1. The average of two positive real numbers is equal to their difference. What is the ratio of the larger number to the smaller one?
Author: Ray Li
2. How many ways are there to arrange the letters 
in a row so that the sequence 
appears at least once?
in a row so that the sequence appears at least once?Author: Ray Li
Author: Ray Li
Clarifications
4. How many positive even numbers have an even number of digits and are less than 10000?
5. Congruent circles 
and 
have radius 
and the center of lies on 
Suppose that and intersect at 
and 
. The line through perpendicular to 
meets and again at 
and 
, respectively. Find the length of 
.
and have radius and the center of lies on Suppose that and intersect at and . The line through perpendicular to meets and again at and , respectively. Find the length of .Author: Ray Li
6. Alice's favorite number has the following properties:
It has 8 distinct digits.
The digits are decreasing when read from left to right.
It is divisible by 180.What is Alice's favorite number?
Author: Anderson Wang
7. A board 
inches long and 
inches high is inclined so that the long side of the board makes a 
degree angle with the ground. The distance from the ground to the highest point on the board can be expressed in the form 
where 
are positive integers and 
is not divisible by the square of any prime. What is 
?
inches long and inches high is inclined so that the long side of the board makes a degree angle with the ground. The distance from the ground to the highest point on the board can be expressed in the form where are positive integers and is not divisible by the square of any prime. What is ?Author: Ray Li
Clarification
8. An 
cube is painted red on 
faces and blue on faces such that no corner is surrounded by three faces of the same color. The cube is then cut into 
unit cubes. How many of these cubes contain both red and blue paint on at least one of their faces?
cube is painted red on faces and blue on faces such that no corner is surrounded by three faces of the same color. The cube is then cut into unit cubes. How many of these cubes contain both red and blue paint on at least one of their faces?Author: Ray Li
Clarification
9. At a certain grocery store, cookies may be bought in boxes of 
or 
What is the minimum positive number of cookies that must be bought so that the cookies may be split evenly among 
people?
or What is the minimum positive number of cookies that must be bought so that the cookies may be split evenly among people?Author: Ray Li
10. A drawer has 
pairs of socks. Three socks are chosen at random. If the probability that there is a pair among the three is 
where 
and 
are relatively prime positive integers, what is 
?
pairs of socks. Three socks are chosen at random. If the probability that there is a pair among the three is where and are relatively prime positive integers, what is ?Author: Ray Li
11. If
and 
can be expressed in the form where 
are relatively prime positive integers, find .
can be expressed in the form where are relatively prime positive integers, find .Author: Ray Li
12. A cross-pentomino is a shape that consists of a unit square and four other unit squares each sharing a different edge with the first square. If a cross-pentomino is inscribed in a circle of radius 
what is 
?
what is ?Author: Ray Li
13. A circle 
has center 
and radius 
A chord 
of also has length r, and the tangents to at and meet at . Ray 
meets at past 
and ray 
meets the circle centered at with radius at 
past 
Compute the degree measure of 
has center and radius A chord of also has length r, and the tangents to at and meet at . Ray meets at past and ray meets the circle centered at with radius at past Compute the degree measure of Author: Ray Li
14. Al told Bob that he was thinking of 
distinct positive integers. He also told Bob the sum of those distinct positive integers. From this information, Bob was able to determine all integers. How many possible sums could Al have told Bob?
distinct positive integers. He also told Bob the sum of those distinct positive integers. From this information, Bob was able to determine all integers. How many possible sums could Al have told Bob?Author: Ray Li
15. Five bricklayers working together finish a job in hours. Working alone, each bricklayer takes at most 
hours to finish the job. What is the smallest number of minutes it could take the fastest bricklayer to complete the job alone?
hours. Working alone, each bricklayer takes at most hours to finish the job. What is the smallest number of minutes it could take the fastest bricklayer to complete the job alone?Author: Ray Li
16. Let 
be a unit cube, with 
and 
opposite square faces, and let 
be the center of face . Rectangular pyramid 
is cut out of the cube. If the surface area of the remaining solid can be expressed in the form 
, where 
and 
are positive integers and is not divisible by the square of any prime, find 
.
