Mock AIME 2 2010 Problems
Mock AIME 2 2010 (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
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Contents
Problem 1
There are lemmings. At each step, we may separate the lemmings into groups of and purge the remainder, separate them into groups of and purge the remainder, or pick one lemming and purge it. Find the smallest number of steps necessary to remove all lemmings.
Problem 2
Let be nonnegative integers such that , and define so that , with for . Given that can take on distinct values, find the remainder when is divided by 1000.
Problem 3
Five gunmen are shooting each other. At the same moment, each randomly chooses one of the other four to shoot. The probability that there are some two people shooting each other can be expressed in the form , where are relatively prime positive integers. Find .
Problem 4
Anderson is bored in physics class. His favorite numbers are , and . He writes , and randomly appends one of his favorite numbers to the end of the decimal he has already written. Since physics class is infinitely long, Anderson writes an infinitely long decimal number. (An example of such a number is ) If the expected value of the number Anderson wrote down is of the form , where and are relatively prime positive integers, find .
Problem 5
Let . Find the three rightmost nonzero digits of the product of the coefficients of .
Problem 6
Let denote the set , and define , where is a subset of the positive integers, to output the greatest common divisor of all elements in , unless is empty, in which case it will output 0. Find the last three digits of , where ranges over all subsets of .
Problem 7
Find the number of functions from to itself such that for all .
Problem 8
In triangle , , , and . In addition, there is a point lying on segment such that . Given that the length of the radius of the circle through and that is tangent to side can be expressed in the form , where and are relatively prime integers, find .
Problem 9
Given that are reals such that , the largest possible value of can be expressed in the form , where and are integers, is a positive integer not divisible by the square of any prime, and is a positive integer such that . Find .
Problem 10
How many positive integers satisfy , where is the number of positive integers less than or equal to relatively prime to ?
Problem 11
Let be a function such that For example, , where we are summing over the triples , and . Find the last three digits of .
Problem 12
Let Find the sum of digits of in base-100.
Problem 13
is inscribed in circle . The radius of is , and . When the incircle of is reflected across segment , it is tangent to . Given that the inradius of can be expressed in the form , where and are positive integers and is not divisible by the square of any prime, find .
Problem 14
Alex and Mitchell decide to play a game. In this game, there are 2010 pieces of candy on a table, and starting with Alex, the two take turns eating some positive integer number of pieces of candy. Since it is bad manners to eat the last candy, whoever eats the last candy loses. The two decide that the amount of candy a person can pick will be a set equal to the positive divisors of a number less than 2010 that each person picks (individually) from the beginning. For example, if Alex picks 19 and Mitchell picks 20, then on each turn, Alex must eat either 1 or 19 pieces, and Mitchell must eat 1, 2, 4, 5, 10, or 20 pieces. Mitchell knows Alex well enough to determine with certainty that Alex will either be immature and pick 69, or be clichéd and pick 42. How many integers can Mitchell pick to guarantee that he will not lose the game?
Problem 15
Given that can be expressed in the form , where and , find the unique integer between 0 and 982 inclusive such that divides . [Note: is a prime.]