# 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:

## Contents

## Problem 1

Find

## Problem 2

Let If is a positive integer, find the sum of all values of such that for some constant .

## Problem 3

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.

## Problem 4

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.

## Problem 5

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

## Problem 6

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

## Problem 7

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 .

## Problem 8

Let , Let be the twelve roots that satisfies , find the least possible value of

## Problem 9

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

## Problem 10

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

## Problem 11

Let a recursive sequence be defined such that , and . Find the last digits of

## Problem 12

Suppose the function . If , and the polynomial contains the points ,, and , find the smallest value of for which

## Problem 13

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.

## Problem 14

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?

## Problem 15

Find the average of all values such that