be a unit cube, with and opposite square faces, and let be the center of face . Rectangular pyramid is cut out of the cube. If the surface area of the remaining solid can be expressed in the form , where and are positive integers and is not divisible by the square of any prime, find .Author: Alex Zhu
17. Each pair of vertices of a regular 10-sided polygon is connected by a line segment. How many unordered pairs of distinct parallel line segments can be chosen from these segments?
Author: Ray Li
18. The sum of the squares of three positive numbers is 
. One of the numbers is equal to the sum of the other two. The difference between the smaller two numbers is 
What is the difference between the cubes of the smaller two numbers?
. One of the numbers is equal to the sum of the other two. The difference between the smaller two numbers is What is the difference between the cubes of the smaller two numbers?Author: Ray Li
Clarification
19. There are 
geese numbered 
standing in a line. The even numbered geese are standing at the front in the order 
where 
is at the front of the line. Then the odd numbered geese are standing behind them in the order, 
where 
is at the end of the line. The geese want to rearrange themselves in order, so that they are ordered 
(1 is at the front), and they do this by successively swapping two adjacent geese. What is the minimum number of swaps required to achieve this formation?
geese numbered standing in a line. The even numbered geese are standing at the front in the order where is at the front of the line. Then the odd numbered geese are standing behind them in the order, where is at the end of the line. The geese want to rearrange themselves in order, so that they are ordered (1 is at the front), and they do this by successively swapping two adjacent geese. What is the minimum number of swaps required to achieve this formation?Author: Ray Li
20. Let 
be a right triangle with a right angle at 
Two lines, one parallel to 
and the other parallel to 
intersect on the hypotenuse 
The lines split the triangle into two triangles and a rectangle. The two triangles have areas and 
What is the area of the rectangle?
be a right triangle with a right angle at Two lines, one parallel to and the other parallel to intersect on the hypotenuse The lines split the triangle into two triangles and a rectangle. The two triangles have areas and What is the area of the rectangle?Author: Ray Li
Author: Alex Zhu
22. Find the largest prime number 
such that when 
is written in base , it has at least trailing zeroes.
such that when is written in base , it has at least trailing zeroes.Author: Alex Zhu
23. Let be an equilateral triangle with side length 
. This triangle is rotated by some angle about its center to form triangle 
The intersection of and 
is an equilateral hexagon with an area that is 
the area of 
The side length of this hexagon can be expressed in the form 
where and are relatively prime positive integers. What is ?
be an equilateral triangle with side length . This triangle is rotated by some angle about its center to form triangle The intersection of and is an equilateral hexagon with an area that is the area of The side length of this hexagon can be expressed in the form where and are relatively prime positive integers. What is ?Author: Ray Li
Author: Alex Zhu
25. Let be the roots of the cubic 
. Given that 
is a cubic polynomial such that 
, 
, 
, and 
,
25. Let
be the roots of the cubic . Given that is a cubic polynomial such that , , , and , find 
.
.Author: Alex Zhu
26. Xavier takes a permutation of the numbers through at random, where each permutation has an equal probability of being selected. He then cuts the permutation into increasing contiguous subsequences, such that each subsequence is as long as possible. Compute the expected number of such subsequences.
through at random, where each permutation has an equal probability of being selected. He then cuts the permutation into increasing contiguous subsequences, such that each subsequence is as long as possible. Compute the expected number of such subsequences.Author: Alex Zhu
Clarification
Author: Ray Li
28. A fly is being chased by three spiders on the edges of a regular octahedron. The fly has a speed of 
meters per second, while each of the spiders has a speed of 
meters per second. The spiders choose their starting positions, and choose the fly's starting position, with the requirement that the fly must begin at a vertex. Each bug knows the position of each other bug at all times, and the goal of the spiders is for at least one of them to catch the fly. What is the maximum so that for any 
the fly can always avoid being caught?
