Difference between revisions of "Arcticturn Prep"
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==Problem 6== | ==Problem 6== | ||
In convex quadrilateral <math>KLMN</math>, side <math>\overline{MN}</math> is perpendicular to diagonal <math>\overline{KM}</math>, side <math>\overline{KL}</math> is perpendicular to diagonal <math>\overline{LN}</math>, <math>MN = 65</math>, and <math>KL = 28</math>. The line through <math>L</math> perpendicular to side <math>\overline{KN}</math> intersects diagonal <math>\overline{KM}</math> at <math>O</math> with <math>KO = 8</math>. Find <math>MO</math>. | In convex quadrilateral <math>KLMN</math>, side <math>\overline{MN}</math> is perpendicular to diagonal <math>\overline{KM}</math>, side <math>\overline{KL}</math> is perpendicular to diagonal <math>\overline{LN}</math>, <math>MN = 65</math>, and <math>KL = 28</math>. The line through <math>L</math> perpendicular to side <math>\overline{KN}</math> intersects diagonal <math>\overline{KM}</math> at <math>O</math> with <math>KO = 8</math>. Find <math>MO</math>. | ||
+ | |||
+ | ==Problem 14== | ||
+ | Find the least odd prime factor of <math>2019^8 + 1</math>. |
Revision as of 11:45, 16 June 2024
Contents
[hide]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 .
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
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).
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
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 .
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
Find the least odd prime factor of .