Difference between revisions of "1985 AHSME Problems"
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<math> x_1=\sqrt[3]{3} </math> | <math> x_1=\sqrt[3]{3} </math> | ||
− | <math> x_2=\sqrt[3]{3}^\sqrt[3]{3} </math> | + | <math> x_2=\sqrt[3]{3}^{\sqrt[3]{3}} </math> |
and in general | and in general | ||
− | <math> x_n=(x_{n-1})^\sqrt[3]{3} </math> for <math> n>1 </math>. | + | <math> x_n=(x_{n-1})^{\sqrt[3]{3}} </math> for <math> n>1 </math>. |
What is the smallest value of <math> n </math> for which <math> x_n </math> is an [[integer]]? | What is the smallest value of <math> n </math> for which <math> x_n </math> is an [[integer]]? | ||
Line 268: | Line 268: | ||
[[1985 AHSME Problems/Problem 27|Solution]] | [[1985 AHSME Problems/Problem 27|Solution]] | ||
+ | |||
==Problem 28== | ==Problem 28== | ||
In <math> \triangle ABC </math>, we have <math> \angle C=3\angle A, a=27, </math> and <math> c=48 </math>. What is <math> b </math>? | In <math> \triangle ABC </math>, we have <math> \angle C=3\angle A, a=27, </math> and <math> c=48 </math>. What is <math> b </math>? |
Revision as of 22:34, 13 March 2015
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
- 30 Problem 30
- 31 See also
Problem 1
If , then
Problem 2
In an arcade game, the "monster" is the shaded sector of a circle of radius cm, as shown in the figure. The missing piece (the mouth) has central angle . What is the perimeter of the monster in cm?
Problem 3
In right with legs and , arcs of circles are drawn, one with center and radius , the other with center and radius . They intersect the hypotenuse at and . Then, has length:
Problem 4
A large bag of coins contains pennies, dimes, and quarters. There are twice as many dimes as pennies and three times as many quarters as dimes. An amount of money which could be in the bag is
$$$$$
Problem 5
Which terms must be removed from the sum
if the sum of the remaining terms is equal to ?
Problem 6
One student in a class of boys and girls is chosen to represent the class. Each student is equally likely to be chosen and the probability that a boy is chosen is of the probability that a girl is chosen. The ratio of the number of boys to the total number of boys and girls is
Problem 7
In some computer languages (such as APL), when there are no parentheses in an algebraic expression, the operations are grouped from left to right. Thus, in such languages means the same as in ordinary algebraic notation. If is evaluated in such a language, the result in ordinary algebraic notation would be
Problem 8
Let and be real numbers with and nonzero. The solution to is less than the solution to if and only if
Problem 9
The odd positive integers , are arranged into five columns continuing with the pattern shown on the right. Counting from the left, the column in which appears in is the
Problem 10
An arbitrary circle can intersect the graph in
Problem 11
How many distinguishable rearrangements of the letters in CONTEST have both the vowels first? (For instance, OETCNST is one such arrangement but OTETSNC is not.)
Problem 12
Let's write and as three distinct prime numbers, where is not a prime. Which of the following is the smallest positive perfect cube having as a divisor?
Problem 13
Pegs are put in a board unit apart both horizontally and vertically. A rubber band is stretched over pegs as shown in the figure, forming a quadrilateral. Its area in square units is
Problem 14
Exactly three of the interior angles of a convex polygon are obtuse. What is the maximum number of sides of such a polygon?
Problem 15
If and are positive numbers such that and , then the value of is:
Problem 16
If and , then the value of is
Problem 17
Diagonal of rectangle is divided into segments of length by parallel lines and that pass through and and are perpendicular to . The area of , rounded to the nearest tenth, is
Problem 18
Six bags of marbles contain and marbles, respectively. One bag contains chipped marbles only. The other bags contain no chipped marbles. Jane takes three of the bags and George takes two of the others. Only the bag of chipped marbles remains. If Jane gets twice as many marbles as George, how many chipped marbles are there?
Problem 19
Consider the graphs and , where is a positive constant and and are real variables. In how many points do the two graphs intersect?
Problem 20
A wooden cube with edge length units (where is an integer ) is painted black all over. By slices parallel to its faces, the cube is cut into smaller cubes each of unit length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is ?
Problem 21
How many integers satisfy the equation
Problem 22
In a circle with center , is a diameter, is a chord, , and . Then the length of is:
Problem 23
If and , where , then which of the following is not correct?
Problem 24
A non-zero digit is chosen in such a way that the probability of choosing digit is . The probability that the digit is chosen is exactly the probability that the digit is chosen in the set
Problem 25
The volume of a certain rectangular solid is , its total surface area is , and its three dimensions are in geometric progression. The sums of the lengths in cm of all the edges of this solid is
Problem 26
Find the least positive integer for which is a non-zero reducible fraction.
Problem 27
Consider a sequence defined by
and in general
for .
What is the smallest value of for which is an integer?
Problem 28
In , we have and . What is ?
Problem 29
In their base representation, the integer consists of a sequence of eights and the integer consists of a sequence of fives. What is the sum of the digits of the base representation of ?
Problem 30
Let be the greatest integer less than or equal to . Then the number of real solutions to is
See also
1986 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1984 AHSME |
Followed by 1986 AHSME | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.