Difference between revisions of "1987 AHSME Problems"
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If <math>(x, y)</math> is a solution to the system | If <math>(x, y)</math> is a solution to the system | ||
− | <math>xy=6 | + | <math>xy=6</math> and <math>x^2y+xy^2+x+y=63</math>, |
find <math>x^2+y^2</math>. | find <math>x^2+y^2</math>. | ||
Revision as of 12:36, 31 March 2018
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
Problem 1
equals
Problem 2
A triangular corner with side lengths is cut from equilateral triangle ABC of side length
.
The perimeter of the remaining quadrilateral is
Problem 3
How many primes less than have
as the ones digit? (Assume the usual base ten representation)
Problem 4
equals
Problem 5
A student recorded the exact percentage frequency distribution for a set of measurements, as shown below.
However, the student neglected to indicate , the total number of measurements. What is the smallest possible value of
?
Problem 6
In the shown,
is some interior point, and
are the measures of angles
in degrees. Solve for
in terms of
and
.
Problem 7
If , which of the four quantities
is the largest?
Problem 8
In the figure the sum of the distances and
is
Problem 9
The first four terms of an arithmetic sequence are . The ratio of
to
is
Problem 10
How many ordered triples of non-zero real numbers have the property that each number is the product of the other two?
Problem 11
Let be a constant. The simultaneous equations
have a solution
inside Quadrant I if and only if
Problem 12
In an office, at various times during the day the boss gives the secretary a letter to type, each time putting the letter on top of the pile in the secretary's in-box. When there is time, the secretary takes the top letter off the pile and types it. If there are five letters in all, and the boss delivers them in the order , which of the following could not be the order in which the secretary types them?
Problem 13
A long piece of paper cm wide is made into a roll for cash registers by wrapping it
times around a cardboard tube of diameter
cm,
forming a roll
cm in diameter. Approximate the length of the paper in meters.
(Pretend the paper forms
concentric circles with diameters evenly spaced from
cm to
cm.)
Problem 14
is a square and
and
are the midpoints of
and
respectively. Then
Problem 15
If is a solution to the system
and
,
find
.
Problem 16
A cryptographer devises the following method for encoding positive integers. First, the integer is expressed in base .
Second, a 1-to-1 correspondence is established between the digits that appear in the expressions in base
and the elements of the set
. Using this correspondence, the cryptographer finds that three consecutive integers in increasing
order are coded as
, respectively. What is the base-
expression for the integer coded as
?
Problem 17
In a mathematics competition, the sum of the scores of Bill and Dick equalled the sum of the scores of Ann and Carol. If the scores of Bill and Carol had been interchanged, then the sum of the scores of Ann and Carol would have exceeded the sum of the scores of the other two. Also, Dick's score exceeded the sum of the scores of Bill and Carol. Determine the order in which the four contestants finished, from highest to lowest. Assume all scores were nonnegative.
Problem 18
It takes algebra books (all the same thickness) and
geometry books (all the same thickness,
which is greater than that of an algebra book) to completely fill a certain shelf.
Also,
of the algebra books and
of the geometry books would fill the same shelf.
Finally,
of the algebra books alone would fill this shelf. Given that
are distinct positive integers,
it follows that
is
Problem 19
Which of the following is closest to ?
Problem 20
Evaluate
Problem 21
There are two natural ways to inscribe a square in a given isosceles right triangle.
If it is done as in Figure 1 below, then one finds that the area of the square is .
What is the area (in
) of the square inscribed in the same
as shown in Figure 2 below?
Problem 22
A ball was floating in a lake when the lake froze. The ball was removed (without breaking the ice),
leaving a hole cm across as the top and
cm deep. What was the radius of the ball (in centimeters)?
Problem 23
If is a prime and both roots of
are integers, then
Problem 24
How many polynomial functions of degree
satisfy
?
Problem 25
is a triangle:
and both the coordinates of
are integers. What is the minimum area
can have?
Problem 26
The amount is split into two nonnegative real numbers uniformly at random,
for instance, into
and
, or into
and
.
Then each number is rounded to its nearest integer, for instance,
and
in the first case above,
and
in the second. What is the probability that the two integers sum to
?
Problem 27
A cube of cheese is cut along the planes
and
. How many pieces are there?
(No cheese is moved until all three cuts are made.)
Problem 28
Let be real numbers. Suppose that all the roots of
are complex numbers lying on a circle
in the complex plane centered at
and having radius
. The sum of the reciprocals of the roots is necessarily
Problem 29
Consider the sequence of numbers defined recursively by and for
by
when
is even
and by
when
is odd. Given that
, the sum of the digits of
is
Problem 30
In the figure, has
and
. A line
, with
on
and
, divides
into two pieces of equal area.
(Note: the figure may not be accurate; perhaps
is on
instead of
.)
The ratio
is