Difference between revisions of "1997 AHSME Problems"
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+ | {{AHSME Problems | ||
+ | |year = 1997 | ||
+ | }} | ||
== Problem 1 == | == Problem 1 == | ||
If <math>\texttt{a}</math> and <math>\texttt{b}</math> are digits for which | If <math>\texttt{a}</math> and <math>\texttt{b}</math> are digits for which | ||
− | <math> \begin{ | + | <math> \begin{array}{ccc}& 2 & a\\ \times & b & 3\\ \hline & 6 & 9\\ 9 & 2\\ \hline 9 & 8 & 9\end{array} </math> |
then <math>\texttt{a+b =}</math> | then <math>\texttt{a+b =}</math> | ||
Line 150: | Line 153: | ||
==Problem 13== | ==Problem 13== | ||
− | How many two-digit positive integers <math>N</math> have the property that the sum of <math>N</math> and the number obtained by reversing the order of the digits of is a perfect square? | + | How many two-digit positive integers <math>N</math> have the property that the sum of <math>N</math> and the number obtained by reversing the order of the digits of <math>N</math> is a perfect square? |
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8 </math> | <math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8 </math> | ||
Line 166: | Line 169: | ||
==Problem 15== | ==Problem 15== | ||
− | Medians <math>BD</math> and <math> | + | Medians <math>BD</math> and <math>CE</math> of triangle <math>ABC</math> are perpendicular, <math>BD=8</math>, and <math>CE=12</math>. The area of triangle <math>ABC</math> is |
<asy> | <asy> | ||
Line 202: | Line 205: | ||
==Problem 17== | ==Problem 17== | ||
− | A line <math> | + | A line <math>x=k</math> intersects the graph of <math>y=\log_5 x</math> and the graph of <math>y=\log_5 (x + 4)</math>. The distance between the points of intersection is <math>0.5</math>. Given that <math>k = a + \sqrt{b}</math>, where <math>a</math> and <math>b</math> are integers, what is <math>a+b</math>? |
<math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10 </math> | <math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10 </math> | ||
Line 359: | Line 362: | ||
==Problem 30== | ==Problem 30== | ||
− | For positive integers <math>n</math>, denote <math>D(n)</math> by the number of pairs of different adjacent digits in the binary (base two) representation of <math>n</math>. For example, <math> D(3) = D(11_{2}) = 0 </math>, <math> D(21) = D(10101_{2}) = 4 </math>, and <math> D(97) = D(1100001_{2}) = 2 </math>. For how many positive integers less than or equal <math>97</math> | + | For positive integers <math>n</math>, denote <math>D(n)</math> by the number of pairs of different adjacent digits in the binary (base two) representation of <math>n</math>. For example, <math> D(3) = D(11_{2}) = 0 </math>, <math> D(21) = D(10101_{2}) = 4 </math>, and <math> D(97) = D(1100001_{2}) = 2 </math>. For how many positive integers less than or equal to <math>97</math> does <math>D(n) = 2</math>? |
<math> \textbf{(A)}\ 16\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 35 </math> | <math> \textbf{(A)}\ 16\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 35 </math> | ||
[[1997 AHSME Problems/Problem 30|Solution]] | [[1997 AHSME Problems/Problem 30|Solution]] | ||
+ | |||
+ | == See also == | ||
+ | |||
+ | * [[AMC 12 Problems and Solutions]] | ||
+ | * [[Mathematics competition resources]] | ||
+ | |||
+ | {{AHSME box|year=1997|before=[[1996 AHSME]]|after=[[1998 AHSME]]}} | ||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 13:38, 19 February 2020
1997 AHSME (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
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 |
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 and
are digits for which
then
Problem 2
The adjacent sides of the decagon shown meet at right angles. What is its perimeter?
Problem 3
If ,
, and
are real numbers such that
![$(x-3)^2 + (y-4)^2 + (z-5)^2 = 0$](http://latex.artofproblemsolving.com/b/5/a/b5a9f6e5a95f76eda476bf20df06fc2bdb1e8fc9.png)
then
Problem 4
If is
larger than
, and
is
larger than
, then
is what percent larger than
?
Problem 5
A rectangle with perimeter is divided into five congruent rectangles as shown in the diagram. What is the perimeter of one of the five congruent rectangles?
