</b> <b>7</b> <b>8</b> <b>9</b> <b>10</b>
687 6 2 6 1 5 2 0 12₅ 1₂ 5 1

<b>11</b> <b>12</b> <b>13</b> <b>14</b> <b>15</b> <b>16</b> <b>17</b> <b>18</b> <b>19</b> <b>20</b>

2 6

4

 3 7 ₅₃₂₁ ₆ _{{ 2}}_ ₁ 3
2

e 12 2 587

<b>1-B</b> <b>2-B</b> <b>3-B</b>

</div>

<!--links-->