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Difference between revisions of "2020 CIME I Problems"
 
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==Problem 1==
 
==Problem 1==
−
A knight begins on the point <math>(0,0)</math> in the coordinate plane. From any point <math>(x,y)</math> the knight moves to either <math>(x+2,y+1)</math> or <math>(x+1,y+2)</math>. Find the number of ways the knight can reach <math>(15,15)</math>.
+
A knight begins on the point <math>(0,0)</math> in the coordinate plane. From any point <math>(x,y)</math> the knight moves to either <math>(x+2,y+1)</math> or <math>(x+1,y+2)</math>. Find the number of ways the knight can reach the point <math>(15,15)</math>.
  
 
[[2020 CIME I Problems/Problem 1 | Solution]]
 
[[2020 CIME I Problems/Problem 1 | Solution]]
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==Problem 3==
 
==Problem 3==
−
In a math competition, all teams must consist of between <math>12</math> and <math>15</math> members,
+
In a math competition, all teams must consist of between <math>12</math> and <math>15</math> members, inclusive. Mr. Beluhov has <math>n > 0</math> 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 <math>n</math>.
−
inclusive. Mr. Beluhov has <math>n > 0</math> 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 <math>n</math>.
 
  
 
[[2020 CIME I Problems/Problem 3 | Solution]]
 
[[2020 CIME I Problems/Problem 3 | Solution]]
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==Problem 6==
 
==Problem 6==
−
Find the number of complex numbers <math>z</math> satisfying <math>|z|=1</math> and <math>z^{850}</math>+z^{350}+1=0<math>.
+
Find the number of complex numbers <math>z</math> satisfying <math>|z|=1</math> and <math>z^{850}+z^{350}+1=0</math>.
  
 
[[2020 CIME I Problems/Problem 6 | Solution]]
 
[[2020 CIME I Problems/Problem 6 | Solution]]
  
 
==Problem 7==
 
==Problem 7==
−
For every positive integer </math>n<math> define <cmath>f(n)=\frac{n}{1 \cdot 3 \cdot 5 \cdots (2n+1)}.</cmath> Suppose that the sum </math>f(1)+f(2)+\cdots+f(2020)<math> can be expressed as </math>\frac{p}{q}<math> for relatively prime integers </math>p<math> and </math>q<math>. Find the remainder when </math>p<math> is divided by </math>1000<math>.
+
For every positive integer <math>n</math>, define <cmath>f(n)=\frac{n}{1 \cdot 3 \cdot 5 \cdots (2n+1)}.</cmath> Suppose that the sum <math>f(1)+f(2)+\cdots+f(2020)</math> can be expressed as <math>\frac{p}{q}</math> for relatively prime integers <math>p</math> and <math>q</math>. Find the remainder when <math>p</math> is divided by <math>1000</math>.
  
 
[[2020 CIME I Problems/Problem 7 | Solution]]
 
[[2020 CIME I Problems/Problem 7 | Solution]]
  
 
==Problem 8==
 
==Problem 8==
−
A person has been declared the first to inhabit a certain planet on day </math>N=0<math>. For each positive integer </math>N>0<math>, if there is a positive number of people on the planet, then either one of the following three occurs, each with probability </math>\frac{1}{3}<math>:
+
A person has been declared the first to inhabit a certain planet on day <math>N=0</math>. For each positive integer <math>N>0</math>, if there is a positive number of people on the planet, then either one of the following three occurs, each with probability <math>\frac{1}{3}</math>:
 +
 
 
:(i) the population stays the same;
 
:(i) the population stays the same;
−
:(ii) the population increases by </math>2^N<math>; or
+
:(ii) the population increases by <math>2^N</math>; or
−
:(iii) the population decreases by </math>2^{N-1}<math>. (If there are no greater than </math>2^{N-1}<math> people on the planet, the population drops to zero, and the process terminates).
+
:(iii) the population decreases by <math>2^{N-1}</math>. (If there are no greater than <math>2^{N-1}</math> people on the planet, the population drops to zero, and the process terminates.)
−
The probability that at some point there are exactly </math>2^{20}+2^{19}+2^{10}+2^9+1<math> people on the planet can be written as </math>\frac{p}{3^q}<math>, where </math>p<math> and </math>q<math> are positive integers such that </math>p<math> is not divisible by </math>3<math>. Find the remainder when </math>p+q<math> is divided by </math>1000$.
+
 
 +
The probability that at some point there are exactly <math>2^{20}+2^{19}+2^{10}+2^9+1</math> people on the planet can be written as <math>\frac{p}{3^q}</math>, where <math>p</math> and <math>q</math> are positive integers such that <math>p</math> isn't divisible by <math>3</math>. Find the remainder when <math>p+q</math> is divided by <math>1000</math>.
 +
 
