1994 AIME Problems
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Contents
Problem 1
The increasing sequence consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term of the sequence is divided by 1000?
Problem 2
A circle with diameter of length 10 is internally tangent at
to a circle of radius 20. Square
is constructed with
and
on the larger circle,
tangent at
to the smaller circle, and the smaller circle outside
. The length of
can be written in the form
, where
and
are integers. Find
.
Problem 3
The function has the property that, for each real number
![$f(x)+f(x-1) = x^2\,$](http://latex.artofproblemsolving.com/6/7/8/67827ba82e465559d945580a479bca7bcce2fdd9.png)
.
If what is the remainder when
is divided by 1000?
Problem 4
Find the positive integer for which
![$\lfloor \log_2{1}\rfloor+\lfloor\log_2{2}\rfloor+\lfloor\log_2{3}\rfloor+\cdots+\lfloor\log_2{n}\rfloor=1994$](http://latex.artofproblemsolving.com/f/3/7/f37b096bac8263b45ed10c89504cd053281387e8.png)
.
(For real ,
is the greatest integer
)
Problem 5
Given a positive integer , let
be the product of the non-zero digits of
. (If
has only one digits, then
is equal to that digit.) Let
![$S=p(1)+p(2)+p(3)+\cdots+p(999)$](http://latex.artofproblemsolving.com/3/4/d/34d0f62dc5536d2351b2e6ecbf4226fcabf7ab4d.png)
.
What is the largest prime factor of ?
Problem 6
The graphs of the equations
![$y=k, \qquad y=\sqrt{3}x+2k, \qquad y=-\sqrt{3}x+2k,$](http://latex.artofproblemsolving.com/1/4/2/1420d57ceea649c3bb54cffc8b0aae09465d2483.png)
are drawn in the coordinate plane for These 63 lines cut part of the plane into equilateral triangles of side
How many such triangles are formed?
Problem 7
For certain ordered pairs of real numbers, the system of equations
![$ax+by=1\,$](http://latex.artofproblemsolving.com/9/4/b/94b5a3e0b1a726c775259827de5d37bb20d31b62.png)
![$x^2+y^2=50\,$](http://latex.artofproblemsolving.com/d/5/b/d5b1e8d558c3818bdd87c0b06261db4f6a0977fe.png)
has at least one solution, and each solution is an ordered pair of integers. How many such ordered pairs
are there?
Problem 8
The points ,
, and
are the vertices of an equilateral triangle. Find the value of
.
Problem 9
A solitarire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no two of which match; otherwise the drawing continues until the bag is empty. The probability that the bag will be emptied is where
and
are relatively prime positive integers. Find
Problem 10
In triangle angle
is a right angle and the altitude from
meets
at
The lengths of the sides of
are integers,
and
, where
and
are relatively prime positive integers. Find
Problem 11
Ninety-four bricks, each measuring are to stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contribues
or
or
to the total height of the tower. How many differnt tower heights can be achieved using all 94 of the bricks?
Problem 12
A fenced, rectangular field measures 24 meters by 52 meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the field. What is the largest number of square test plots into which the field can be partitioned using all or some of the 1994 meters of fence?
Problem 13
The equation
![$x^{10}+(13x-1)^{10}=0\,$](http://latex.artofproblemsolving.com/7/3/e/73ef5b72d76840f828b8eded9fe5a63bba1d2958.png)
has 10 complex roots where the bar denotes complex conjugation. Find the value of
![$\frac 1{r_1\overline{r_1}}+\frac 1{r_2\overline{r_2}}+\frac 1{r_3\overline{r_3}}+\frac 1{r_4\overline{r_4}}+\frac 1{r_5\overline{r_5}}.$](http://latex.artofproblemsolving.com/3/d/f/3df94cc625c07e4055d82e8eef058b98b14fde99.png)
Problem 14
A beam of light strikes at point
with angle of incidence
and reflects with an equal angle of reflection as shown. The light beam continues its path, reflecting off line segments
and
according to the rule: angle of incidence equals angle of reflection. Given that
and
determine the number of times the light beam will bounce off the two line segments. Include the first reflection at
in your count.
Problem 15
Given a point on a triangular piece of paper
consider the creases that are formed in the paper when
and
are folded onto
Let us call
a fold point of
if these creases, which number three unless
is one of the vertices, do not intersect. Suppose that
and
Then the area of the set of all fold points of
can be written in the form
where
and
are positive integers and
is not divisible by the square of any prime. What is
?