2020 CIME II Problems
2020 CIME II (Answer Key) | AoPS Contest Collections | ||
Instructions
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Contents
Problem 1
Let be a triangle. The bisector of intersects at , and the bisector of intersects at . If , , and , then the perimeter of can be expressed in the form , where and are relatively prime positive integers. Find .
Problem 2
Find the number of nonempty subsets of such that has an even number of elements, and the product of the elements of is even.
Problem 3
In a jar there are blue jelly beans and green jelly beans. Then, of the blue jelly beans are removed and of the green jelly beans are removed. If afterwards the total number of jelly beans is of the original number of jelly beans, then determine the percent of the remaining jelly beans that are blue.
Problem 4
The probability a randomly chosen positive integer has more digits when written in base than when written in base can be expressed in the form , where and are relatively prime positive integers. Find .
Problem 5
A positive integer is said to be -consecutive if it can be written as the sum of consecutive positive integers. Find the number of positive integers less than that are either -consecutive or -consecutive (or both), but not -consecutive.
Problem 6
An infinite number of buckets, labeled , , , , lie in a line. A red ball, a green ball, and a blue ball are each tossed into a bucket, such that for each ball, the probability the ball lands in bucket is . Given that all three balls land in the same bucket and that is even, then the expected value of can be expressed as , where and are relatively prime positive integers. Find .
Problem 7
Let be a triangle with , , and . If is a point on the interior of segment such that the length is an integer, then the average of all distinct possible values of can be expressed in the form , where and are relatively prime positive integers. Find . [asy]
size(3.5cm); defaultpen(fontsize(10pt)); pair A,B,C,M; A=dir(95); B=dir(-117); C=dir(-63); M=(2B+3C)/5;
draw(A--B--C--A); draw(A--M,dashed); dot("",A,N); dot("",B,SW); dot("",C,SE); dot("",M,NW); label("",A--B,W); label("",A--C,E); label("",B--C,S); [/asy]
Problem 8
A committee has an oligarchy, consisting of of the members of the committee. Suppose that of the work is done by the oligarchy. If the average amount of work done by a member of the oligarchy is times the amount of work done by a nonmember of the oligarchy, find the maximum possible value of .
Problem 9
Let . There are real numbers such that Find the remainder when is divided by .
Problem 10
Over all ordered triples of positive integers for which , compute the sum of all values of .
Problem 11
Let be a parallelogram such that , , and . Two externally tangent circles of radius are positioned in the interior of the parallelogram. The largest possible value of is , where and are positive integers. Find .
Problem 12
Positive integers , , satisfy \begin{align*}
\operatorname{lcm}(\gcd(a,b),c)&=180,\\ \operatorname{lcm}(\gcd(b,c),a)&=360,\\ \operatorname{lcm}(\gcd(c,a),b)&=540. \end{align*}Find the minimum possible value of .
Problem 13
A number is increasing if its digits, read from left to right, are strictly increasing. For instance, and are increasing while is not. Find the smallest positive integer not expressible as the sum of three or fewer increasing numbers.
Problem 14
A positive integer is lexicographically smaller than a positive integer if for some positive integer , the th digit of from the left is less than the th digit of from the left, but for all positive integers , the th digit of is equal to the th digit of from the left. Say the th digit of a positive integer with less than digits is . For instance, is lexicographically smaller than , which is in turn lexicographically smaller than .
Let denote the number of positive integers for which there exists an integer such that when the elements of the set are sorted lexicographically from least to greatest, is the th number in this list. Find the remainder when is divided by .
Problem 15
Let be a regular -gon with area , and let for all integers . Let be the sum of the squares all positive integers such that for all , and ; for all , lines and are not parallel, do not coincide, and intersect at a point ; and the points , , , form a polygon with positive, rational area. Find the remainder when divided by .
See also
2020 CIME II (Problems • Answer Key • Resources) | ||
Preceded by 2020 CIME I Problems |
Followed by 2021 CIME I Problems | |
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All CIME Problems and Solutions |
The problems on this page are copyrighted by the MAC's Christmas Mathematics Competitions.