20011The sum of two numbers is S .Suppo 3is added to each number and then each of the resulting numbers is doubled.What is the sum of the final two numbers?
(A)2S +3(B)3S +2(C)3S +6(D)2S +6(E)2S +122Let P (n )and S (n )denote the product and the sum,respectively,of the digits of the integer n .For example,P (23)=6and S (23)=5.Suppo N is a two-digit number such that N =P (N )+S (N ).What is the units digit of N ?
(A)2(B)3(C)6(D)8(E)93The state income tax where Kristin lives is levied at the rate of p %of the first $28000of annual income plus (p +2)%of any amount above $28000.Kristin noticed that the state income tax she paid amounted to (p +0.25)%of her annual income.What was her annual income?
(A)$28000(B)$32000(C)$35000(D)$42000(E)$560004The mean of three numbers is 10more than the least of the numbers and 15less than the greatest.The median of the three numbers is 5.What is their sum?
(A)5(B)20(C)25(D)30(E)365What is the product of all odd positive integers less than 10000?
(A)10000!
(5000!)2(B)10000!
25000(C)9999!
25000(D)10000!
25000·5000!(E)5000!
250006A telephone number has the form ABC −DEF −GHIJ,where each letter reprents a different digit.The digits in each part of the numbers are in decreasing order;that is,A >B >C ,D >E >F ,and G >H >I >J.Furthermore,D ,E ,and F are concutive even digits;G ,H ,I ,and J are concutive odd digits;and A +B +C =9.Find A.
(A)4(B)5(C)6(D)7(E)87A charity lls 140benefit tickets for a total of $2001.Some tickets ll for full price (a whole dollar amount),and the rest lls for half price.How much money is raid by the full-price tickets?
(A)$782(B)$986(C)$1158(D)$1219(E)$14498Which of the cones can be formed from a 252◦ctor of a circle of radius 10by aligning the two straight sides?
(A)A cone with slant height of 10and radius 6(B)A cone with height of 10and radius 6
(C)A cone with slant height of 10and radius 7(D)A cone with height of 10and radius 7(E)A cone with slant height of 10and radius 8This file was downloaded from the AoPS −MathLinks Math Olympiad Resources Page Page /
2001
9Let f be a function satisfying f(xy)=f(x)/y for all positive real numbers x and y.If f(500)=3,what is the value of f(600)?
(A)1(B)2(C)5
2
(D)3(E)
18
5艺术节活动方案
10The plane is tiled by congruent squares and congruent pentagons as indicated.The drawing below illustrates one square.The percent of the plane that is enclod by the pentagon is clost to
(A)50(B)52(C)54(D)56(E)58
11A box contains exactlyfive chips,three red and two white.Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn.What is the probability that the last chip drawn is white?
(A)3
10
(B)
2
5
(C)
1
2
(D)
3
5
(E)
7
10
宝宝舌头发紫图片12How many positive integers not exceeding2001are multiple of3or4but not5?
(A)768(B)801(C)934(D)1067(E)1167
13The parabola with equation y=ax2+bx+c and vertex(h,k)is reflected about the line y=k.This results in the parabola with equation y=dx2+ex+f.Which of the following equals a+b+c+d+e+f?
(A)2b(B)2c(C)2a+2b(D)2h(E)2k
14Given the nine-sided regular polygon A1A2A3A4A5A6A7A8A9,how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the t{A1,A2,...A9}?
(A)30(B)36(C)63(D)66(E)72
15An inct lives on the surface of a regular tetrahedron with edges of length1.It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge.What is the length of the shortest such trip?(Note:Two edges of a tetrahedron are opposite if they have no common endpoint.)
(A)1
2
√
3(B)1(C)
竹荪鸡汤禁忌
√
2(D)
3
鹊山2
(E)2
16A spider has one sock and one shoe for each of its eight legs.In how many different orders can the spider put on its socks and shoes,assuming that,on each leg,the sock must be put on before the shoe?
(A)8!(B)28·8!(C)(8!)2(D)16!
28
(E)16!
2001
17A point P is lected at random from the interior of the pentagon with vertices A=(0,2),B= (4,0),C=(2π+1,0),D=(2π+1,4),and E=(0,4).What is the probability that∠AP B is obtu?
(A)1
桑螵蛸的功效与作用
5
(B)
1
4
(C)
5
16
(D)
3
8
(E)
1
2
18A circle centered at A with a radius of1and a circle centered at B with a radius of4are externally tangent.A third circle is tangent to thefirst two and to one of their common external tangents as shown.The radius of the third circle is
(A)1
3
(B)
2
5
(C)
5
12
(D)
什么是本科提前批4
9
(E)
1
韦后2
19The polynomial P(x)=x3+ax2+bx+c has the property that the mean of its zeros,the product of its z
eros,and the sum of its coefficients are all equal.If the y-intercept of the graph of y=P(x)is2,what is b?
(A)−11(B)−10(C)−9(D)1(E)5
20Points A=(3,9),B=(1,1),C=(5,3),and D=(a,b)lie in thefirst quadrant and are the vertices of quadrilateral ABCD.The quadrilateral formed by joining the midpoints of AB,BC,CD,and DA is a square.What is the sum of the coordinates of point D?
(A)7(B)9(C)10(D)12(E)16
21Four positive integers a,b,c,and d have a product of8!and satisfy
ab+a+b=524
bc+b+c=146,and
cd+c+d=104.
What is a−d?
(A)4(B)6(C)8(D)10(E)12
22In rectangle ABCD,points F and G lie on AB so that AF=F G=GB and E is the midpoint of DC.Also,AC intercts EF at H and EG at J.The area of the rectangle ABCD is70.Find the area of triangle EHJ.
(A)5
2
(B)
35
12
(C)3(D)
7
2
(E)
西游记女儿国国王扮演者
35
8
23A polynomial of degree four with leading coefficient1and integer coefficients has two zeros, both of which are integers.Which of the following can also be a zero of the polynomial?
(A)1+i
√
11
2
(B)
1+i
2
(C)
1
2
+i(D)1+
i
2
(E)
1+i
√
13
2
200124In ABC ,∠ABC =45◦.Point D is on BC so that 2·BD =CD and ∠DAB =15◦.Find
∠ACB .
(A)54◦(B)60◦(C)72◦(D)75◦(E)90◦25Consider quences of positive real numbers of the form x,2000,y,...,in which every term
after the first is 1less than the product of its two immediate neighbors.For how many different values of x does the term 2001appear somewhere in the quence?
(A)1(B)2(C)3(D)4(E)more than 4
