2020 CIME I Problems
2020 CIME I (Answer Key) | AoPS Contest Collections | ||
Instructions
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Contents
Problem 1
A knight begins on the point in the coordinate plane. From any point the knight moves to either or . Find the number of ways the knight can reach the point .
Problem 2
At the local Blast Store, there are sufficiently many items with a price of for each nonnegative integer . A sales tax of is applied on all items. If the total cost of a purchase, after tax, is an integer number of cents, find the minimum possible number of items in the purchase.
Problem 3
In a math competition, all teams must consist of between and members, inclusive. Mr. Beluhov has students and he realizes that he cannot form teams so that each of his students is on exactly one team. Find the sum of all possible values of .
Problem 4
There exists a unique positive real number satisfying Given that can be written in the form for integers with , find .
Problem 5
Let be a rectangle with sides and let be the reflection of over . If and the area of is , find the area of .
Problem 6
Find the number of complex numbers satisfying and .
Problem 7
For every positive integer , define Suppose that the sum can be expressed as for relatively prime integers and . Find the remainder when is divided by .
Problem 8
A person has been declared the first to inhabit a certain planet on day . For each positive integer , if there is a positive number of people on the planet, then either one of the following three occurs, each with probability :
- (i) the population stays the same;
- (ii) the population increases by ; or
- (iii) the population decreases by . (If there are no greater than people on the planet, the population drops to zero, and the process terminates.)
The probability that at some point there are exactly people on the planet can be written as , where and are positive integers such that isn't divisible by . Find the remainder when is divided by .
Problem 9
Let be a cyclic quadrilateral with . Let be the point on such that . Then can be expressed in the form , where and are relatively prime positive integers. Find .
Problem 10
Let be the divisors of a positive integer . Let be the sum of all positive integers satisfying Find the remainder when is divided by .
Problem 11
An of a triangle is a circle tangent to one of the sides of the triangle and the extensions of the other two sides. Let be a triangle with and let denote the radii of the excircles opposite to , respectively. If and , then can be expressed in the form , where and are positive integers and isn't divisible by the square of any prime. Find .
Problem 12
Define a sequence by where is expressed in binary. Let be the sum of the digits when is expressed in binary. Find the remainder when is divided by .
Problem 13
Chris writes on a piece of paper the positive integers from to in that order. Then, he randomly writes either or between every two adjacent numbers, each with equal probability. The expected value of the expression he writes can be expressed as for relatively prime positive integers and . Find the remainder when is divided by .
Problem 14
Let ABC be a triangle with sides . Denote by and the circumcenter and incenter of , respectively. The incircle of touches at , and line intersects the circumcircle of again at . Then the length of can be expressed in the form , where are positive integers, and are relatively prime, and is not divisible by the square of any prime. Find .
Problem 15
Find the number of integer sequences such that
- (1) and ,
- (2) for all , and
- (3) there do not exist such that is divisible by .
2020 CIME I (Problems • Answer Key • Resources) | ||
Preceded by 2019 CIME II Problems |
Followed by 2020 CIME II Problems | |
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