Arcticturn Prep
Contents
[hide]- 1 Problem 5*
- 2 Problem 6 (check)
- 3 Problem 9
- 4 Problem 13*
- 5 Problem 6 (check)
- 6 Problem 9*
- 7 Problem 10*
- 8 Problem 5 (check)
- 9 Problem 9
- 10 Problem 11 (check)
- 11 Problem 13
- 12 Problem 5 (check)
- 13 Problem 6
- 14 Problem 14* (check)
- 15 Problem 8 (almost)
- 16 Problem 10
- 17 Problem 6
- 18 Problem 7
- 19 Problem 9
- 20 Problem 12
- 21 Problem 8
- 22 Problem 11
- 23 Problem 6
- 24 Problem 5
- 25 Problem 13
- 26 Problem 7
- 27 Problem 9
- 28 Problem 8
- 29 Problem 8
- 30 Problem 11
- 31 Problem 6
- 32 Problem 7
- 33 Problem 9
- 34 Problem 9
- 35 Problem 9
- 36 Problem 6
- 37 Problem 15
- 38 Problem 8
Problem 5*
Suppose that ,
, and
are complex numbers such that
,
, and
, where
. Then there are real numbers
and
such that
. Find
.
Problem 6 (check)
A real number is chosen randomly and uniformly from the interval
. The probability that the roots of the polynomial
are all real can be written in the form
, where
and
are relatively prime positive integers. Find
.
Problem 9
Octagon with side lengths
and
is formed by removing 6-8-10 triangles from the corners of a
rectangle with side
on a short side of the rectangle, as shown. Let
be the midpoint of
, and partition the octagon into 7 triangles by drawing segments
,
,
,
,
, and
. Find the area of the convex polygon whose vertices are the centroids of these 7 triangles.
Note: Homothety
Problem 13*
Misha rolls a standard, fair six-sided die until she rolls 1-2-3 in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is where
and
are relatively prime positive integers. Find
.
Note: Find relation between odd & even.
Problem 6 (check)
Let be the number of complex numbers
with the properties that
and
is a real number. Find the remainder when
is divided by
.
Problem 9*
Find the number of four-element subsets of with the property that two distinct elements of a subset have a sum of
, and two distinct elements of a subset have a sum of
. For example,
and
are two such subsets.
Problem 10*
The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point . At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the path
, which has
steps. Let
be the number of paths with
steps that begin and end at point
Find the remainder when
is divided by
.
Problem 5 (check)
A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are ,
,
,
,
, and
. Find the greatest possible value of
.
Problem 9
A special deck of cards contains cards, each labeled with a number from
to
and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, the probability that Sharon can discard one of her cards and
have at least one card of each color and at least one card with each number is
, where
and
are relatively prime positive integers. Find
.
Problem 11 (check)
Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way).
Note: Complimentary counting + PiE
Problem 13
For each integer , let
be the number of
-element subsets of the vertices of a regular
-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of
such that
.
Problem 5 (check)
A moving particle starts at the point and moves until it hits one of the coordinate axes for the first time. When the particle is at the point
, it moves at random to one of the points
,
, or
, each with probability
, independently of its previous moves. The probability that it will hit the coordinate axes at
is
, where
and
are positive integers, and
is not divisible by
. Find
.
Note: recursion with probability
Problem 6
In convex quadrilateral , side
is perpendicular to diagonal
, side
is perpendicular to diagonal
,
, and
. The line through
perpendicular to side
intersects diagonal
at
with
. Find
.
Problem 14* (check)
Find the least odd prime factor of .
Note: Use FLT
Problem 8 (almost)
Find the number of sets of three distinct positive integers with the property that the product of
and
is equal to the product of
and
.
Problem 10
Triangle is inscribed in circle
. Points
and
are on side
with
. Rays
and
meet
again at
and
(other than
), respectively. If
and
, then
, where
and
are relatively prime positive integers. Find
.
