Difference between revisions of "2020 CIME II Problems"
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==Problem 2== | ==Problem 2== | ||
− | Find the number of nonempty subsets <math>S</math> of <math>\{1,2,3, | + | Find the number of nonempty subsets <math>S</math> of <math>\{1,2,3,4,5,6,7,8,9,10\}</math> such that <math>S</math> has an even number of elements, and the product of the elements of <math>S</math> is even. |
[[2020 CIME II Problems/Problem 2|Solution]] | [[2020 CIME II Problems/Problem 2|Solution]] | ||
Line 34: | Line 34: | ||
Let <math>ABC</math> be a triangle with <math>AB=340</math>, <math>BC=146</math>, and <math>CA=390</math>. If <math>M</math> is a point on the interior of segment <math>BC</math> such that the length <math>AM</math> is an integer, then the average of all distinct possible values of <math>AM</math> can be expressed in the form <math>\tfrac pq</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. | Let <math>ABC</math> be a triangle with <math>AB=340</math>, <math>BC=146</math>, and <math>CA=390</math>. If <math>M</math> is a point on the interior of segment <math>BC</math> such that the length <math>AM</math> is an integer, then the average of all distinct possible values of <math>AM</math> can be expressed in the form <math>\tfrac pq</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. | ||
− | + | <asy> | |
size(3.5cm); defaultpen(fontsize(10pt)); | size(3.5cm); defaultpen(fontsize(10pt)); | ||
pair A,B,C,M; | pair A,B,C,M; | ||
Line 44: | Line 44: | ||
draw(A--B--C--A); | draw(A--B--C--A); | ||
draw(A--M,dashed); | draw(A--M,dashed); | ||
− | dot(" | + | dot("$A$",A,N); |
− | dot(" | + | dot("$B$",B,SW); |
− | dot(" | + | dot("$C$",C,SE); |
− | dot(" | + | dot("$M$",M,NW); |
− | label(" | + | label("$340$",A--B,W); |
− | label(" | + | label("$390$",A--C,E); |
− | label(" | + | label("$146$",B--C,S); |
− | + | </asy> | |
[[2020 CIME II Problems/Problem 7|Solution]] | [[2020 CIME II Problems/Problem 7|Solution]] | ||
Line 76: | Line 76: | ||
==Problem 12== | ==Problem 12== | ||
− | Positive integers <math>a</math>, <math>b</math>, <math>c</math> satisfy <cmath>lcm(gcd(a,b),c)=180,</cmath> <cmath>lcm(gcd(b,c),a)=360,</cmath> <cmath>lcm(gcd(c,a),b)=540.</cmath> | + | Positive integers <math>a</math>, <math>b</math>, <math>c</math> satisfy <cmath>\mathbf{lcm} (\mathbf{gcd} (a,b),c)=180,</cmath> <cmath>\mathbf{lcm} (\mathbf{gcd} (b,c),a)=360,</cmath> <cmath>\mathbf{lcm} (\mathbf{gcd} (c,a),b)=540.</cmath> |
Find the minimum possible value of <math>a+b+c</math>. | Find the minimum possible value of <math>a+b+c</math>. | ||
Latest revision as of 18:32, 7 January 2021
2020 CIME II (Answer Key) | AoPS Contest Collections | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
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
.
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
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 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All CIME Problems and Solutions |
The problems on this page are copyrighted by the MAC's Christmas Mathematics Competitions.