活动庆典
19921Find the sum of all positive rational numbers that are less than 10and that have denominator 30when written in lowest terms.2A positive integer is called ascending if,in its decimal reprentation,there are at least two digits and each digit is less than any digit to its right.How many ascending positive integers are there?3A tennis player computes her win ratio by dividing the number of matches she has won by the total number of matches she has played.At the start of a weekend,her win ratio is exactly .500.During the weekend,she plays four matches,winning three and losing one.At the end of the weekend,her win ratio is greater than .503.What’s the largest number of matches she could’ve won before the weekend began?4In Pascal’s Triangle,each entry is the sum of the two entries above it.In which row of Pascal’s Triangle do three concutive entries occur that are in the ratio 3:4:5?
[img]6590[/img]5Let S be the t of all rational numbers r ,0<r <1,that have a repeating decimal expansion in the form =0.abc ,where the digits a ,b ,and c are not necessarily distinct.To write the elements of S as fractions in lowest terms,how many different numerators are required?6For how many pairs of concutive integers in {1000,1001,1002,...,2000}is no carrying required when the two integers are added?7Faces ABC and BCD of tetrahedron ABCD meet at an angle of 30◦.The area of face ABC is 120,the area of face BCD is 80,and BC =10.Find the volume of the tetrahedron.8For any quence of real numbers A =(a 1,a 2,a 3,...),define ∆A to be the quence (a 2−a 1,a 3−a 2,a 4−a 3,...)
,who n th term is a n +1−a n .Suppo that all of the terms of thet quence ∆(∆A )are 1,and that a 19=a 92=0.Find a 1.9Trapezoid ABCD has sides AB =92,BC =50,CD =19,and AD =70,with AB parallel to CD .A circle with center P on AB is drawn tangent to BC and AD .Given that AP =m n ,where m and n are relatively prime positive integers,find m +n .10Consider the region A in the complex plane that consists of all points z such that both z 40and 40z have real and imaginary parts between 0and 1,inclusive.What is the integer that is
nearest the area of A ?11Lines l 1and l 2both pass through the origin and make first-quadrant angles of π70and π54radians,respectively,with the positive x-axis.For any line l ,the transformation R (l )produces
散点图
another line as follows:l is reflected in l 1,and the resulting line is reflected in l 2.Let /This file was downloaded from the AoPS Math Olympiad Resources Page
Page 1
1992
R(1)(l)=R(l)and R(n)(l)=R
R(n−1)(l)
.Given that l is the line y=19
92
x,find the smallest
positive integer m for which R(m)(l)=l.
12In a game of Chomp,two players alternately take bites from a5-by-7grid of unit squares.To take a bite,a player choos one of the remaining squares,then removes(”eats”)all squares in the quadrant defined by the left edge(extended upward)and the lower edge(extended rightward)of the chon square.For example,the bite determined by the shaded square in the diagram would remove the shaded square and the four squares marked by×.(The squares with two or more dotted edges have been removed form the original board in previous moves.) [img]6591[/img]
The object of the game is to make one’s opponent take the last bite.The diagram shows one of the many subts of the t of35unit squares that can occur during the game of Chomp.
How many different subts are there in all?Include the full board and empty board in your count.
13Triangle ABC has AB=9and BC:AC=40:41.What’s the largest area that this triangle can have?
童年的春节14In triangle ABC,A ,B ,and C are on the sides BC,AC,and AB,respectively.Given that
AA ,BB ,and CC are concurrent at the point O,and that AO
OA +BO
OB
+CO
OC
可汗学院
=92,find
AO OA ·BO
OB
jtag
·CO
黑米红豆粥的功效OC
.
15Define a positive integer n to be a factorial tail if there is some positive integer m such that the decimal reprentation of m!ends with exactly n zeroes.How many positive integers less than1992are not factorial tails?
上行文格式标准模板>惠州特色美食/
Thisfile was downloaded from the AoPS Math Olympiad Resources Page Page2