Problem 6
In let
be the center of the inscribed circle, and let the bisector of
intersect
at
. The line through
and
intersects the circumscribed circle of
at the two points
and
. If
and
, then
, where
and
are relatively prime positive integers. Find
.
Note: angle chase, then angle bisectors.
Problem 7
For integers and
consider the complex number
Find the number of ordered pairs of integers
such that this complex number is a real number.
Note: and beware of absolute value sign
Problem 9
Triangle has
and
. This triangle is inscribed in rectangle
with
on
and
on
. Find the maximum possible area of
.
Use:
Problem 12
Find the least positive integer such that
is a product of at least four not necessarily distinct primes.
Note: should be multiple of
.
Problem 8
Let and
be positive integers satisfying
. The maximum possible value of
is
, where
and
are relatively prime positive integers. Find
.
Note: SFFT
Problem 11
The circumcircle of acute has center
. The line passing through point
perpendicular to
intersects lines
and
at
and
, respectively. Also
,
,
, and
, where
and
are relatively prime positive integers. Find
.
Note: easy but pay attention to the wording
Problem 6
Point and
are equally spaced on a minor arc of a circle. Points
and
are equally spaced on a minor arc of a second circle with center
as shown in the figure below. The angle
exceeds
by
. Find the degree measure of
.
Note: Arc angles
Problem 5
Real numbers and
are roots of
, and
and
are roots of
. Find the sum of all possible values of
.
Problem 13
With all angles measured in degrees, the product , where
and
are integers greater than 1. Find
.
Problem 7
Let and
be complex numbers such that
and
. Let
. The maximum possible value of
can be written as
, where
and
are relatively prime positive integers. Find
. (Note that
, for
, denotes the measure of the angle that the ray from
to
makes with the positive real axis in the complex plane.)
Problem 9
Let be the three real roots of the equation
. Find
.
Problem 8
The domain of the function is a closed interval of length
, where
and
are positive integers and
. Find the remainder when the smallest possible sum
is divided by
.
Note: stupid problem - need to test
Problem 8
The complex numbers and
satisfy the system
Find the smallest possible value of
.
Problem 11
A frog begins at and makes a sequence of jumps according to the following rule: from
the frog jumps to
which may be any of the points
or
There are
points
with
that can be reached by a sequence of such jumps. Find the remainder when
is divided by
Note: Don't silly answer extraction
Problem 6
Find the smallest positive integer with the property that the polynomial
can be written as a product of two nonconstant polynomials with integer coefficients.
Note:
Problem 7
Define an ordered triple of sets to be
if
and
. For example,
is a minimally intersecting triple. Let
be the number of minimally intersecting ordered triples of sets for which each set is a subset of
. Find the remainder when
is divided by
.
Problem 9
Let be a real solution of the system of equations
,
,
. The greatest possible value of
can be written in the form
, where
and
are relatively prime positive integers. Find
.
Problem 9
Ten identical crates each of dimensions ft
ft
ft. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let
be the probability that the stack of crates is exactly
ft tall, where
and
are relatively prime positive integers. Find
.
Problem 9
Rectangle is given with
and
Points
and
lie on
and
respectively, such that
The inscribed circle of triangle
is tangent to
at point
and the inscribed circle of triangle
is tangent to
at point
Find
Note: Don't trust diagrams.
Problem 6
A frog is placed at the origin on the number line, and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of , or to the closest point with a greater integer coordinate that is a multiple of
. A move sequence is a sequence of coordinates which correspond to valid moves, beginning with
, and ending with
. For example,
is a move sequence. How many move sequences are possible for the frog?
Note: 0 => 13 => 26 => 39
Problem 15
The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths ,
, and
, as shown, is
, where
,
, and
are positive integers,
and
are relatively prime, and
is not divisible by the square of any prime. Find
.
Problem 8
Let be the sum of the base
logarithms of all the proper divisors of
. What is the integer nearest to
?