2020 CIME I Problems

2020 CIME I (Answer Key) Printable version \| AoPS Contest Collections
Instructions 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. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
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Problem 1

A knight begins on the point $(0,0)$ in the coordinate plane. From any point $(x,y)$ the knight moves to either $(x+2,y+1)$ or $(x+1,y+2)$ . Find the number of ways the knight can reach the point $(15,15)$ .

Solution

Problem 2

At the local Blast Store, there are sufficiently many items with a price of $$n.99$ for each nonnegative integer $n$ . A sales tax of $7.5\%$ 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.

Solution

Problem 3

In a math competition, all teams must consist of between $12$ and $15$ members, inclusive. Mr. Beluhov has $n > 0$ 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 $n$ .

Solution

Problem 4

There exists a unique positive real number $x$ satisfying $x=\sqrt{x^2+\frac{1}{x^2}} - \sqrt{x^2-\frac{1}{x^2}}.$ Given that $x$ can be written in the form $x=2^\frac{m}{n} \cdot 3^\frac{-p}{q}$ for integers $m, n, p, q$ with $\gcd(m, n) = \gcd(p, q) = 1$ , find $m+n+p+q$ .

Solution

Problem 5

Let $ABCD$ be a rectangle with sides $AB>BC$ and let $E$ be the reflection of $A$ over $\overline{BD}$ . If $EC=AD$ and the area of $ECBD$ is $144$ , find the area of $ABCD$ .

Solution

Problem 6

Find the number of complex numbers $z$ satisfying $|z|=1$ and $z^{850}+z^{350}+1=0$ .

Solution

Problem 7

For every positive integer $n$ , define $f(n)=\frac{n}{1 \cdot 3 \cdot 5 \cdots (2n+1)}.$ Suppose that the sum $f(1)+f(2)+\cdots+f(2020)$ can be expressed as $\frac{p}{q}$ for relatively prime integers $p$ and $q$ . Find the remainder when $p$ is divided by $1000$ .

Solution

Problem 8

A person has been declared the first to inhabit a certain planet on day $N=0$ . For each positive integer $N=0$ , if there is a positive number of people on the planet, then either one of the following three occurs, each with probability $\frac{1}{3}$ :

(i) the population stays the same;

(ii) the population increases by $2^N$ ; or

(iii) the population decreases by $2^{N-1}$ . (If there are no greater than $2^{N-1}$ people on the planet, the population drops to zero, and the process terminates.)

The probability that at some point there are exactly $2^{20}+2^{19}+2^{10}+2^9+1$ people on the planet can be written as $\frac{p}{3^q}$ , where $p$ and $q$ are positive integers such that $p$ isn't divisible by $3$ . Find the remainder when $p+q$ is divided by $1000$ .

Solution

Problem 9

Let $ABCD$ be a cyclic quadrilateral with $AB=6, AC=8, BD=5, CD=2$ . Let $P$ be the point on $\overline{AD}$ such that $\angle APB = \angle CPD$ . Then $\frac{BP}{CP}$ can be expressed in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Solution

Problem 10

Let $1=d_1<d_2<\cdots<d_k=n$ be the divisors of a positive integer $n$ . Let $S$ be the sum of all positive integers $n$ satisfying $n=d_1^1+d_2^2+d_3^3+d_4^4.$ Find the remainder when $S$ is divided by $1000$ .

Solution

Problem 11

An $excircle$ of a triangle is a circle tangent to one of the sides of the triangle and the extensions of the other two sides. Let $ABC$ be a triangle with $\angle ACB = 90$ and let $r_A, r_B, r_C$ denote the radii of the excircles opposite to $A, B, C$ , respectively. If $r_A=9$ and $r_B=11$ , then $r_C$ can be expressed in the form $m+\sqrt{n}$ , where $m$ and $n$ are positive integers and $n$ isn't divisible by the square of any prime. Find $m+n$ .

Solution

Problem 12

Define a sequence $a_0, a_1, a_2, ...$ by $a_i = \underbrace{1\ldots 1}_{2^i 1's} \underbrace{0\ldots 0}_{(2^i-1) 0's} 1_2,$ where $a_i$ is expressed in binary. Let $S$ be the sum of the digits when $a_0 a_1 a_2 \cdots a_{10}$ is expressed in binary. Find the remainder when $S$ is divided by $1000$ .

Solution

Problem 13

Chris writes on a piece of paper the positive integers from $1$ to $8$ in that order. Then, he randomly writes either $+$ or $\times$ between every two adjacent numbers, each with equal probability. The expected value of the expression he writes can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p$ and $q$ . Find the remainder when $p+q$ is divided by $1000$ .

Solution

Problem 14

Let ABC be a triangle with sides $AB = 5, BC = 7, CA = 8$ . Denote by $O$ and $I$ the circumcenter and incenter of $\triangle ABC$ , respectively. The incircle of $\triangle ABC$ touches $\overline{BC}$ at $D$ , and line $OD$ intersects the circumcircle of $\triangle AID$ again at $K$ . Then the length of $DK$ can be expressed in the form $\frac{m \sqrt n}{p}$ , where $m, n, p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+p$ .

Solution

Problem 15

Find the number of integer sequences $a_1, a_2, \ldots, a_6$ such that

(1) $0 \le a_1 < 6$ and $12 \le a_6 < 18$ ,

(2) $1 \le a_{k+1}-a_k < 6$ for all $1 \le k < 6$ , and

(3) there do not exist $1 \le i < j \le 6$ such that $a_j-a_i$ is divisible by $6$ .

Solution

2020 CIME I (Problems • Answer Key • Resources)
Preceded by 2019 CIME II Problems	Followed by 2020 CIME II Problems
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2020 CIME I Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15