What is the value of
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The sum of three numbers is The first number is
times the third number, and the third number is
less than the second number. What is the absolute value of the difference between the first and second numbers?
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Five rectangles, ,
,
,
, and
, are arranged in a square as shown below. These rectangles have dimensions
,
,
,
, and
, respectively. (The figure is not drawn to scale.) Which of the five rectangles is the shaded one in the middle?
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The least common multiple of a positive integer and
is
, and the greatest common divisor of
and
is
. What is the sum of the digits of
?
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The between points
and
in the coordinate plane is given by
For how many points
with integer coordinates is the taxicab distance between
and the origin less than or equal to
?
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A data set consists of (not distinct) positive integers:
,
,
,
,
, and
. The average (arithmetic mean) of the
numbers equals a value in the data set. What is the sum of all possible values of
?
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A rectangle is partitioned into regions as shown. Each region is to be painted a solid color – red, orange, yellow, blue, or green – so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?
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The infinite productevaluates to a real number. What is that number?
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On Halloween children walked into the principal’s office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order.
“Are you a truth-teller?” The principal gave a piece of candy to each of the children who answered yes.
“Are you an alternater?” The principal gave a piece of candy to each of the children who answered yes.
“Are you a liar?” The principal gave a piece of candy to each of the children who answered yes.
How many pieces of candy in all did the principal give to the children who always tell the truth?
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How many ways are there to split the integers through
into
pairs such that in each pair, the greater number is at least
times the lesser number?
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What is the product of all real numbers such that the distance on the number line between
and
is twice the distance on the number line between
and
?
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Let be the midpoint of
in regular tetrahedron
. What is
?
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Let be the region in the complex plane consisting of all complex numbers
that can be written as the sum of complex numbers
and
, where
lies on the segment with endpoints
and
, and
has magnitude at most
. What integer is closest to the area of
?
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What is the value ofwhere
denotes the base-ten logarithm?
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The roots of the polynomial are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by
units. What is the volume of the new box?
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A triangular number is a positive integer that can be expressed in the form , for some positive integer
. The three smallest triangular numbers that are also perfect squares are
and
. What is the sum of the digits of the fourth smallest triangular number that is also a perfect square?
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Suppose is a real number such that the equation
has more than one solution in the interval
. The set of all such
that can be written in the form
where
and
are real numbers with
. What is
?
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Let be the transformation of the coordinate plane that first rotates the plane
degrees counterclockwise around the origin and then reflects the plane across the
-axis. What is the least positive integer
such that performing the sequence of transformations
returns the point
back to itself?
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Suppose that cards numbered
are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards
are picked up on the first pass,
and
on the second pass,
on the third pass,
on the fourth pass, and
on the fifth pass. For how many of the
possible orderings of the cards will the
cards be picked up in exactly two passes?
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Isosceles trapezoid has parallel sides
and
with
and
There is a point
in the plane such that
and
What is
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LetWhich of the following polynomials is a factor of
?
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Let be a real number, and let
and
be the two complex numbers satisfying the equation
. Points
,
,
, and
are the vertices of (convex) quadrilateral
in the complex plane. When the area of
obtains its maximum possible value,
is closest to which of the following?
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Let and
be the unique relatively prime positive integers such that
Let
denote the least common multiple of the numbers
. For how many integers with
is
?
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How many strings of length formed from the digits
,
,
,
,
are there such that for each
, at least
of the digits are less than
? (For example,
satisfies this condition because it contains at least
digit less than
, at least
digits less than
, at least
digits less than
, and at least
digits less than
. The string
does not satisfy the condition because it does not contain at least
digits less than
.)
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A circle with integer radius is centered at
. Distinct line segments of length
connect points
to
for
and are tangent to the circle, where
,
, and
are all positive integers and
. What is the ratio
for the least possible value of
?
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