 +
[[2020 CIME I Problems/Problem 8 | Solution]]
 +
 
 +
==Problem 9==
 +
Let <math>ABCD</math> be a cyclic quadrilateral with <math>AB=6, AC=8, BD=5, CD=2</math>. Let <math>P</math> be the point on <math>\overline{AD}</math> such that <math>\angle APB = \angle CPD</math>. Then <math>\frac{BP}{CP}</math> can be expressed in the form <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
 +
 
 +
[[2020 CIME I Problems/Problem 9 | Solution]]
 +
 
 +
==Problem 10==
 +
Let <math>1=d_1<d_2<\cdots<d_k=n</math> be the divisors of a positive integer <math>n</math>. Let <math>S</math> be the sum of all positive integers <math>n</math> satisfying <cmath>n=d_1^1+d_2^2+d_3^3+d_4^4.</cmath> Find the remainder when <math>S</math> is divided by <math>1000</math>.
 +
 
 +
[[2020 CIME I Problems/Problem 10 | Solution]]
 +
 
 +
==Problem 11==
 +
An <math>excircle</math> of a triangle is a circle tangent to one of the sides of the triangle and the extensions of the other two sides. Let <math>ABC</math> be a triangle with <math>\angle ACB = 90</math> and let <math>r_A, r_B, r_C</math> denote the radii of the excircles opposite to <math>A, B, C</math>, respectively. If <math>r_A=9</math> and <math>r_B=11</math>, then <math>r_C</math> can be expressed in the form <math>m+\sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers and <math>n</math> isn't divisible by the square of any prime. Find <math>m+n</math>.
 +
 
 +
[[2020 CIME I Problems/Problem 11 | Solution]]
 +
 
 +
==Problem 12==
 +
Define a sequence <math>a_0, a_1, a_2, ...</math> by <cmath>a_i=\underbrace{1\ldots1}_{2^{i}\text{ 1's}}\underbrace{0\ldots0}_{(2^i-1)\text{ 0's}}1_2,</cmath> where <math>a_i</math> is expressed in binary. Let <math>S</math> be the sum of the digits when <math>a_0 a_1 a_2 \cdots a_{10}</math> is expressed in binary. Find the remainder when <math>S</math> is divided by <math>1000</math>.
 +
 
 +
[[2020 CIME I Problems/Problem 12 | Solution]]
 +
 
 +
==Problem 13==
 +
Chris writes on a piece of paper the positive integers from <math>1</math> to <math>8</math> in that order. Then, he randomly writes either <math>+</math> or <math>\times</math> between every two adjacent numbers, each with equal probability. The expected value of the expression he writes can be expressed as <math>\frac{p}{q}</math> for relatively prime positive integers <math>p</math> and <math>q</math>. Find the remainder when <math>p+q</math> is divided by <math>1000</math>.
 +
 
 +
[[2020 CIME I Problems/Problem 13 | Solution]]
 +
 
 +
==Problem 14==
 +
Let ABC be a triangle with sides <math>AB = 5, BC = 7, CA = 8</math>. Denote by <math>O</math> and <math>I</math> the circumcenter and incenter of <math>\triangle ABC</math>, respectively. The incircle of <math>\triangle ABC</math> touches <math>\overline{BC}</math> at <math>D</math>, and line <math>OD</math> intersects the circumcircle of <math>\triangle AID</math> again at <math>K</math>. Then the length of <math>DK</math> can be expressed in the form <math>\frac{m \sqrt n}{p}</math>, where <math>m, n, p</math> are positive integers, <math>m</math> and <math>p</math> are relatively prime, and <math>n</math> is not divisible by the square of any prime. Find <math>m+n+p</math>.
 +
 
 +
[[2020 CIME I Problems/Problem 14 | Solution]]
 +
 
 +
==Problem 15==
 +
Find the number of integer sequences <math>a_1, a_2, \ldots, a_6</math> such that
 +
:(1) <math>0 \le a_1 < 6</math> and <math>12 \le a_6 < 18</math>,
 +
:(2) <math>1 \le a_{k+1}-a_k < 6</math> for all <math>1 \le k < 6</math>, and
 +
:(3) there do not exist <math>1 \le i < j \le 6</math> such that <math>a_j-a_i</math> is divisible by <math>6</math>.
 +
 
 +
[[2020 CIME I Problems/Problem 15 | Solution]]
 +
 
 +
{{CIME box|year=2020|n=I|before=[[2019 CIME II Problems]]|after=[[2020 CIME II Problems]]}}

Latest revision as of 17:00, 31 August 2020

2020 CIME I (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

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 \[a_i=\underbrace{1\ldots1}_{2^{i}\text{ 1's}}\underbrace{0\ldots0}_{(2^i-1)\text{ 0's}}1_2,\] 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
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All CIME Problems and Solutions
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