1999 AHSME Problems
1999 AHSME (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
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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
Problem 2
Which of the following statements is false?
Problem 3
The number halfway between and is
Problem 4
Find the sum of all prime numbers between and that are simultaneously greater than a multiple of and less than a multiple of .
Problem 5
The marked price of a book was less than the suggested retail price. Alice purchased the book for half the marked price at a Fiftieth Anniversary sale. What percent of the suggested retail price did Alice pay?
Problem 6
What is the sum of the digits of the decimal form of the product ?
Problem 7
What is the largest number of acute angles that a convex hexagon can have?
Problem 8
At the end of 1994 Walter was half as old as his grandmother. The sum of the years in which they were born is 3838. How old will Walter be at the end of 1999?
Problem 9
Before Ashley started a three-hour drive, her car's odometer reading was 29792, a palindrome. (A palindrome is a number that reads the same way from left to right as it does from right to left). At her destination, the odometer reading was another palindrome. If Ashley never exceeded the speed limit of 75 miles per hour, which of the following was her greatest possible average speed?
Problem 10
A sealed envelope contains a card with a single digit on it. Three of the following statements are true, and the other is false.
I. The digit is 1.
II. the digit is not 2.
III. The digit is 3.
IV. The digit is not 4.
Which one of the following must necessarily be correct?
Problem 11
The student lockers at Olympic High are numbered consecutively beginning with locker number . The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number and four cents to label locker number . If it costs to label all the lockers, how many lockers are there at the school?
Problem 12
What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions and , each with leading coefficient ?
Problem 13
Define a sequence of real numbers by and for all . Then equals
Problem 14
Four girls - Mary, Aline, Tina, and Hana - sang songs in a concert as trios, with one girl sitting out each time. Hanna sang 7 songs, which was more than any other girl, and Mary sang 4 songs, which was fewer than any other girl. How many songs did these trios sing?
Problem 15
Let be a real number such that . Then
Problem 16
What is the radius of a circle inscribed in a rhombus with diagonals of length and ?
Problem 17
Let be a polynomial such that when is divided by , the remainder is , and when is divided by , the remainder is . What is the remainder when is divided by ?
Problem 18
How many zeros does have on the interval ?
Problem 19
Consider all triangles satisfying in the following conditions: , is a point on for which , and are integers, and . Among all such triangles, the smallest possible value of is
Problem 20
The sequence satisfies , and, for all , is the arithmetic mean of the first terms. Find .
Problem 21
A circle is circumscribed about a triangle with sides and thus dividing the interior of the circle into four regions. Let and be the areas of the non-triangular regions, with be the largest. Then
Problem 22
The graphs of and intersect at points and . Find .
Problem 23
The equiangular convex hexagon has and The area of the hexagon is
Problem 24
Six points on a circle are given. Four of the chords joining pairs of the six points are selected at random. What is the probability that the four chords form a convex quadrilateral?
Problem 25
There are unique integers such that
where for . Find .
Problem 26
Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length . The polygons meet at a point in such a way that the sum of the three interior angles at is . Thus the three polygons form a new polygon with as an interior point. What is the largest possible perimeter that this polygon can have?
Problem 27
In triangle , and . Then in degrees is
Problem 28
Let be a sequence of integers such that
;
.
Let and be the minimal and maximal possible values of , respectively. Then
Problem 29
A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point is selected at random inside the circumscribed sphere. The probability that lies inside one of the five small spheres is closest to
Problem 30
The number of ordered pairs of integers for which and
is equal to
See Also
1999 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1998 AHSME |
Followed by Last AHSME, see 2000 AMC 12 | |
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All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.