meters per second, while each of the spiders has a speed of meters per second. The spiders choose their starting positions, and choose the fly's starting position, with the requirement that the fly must begin at a vertex. Each bug knows the position of each other bug at all times, and the goal of the spiders is for at least one of them to catch the fly. What is the maximum so that for any the fly can always avoid being caught?Author: Anderson Wang
29. How many positive integers with 
are there such that the coefficient of 
in the expansion of
29. How many positive integers
with are there such that the coefficient of in the expansion ofis zero?
Author: Ray Li
30. The Lattice Point Jumping Frog jumps between lattice points in a coordinate plane that are exactly unit apart. The Lattice Point Jumping Frog starts at the origin and makes 
jumps, ending at the origin. Additionally, it never lands on a point other than the origin more than once. How many possible paths could the frog have taken?
unit apart. The Lattice Point Jumping Frog starts at the origin and makes jumps, ending at the origin. Additionally, it never lands on a point other than the origin more than once. How many possible paths could the frog have taken?Author: Ray Li
Clarifications
31. Let be a triangle inscribed in circle 
, centered at with radius 
Let be the midpoint of , 
be the midpoint of , and be the point where line intersects . Given that lines 
and 
concur on and that 
, find the length of segment 
.
be a triangle inscribed in circle , centered at with radius Let be the midpoint of , be the midpoint of , and be the point where line intersects . Given that lines and concur on and that , find the length of segment .Author: Alex Zhu
What is 
?
?Author: Ray Li
33. You are playing a game in which you have envelopes, each containing a uniformly random amount of money between 
and 
dollars. (That is, for any real 
, the probability that the amount of money in a given envelope is between and is 
.) At any step, you take an envelope and look at its contents. You may choose either to keep the envelope, at which point you finish, or discard it and repeat the process with one less envelope. If you play to optimize your expected winnings, your expected winnings will be . What is 
the greatest integer less than or equal to ?
envelopes, each containing a uniformly random amount of money between and dollars. (That is, for any real , the probability that the amount of money in a given envelope is between and is .) At any step, you take an envelope and look at its contents. You may choose either to keep the envelope, at which point you finish, or discard it and repeat the process with one less envelope. If you play to optimize your expected winnings, your expected winnings will be . What is the greatest integer less than or equal to ?Author: Alex Zhu
34. 
are real numbers satisfying
are real numbers satisfying
Given that 
can be expressed in the form , where are relatively prime positive integers, compute .
can be expressed in the form , where are relatively prime positive integers, compute .Author: Alex Zhu
35. Let 
be the number of 1's in the binary representation of . Find the number of ordered pairs of integers 
with 
and 
.
be the number of 1's in the binary representation of . Find the number of ordered pairs of integers with and .Author:Anderson Wang
36. Let 
be the number of solutions to 
, where 
and 
are elements of the set 
and 
and 
are elements of the set 
. Find the number of for which is odd.
be the number of solutions to , where and are elements of the set and and are elements of the set . Find the number of for which is odd.Author: Alex Zhu
Clarification
37. In triangle , 
and 
. Suppose there exists a point in the interior of triangle such that 
, and that there are points and on segments and , such that 
and 
. Let 
meet 
at 
and let
meet at 
If is the midpoint of , compute the degree measure of 
, and . Suppose there exists a point in the interior of triangle such that , and that there are points and on segments and , such that and . Let meet at and let meet at If is the midpoint of , compute the degree measure of Authors: Alex Zhu and Ray Li
38. Let 
denote the sum of the 2011th powers of the roots of the polynomial
denote the sum of the 2011th powers of the roots of the polynomial 
. How many ones are in the binary expansion of ?
