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The median of the list is
. What is the mean?
A number is
more than the product of its reciprocal and its additive inverse. In which interval does the number lie?
The sum of two numbers is . Suppose
is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?
What is the maximum number for the possible points of intersection of a circle and a triangle?
How many of the twelve pentominoes pictured below have at least one line of symmetry?
Let and
denote the product and the sum, respectively, of the digits of the integer
. For example,
and
. Suppose
is a two-digit number such that
. What is the units digit of
?
When the decimal point of a certain positive decimal number is moved four places to the right, the new number is four times the reciprocal of the original number. What is the original number?
Wanda, Darren, Beatrice, and Chi are tutors in the school math lab. Their schedule is as follows: Darren works every third school day, Wanda works every fourth school day, Beatrice works every sixth school day, and Chi works every seventh school day. Today they are all working in the math lab. In how many school days from today will they next be together tutoring in the lab?
The state income tax where Kristin lives is levied at the rate of of the first
of annual income plus
of any amount above
. Kristin noticed that the state income tax she paid amounted to
of her annual income. What was her annual income?
If ,
, and
are positive with
,
, and
, then
is
Consider the dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square contains unit squares. The second ring contains
unit squares. If we continue this process, the number of unit squares in the
ring is
Suppose that is the product of three consecutive integers and that
is divisible by
. Which of the following is not necessarily a divisor of
?
A telephone number has the form , where each letter represents a different digit. The digits in each part of the numbers are in decreasing order; that is,
,
, and
. Furthermore,
,
, and
are consecutive even digits;
,
,
, and
are consecutive odd digits; and
. Find
.
A charity sells benefit tickets for a total of
. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets?
A street has parallel curbs feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is
feet and each stripe is
feet long. Find the distance, in feet, between the stripes?
The mean of three numbers is more than the least of the numbers and
less than the greatest. The median of the three numbers is
. What is their sum?
Which of the cones listed below can be formed from a sector of a circle of radius
by aligning the two straight sides?
The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to
Pat wants to buy four donuts from an ample supply of three types of donuts: glazed, chocolate, and powdered. How many different selections are possible?
A regular octagon is formed by cutting an isosceles right triangle from each of the corners of a square with sides of length . What is the length of each side of the octagon?
A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter and altitude
, and the axes of the cylinder and cone coincide. Find the radius of the cylinder.
In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by ,
,
,
, and
. Find
.
A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?
In trapezoid ,
and
are perpendicular to
, with
,
, and
. What is
?
How many positive integers not exceeding are multiples of
or
but not
?
And we have
Plugging in, we get
Simplifying, we get
Factoring out the , we get
Obviously, we see that the fraction has to simplify to an integer.
Hence, has to be a factor of 4002.
By inspection, we see that the prime factorization of
We see that through inspection. We also find that
Hence, the price of full tickets out of is
.
The magic sum is determined by the bottom row. .
Solving for :
.
To find our answer, we need to find .
.
A nice thing to know is that any numbers that goes through the middle forms an arithmetic sequence.
Using this, we know that , or
because
would be the average.
We also know that because is the average the magic sum would be
, so we can also write the equation
using the bottom row.
Solving for x in this system we get , so now using the arithmetic sequence knowledge we find that
and
.
Adding these we get .
Take a good-sized sample of consecutive integers; for example, the first 25 positive integers. Determine that the numbers 3, 4, 6, 8, 9, 12, 16, 18, 21, and 24 exhibit the properties given in the question. 25 is a divisor of 2000, so there are numbers satisfying the given conditions between 1 and 2000. Since 2001 is a multiple of 3, add 1 to 800 to get
.
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