Difference between revisions of "Arcticturn Prep"
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<asy> size(6cm); draw(unitcircle); draw(scale(2) * unitcircle); for(int d = 90; d < 360 + 90; d += 72){ draw(2 * dir(d) -- dir(d)); } dot(1 * dir( 90), linewidth(5)); dot(1 * dir(162), linewidth(5)); dot(1 * dir(234), linewidth(5)); dot(1 * dir(306), linewidth(5)); dot(1 * dir(378), linewidth(5)); dot(2 * dir(378), linewidth(5)); dot(2 * dir(306), linewidth(5)); dot(2 * dir(234), linewidth(5)); dot(2 * dir(162), linewidth(5)); dot(2 * dir( 90), linewidth(5)); label("$A$", 1 * dir( 90), -dir( 90)); label("$B$", 1 * dir(162), -dir(162)); label("$C$", 1 * dir(234), -dir(234)); label("$D$", 1 * dir(306), -dir(306)); label("$E$", 1 * dir(378), -dir(378)); label("$F$", 2 * dir(378), dir(378)); label("$G$", 2 * dir(306), dir(306)); label("$H$", 2 * dir(234), dir(234)); label("$I$", 2 * dir(162), dir(162)); label("$J$", 2 * dir( 90), dir( 90)); </asy> | <asy> size(6cm); draw(unitcircle); draw(scale(2) * unitcircle); for(int d = 90; d < 360 + 90; d += 72){ draw(2 * dir(d) -- dir(d)); } dot(1 * dir( 90), linewidth(5)); dot(1 * dir(162), linewidth(5)); dot(1 * dir(234), linewidth(5)); dot(1 * dir(306), linewidth(5)); dot(1 * dir(378), linewidth(5)); dot(2 * dir(378), linewidth(5)); dot(2 * dir(306), linewidth(5)); dot(2 * dir(234), linewidth(5)); dot(2 * dir(162), linewidth(5)); dot(2 * dir( 90), linewidth(5)); label("$A$", 1 * dir( 90), -dir( 90)); label("$B$", 1 * dir(162), -dir(162)); label("$C$", 1 * dir(234), -dir(234)); label("$D$", 1 * dir(306), -dir(306)); label("$E$", 1 * dir(378), -dir(378)); label("$F$", 2 * dir(378), dir(378)); label("$G$", 2 * dir(306), dir(306)); label("$H$", 2 * dir(234), dir(234)); label("$I$", 2 * dir(162), dir(162)); label("$J$", 2 * dir( 90), dir( 90)); </asy> | ||
+ | |||
+ | ==Problem 5== | ||
+ | |||
+ | A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are <math>189</math>, <math>320</math>, <math>287</math>, <math>234</math>, <math>x</math>, and <math>y</math>. Find the greatest possible value of <math>x+y</math>. |
Revision as of 20:30, 15 June 2024
Contents
Problem 5
Suppose that ,
, and
are complex numbers such that
,
, and
, where
. Then there are real numbers
and
such that
. Find
.
Problem 6
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.
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
.
Problem 6
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
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
.