Problem 6
Consider the sequence
whose th term is
. What is the average of the first
terms of the sequence?
Problem 7
The sum of seven integers is . What is the maximum number of the seven integers that can be larger than
?
Problem 8
Mientka Publishing Company prices its bestseller Where's Walter? as follows:
where is the number of books ordered, and
is the cost in dollars of
books. Notice that
books cost less than
books. For how many values of
is it cheaper to buy more than
books than to buy exactly
books?
Problem 9
In the figure, is a
square,
is the midpoint of
, and
is on
. If
is perpendicular to
, then the area of quadrilateral
is
Problem 10
Two six-sided dice are fair in the sense that each face is equally likely to turn up. However, one of the dice has the replaced by
and the other die has the
replaced by
. When these dice are rolled, what is the probability that the sum is an odd number?
Problem 11
In the sixth, seventh, eighth, and ninth basketball games of the season, a player scored ,
,
, and
points, respectively. Her points-per-game average was higher after nine games than it was after the first five games. If her average after ten games was greater than
, what is the least number of points she could have scored in the tenth game?
Problem 12
If and
are real numbers and
, then the line whose equation is
cannot contain the point
Problem 13
How many two-digit positive integers have the property that the sum of
and the number obtained by reversing the order of the digits of
is a perfect square?
Problem 14
The number of geese in a flock increases so that the difference between the populations in year and year
is directly proportional to the population in year
. If the populations in the years
,
, and
were
,
, and
, respectively, then the population in
was
Problem 15
Medians and
of triangle
are perpendicular,
, and
. The area of triangle
is
Problem 16
The three row sums and the three column sums of the array
are the same. What is the least number of entries that must be altered to make all six sums different from one another?
Problem 17
A line intersects the graph of
and the graph of
. The distance between the points of intersection is
. Given that
, where
and
are integers, what is
?
Problem 18
A list of integers has mode and mean
. The smallest number in the list is
. The median
of the list is a member of the list. If the list member
were replaced by
, the mean and median of the new list would be
and
, respectively. If were
instead replaced by
, the median of the new list would be
. What is
?
Problem 19
A circle with center is tangent to the coordinate axes and to the hypotenuse of the
-
-
triangle
as shown, where
. To the nearest hundredth, what is the radius of the circle?
Problem 20
Which one of the following integers can be expressed as the sum of consecutive positive integers?
Problem 21
For any positive integer , let
What is ?
Problem 22
Ashley, Betty, Carlos, Dick, and Elgin went shopping. Each had a whole number of dollars to spend, and together they had dollars. The absolute difference between the amounts Ashley and Betty had to spend was
dollars. The absolute difference between the amounts Betty and Carlos had was
dollars, between Carlos and Dick was
dollars, between Dick and Elgin was
dollars, and between Elgin and Ashley was
dollars. How many dollars did Elgin have?
Problem 23
In the figure, polygons ,
, and
are isosceles right triangles;
,
, and
are squares with sides of length
; and
is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is
Problem 24
A rising number, such as , is a positive integer each digit of which is larger than each of the digits to its left. There are
five-digit rising numbers. When these numbers are arranged from smallest to largest, the
number in the list does not contain the digit
Problem 25
Let be a parallelogram and let
,
,
, and
be parallel rays in space on the same side of the plane determined by
. If
,
,
, and
and
and
are the midpoints of
and
, respectively, then
Problem 26
Triangle and point
in the same plane are given. Point
is equidistant from
and
, angle
is twice angle
, and
intersects
at point
. If
and
, then
Problem 27
Consider those functions that satisfy
for all real
. Any such function is periodic, and there is a least common positive period
for all of them. Find
.
Problem 28
How many ordered triples of integers satisfy
and
?
Problem 29
Call a positive real number special if it has a decimal representation that consists entirely of digits and
. For example,
and
are special numbers. What is the smallest
such that
can be written as a sum of
special numbers?
Problem 30
For positive integers , denote
by the number of pairs of different adjacent digits in the binary (base two) representation of
. For example,
,
, and
. For how many positive integers less than or equal to
does
?
See also
1997 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1996 AHSME |
Followed by 1998 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.