?Author: Alex Zhu
39. For positive integers 
let 
denote the largest integer 
such that 
divides 
Find the number of subsets (possibly containing 0 or 1 elements) of 
such that for any distinct 
, 
is even.
let denote the largest integer such that divides Find the number of subsets (possibly containing 0 or 1 elements) of such that for any distinct , is even.Author: Alex Zhu
Clarification
Find the largest possible value of 
.
.Author: Alex Zhu
41. Find the remainder when
is divided by 2016.
Author: Alex Zhu
42. In triangle 
and 
Let be a point outside triangle such that 
and 
Suppose that 
and that 
If 
can be expressed in the form 
where are pairwise relatively prime integers, find .
and Let be a point outside triangle such that and Suppose that and that If can be expressed in the form where are pairwise relatively prime integers, find .Author: Ray Li
43. An integer is selected at random between 1 and 
inclusive. The probability that 
is divisible by can be expressed in the form , where and are relatively prime positive integers. Find .
is selected at random between 1 and inclusive. The probability that is divisible by can be expressed in the form , where and are relatively prime positive integers. Find .Author: Alex Zhu
44. Given a set of points in space, a jump consists of taking two points, and and replacing with the reflection of over 
. Find the smallest number such that for any set of lattice points in -dimensional-space, it is possible to perform a finite number of jumps so that some two points coincide.
and and replacing with the reflection of over . Find the smallest number such that for any set of lattice points in -dimensional-space, it is possible to perform a finite number of jumps so that some two points coincide.Author: Anderson Wang
45. Let 
be 5 distinguishable keys, and let 
be distinguishable doors. For 
, key 
opens doors 
and 
(where 
) and can only be used once. The keys and doors are placed in some order along a hallway. Keyha walks into the hallway, picks a key
and opens a door with it, such that she never obtains a key before all the doors in front of it are unlocked. In how many orders can the keys and doors be placed such that Key$ha can open all of the doors?
be 5 distinguishable keys, and let be distinguishable doors. For , key opens doors and (where ) and can only be used once. The keys and doors are placed in some order along a hallway. Keyha walks into the hallway, picks a keyand opens a door with it, such that she never obtains a key before all the doors in front of it are unlocked. In how many orders can the keys and doors be placed such that Key$ha can open all of the doors?
Author: Mitchell Lee
Clarifications
46. If 
is a function from the set of positive integers to itself such that 
for all natural , and 
for all naturals and 
. Find the number of possible values of 
.
is a function from the set of positive integers to itself such that for all natural , and for all naturals and . Find the number of possible values of .Author: Alex Zhu
47. Let 
be an isosceles trapezoid with bases 
and 
and legs 
A circle with center passes through 
and 
Let be the midpoint of segment 
and ray 
meet again at 
Let be the midpoint of 
and be the intersection of with 
Let be the intersection of ray 
with ray 
There is a point 
on the circumcircle of 
such that 
The length of 
can be expressed in the form where and are relatively prime positive integers. What is ?
be an isosceles trapezoid with bases and and legs A circle with center passes through and Let be the midpoint of segment and ray meet again at Let be the midpoint of and be the intersection of with Let be the intersection of ray with ray There is a point on the circumcircle of such that The length of can be expressed in the form where and are relatively prime positive integers. What is ?Author: Ray Li
Author: Alex Zhu
49. Find the magnitude of the product of all complex numbers such that the recurrence defined by 
, 
, and 
also satisfies 
.
such that the recurrence defined by , , and also satisfies .Author: Alex Zhu
50. In tetrahedron 
, the circumcircles of faces 
, 
, and 
each have radius 
. The inscribed sphere of , centered at 
, has radius 
Additionally, 
. Let be the largest possible value of the circumradius of face . Given that can be expressed in the form 
, where and are relatively prime positive integers, find .
, the circumcircles of faces , , and each have radius . The inscribed sphere of , centered at , has radius Additionally, . Let be the largest possible value of the circumradius of face . Given that can be expressed in the form , where and are relatively prime positive integers, find .Author: Alex Zhu
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