G285 2021 Summer Problem Set
Welcome to the Birthday Problem Set! In this set, there are multiple choice AND free-response questions. Feel free to look at the solutions if you are stuck:
Let If is a positive integer, find the sum of all values of such that for some constant .
groups of molecules are gathered in a lab. The scientists in the lab randomly assign the molecules into groups of . Within these groups, there will be distinguishable labels (Strong acid, weak acid, strong base, weak base, nonelectrolyte), and each molecule will randomly be assigned a label such that teams can be empty, and each label is unique in the group. Find the number of ways that the molecules can be arranged by the scientists.
people are attending a hotel conference, of which are executives, and of which are speakers. Each person is designated a seat at one of round tables, each containing seats. If executives must sit at least one speaker and executive, there are ways the people can be seated. Find . Assume seats, people, and table rotations are distinguishable.
Suppose is an equilateral triangle. Let points and lie on the extensions of and respectively such that and . If there exists a point outside of such that , and there exists a point outside outside of such that , the area can be represented as , where and are squarefree,. Find
Let be a rectangle with and . Let points and lie on such that is the midpoint of and lies on . Let point be the center of the circumcircle of quadrilateral such that and lie on the circumcircle of and respectively, along with and . If the shortest distance between and is , and are degenerate, and , find
Geometry285 is playing the game "Guess And Choose". In this game, Geometry285 selects a subset of not necessarily distinct integers from the set such that the sum of all elements in is . Each distinct is selected chronologically and placed in , such that , , , and so on. Then, the elements are randomly arranged. Suppose represents the total number of outcomes that a subset containing integers sums to . If distinct permutations of the same set are considered unique, find the remainder when is divided by .
Let , Let be the twelve roots that satisfies , find the least possible value of
Let circles and with centers and concur at points and such that , . Suppose a point on the extension of is formed such that and lines and intersect and at and respectively. If , the value of can be represented as , where and are relatively prime positive integers, and is square free. Find
Let for . Suppose makes for distinct prime factors . If for is where must satisfy that is an integer, and is divisible by the th and th triangular number. Find
Let a recursive sequence be defined such that , and . Find the last digits of
Suppose the function . If , and the polynomial contains the points ,, and , find the smallest value of for which
Let circles ,, and concur at , where is the common chord shared by , is the common chord shared by , and lies on the common internal tangent of . Let the extension of and intersect and again at and respectively. If , prove is a parallelogram.
Bobby the frog is hopping around the unit circle. Suppose Bobby starts at . After every th minute, Bobby moves to such that , and is an th root of unity for . Suppose Bobby is unidirectional for every minutes, and randomly chooses to reverse his direction after each cycle. In how many ways can Bobby travel around the unit circle exactly times?
Find the average of all values such that