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Difference between revisions of "2020 CIME II Problems"
Line 76: Line 76:
  
 
==Problem 12==
 
==Problem 12==
−
Positive integers <math>a</math>, <math>b</math>, <math>c</math> satisfy
+
Positive integers <math>a</math>, <math>b</math>, <math>c</math> satisfy <cmath>\lcm(\gcd(a,b),c)=180, \lcm(\gcd(b,c),a)=360, \lcm(\gcd(c,a),b)=540.</cmath>
−
\begin{align*}
+
Find the minimum possible value of <math>a+b+c</math>.
−
            \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 <math>a+b+c</math>.
 
  
 
[[2020 CIME II Problems/Problem 12|Solution]]
 
[[2020 CIME II Problems/Problem 12|Solution]]

Revision as of 16:28, 5 September 2020

2020 CIME II (Answer Key)
Printable version | AoPS Contest Collections

Instructions

  1. This is a 15-question, 3-hour examination. All answers are integers ranging from $000$ to $999$, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.
  2. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

Problem 1

Let $ABC$ be a triangle. The bisector of $\angle ABC$ intersects $\overline{AC}$ at $E$, and the bisector of $\angle ACB$ intersects $\overline{AB}$ at $F$. If $BF=1$, $CE=2$, and $BC=3$, then the perimeter of $\triangle ABC$ can be expressed in the form $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 2

Find the number of nonempty subsets $S$ of $\{1,2,3,\ldots,10\}$ such that $S$ has an even number of elements, and the product of the elements of $S$ is even.

Solution

Problem 3

In a jar there are blue jelly beans and green jelly beans. Then, $15\%$ of the blue jelly beans are removed and $40\%$ of the green jelly beans are removed. If afterwards the total number of jelly beans is $80\%$ of the original number of jelly beans, then determine the percent of the remaining jelly beans that are blue.

Solution

Problem 4

The probability a randomly chosen positive integer $N<1000$ has more digits when written in base $7$ than when written in base $8$ can be expressed in the form $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 5

A positive integer $n$ is said to be $k$-consecutive if it can be written as the sum of $k$ consecutive positive integers. Find the number of positive integers less than $1000$ that are either $9$-consecutive or $11$-consecutive (or both), but not $10$-consecutive.

Solution

Problem 6

An infinite number of buckets, labeled $1$, $2$, $3$, $\ldots$, 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 $k$ is $2^{-k}$. Given that all three balls land in the same bucket $B$ and that $B$ is even, then the expected value of $B$ can be expressed as $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 7

Let $ABC$ be a triangle with $AB=340$, $BC=146$, and $CA=390$. If $M$ is a point on the interior of segment $BC$ such that the length $AM$ is an integer, then the average of all distinct possible values of $AM$ can be expressed in the form $\tfrac pq$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

[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("$A$",A,N); dot("$B$",B,SW); dot("$C$",C,SE); dot("$M$",M,NW); label("$340$",A--B,W); label("$390$",A--C,E); label("$146$",B--C,S); [/asy]

Solution

Problem 8

A committee has an oligarchy, consisting of $A\%$ of the members of the committee. Suppose that $B\%$ of the work is done by the oligarchy. If the average amount of work done by a member of the oligarchy is $16$ times the amount of work done by a nonmember of the oligarchy, find the maximum possible value of $B-A$.

Solution

Problem 9

Let $f(x)=x^2-2$. There are $N$ real numbers $x$ such that \[\underbrace{f(f(\ldots f}_{2019\text{ times}}(x)\ldots))=\underbrace{f(f(\ldots f}_{2020\text{ times}}(x)\ldots)).\]Find the remainder when $N$ is divided by $1000$.

Solution

Problem 10

Over all ordered triples of positive integers $(a,b,c)$ for which $a+b+c^2=abc$, compute the sum of all values of $a^3+b^2+c$.

Solution

Problem 11

Let $ABCD$ be a parallelogram such that $AB=40$, $BC=60$, and $BD=50$. Two externally tangent circles of radius $r$ are positioned in the interior of the parallelogram. The largest possible value of $r$ is $\sqrt m-\sqrt n$, where $m$ and $n$ are positive integers. Find $m+n$.

Solution

Problem 12

Positive integers $a$, $b$, $c$ satisfy 

\[\lcm(\gcd(a,b),c)=180, \lcm(\gcd(b,c),a)=360, \lcm(\gcd(c,a),b)=540.\] (Error compiling LaTeX. Unknown error_msg)

Find the minimum possible value of $a+b+c$.

Solution

Problem 13

A number is increasing if its digits, read from left to right, are strictly increasing. For instance, $5$ and $39$ are increasing while $224$ is not. Find the smallest positive integer not expressible as the sum of three or fewer increasing numbers.

Solution

Problem 14

A positive integer $x$ is lexicographically smaller than a positive integer $y$ if for some positive integer $i$, the $i$th digit of $x$ from the left is less than the $i$th digit of $y$ from the left, but for all positive integers $j<i$, the $j$th digit of $x$ is equal to the $j$th digit of $y$ from the left. Say the $i$th digit of a positive integer with less than $i$ digits is $-1$. For instance, $11$ is lexicographically smaller than $110$, which is in turn lexicographically smaller than $12$.

Let $A$ denote the number of positive integers $m$ for which there exists an integer $n\ge2020$ such that when the elements of the set $\{1,2,\ldots,n\}$ are sorted lexicographically from least to greatest, $m$ is the $2020$th number in this list. Find the remainder when $A$ is divided by $1000$.

Solution

Problem 15

Let $P_1P_2\cdots P_{72}$ be a regular $72$-gon with area $1$, and let $P_i=P_{i+72}$ for all integers $i$. Let $S$ be the sum of the squares all positive integers $a<72$ such that for all $i$, $P_{i-3a}\ne P_{i+a}$ and $P_{i-a}\ne P_{i+3a}$; for all $i$, lines $P_{i-3a}P_{i+a}$ and $P_{i-a}P_{i+3a}$ are not parallel, do not coincide, and intersect at a point $Q_i$; and the points $Q_1$, $Q_2$, $\ldots$, $Q_{72}$ form a polygon with positive, rational area. Find the remainder when $S$ divided by $1000$.

Solution

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

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