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Difference between revisions of "2020 CIME I Problems"
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==Problem 3==
 
==Problem 3==
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In a math competition, all teams must consist of between <math>12</math> and <math>15</math> members,
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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>.
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inclusive. Mr. Beluhov has <math>n > 0</math> students and he realizes that he cannot form
 
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teams so that each of his students is on exactly one team. Find the sum of all
 
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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 11==
 
==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>.
 
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>.
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 +
[[2020 CIME I Problems/Problem 11 | Solution]]
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 +
==Problem 12==
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Define a sequence <math>a_0, a_1, a_2, ...</math> by <math></math>a_i = \underbrace{1 \ldots 1}_\text{<math>2^i</math> 1's} \underbrace{0 \ldots 0}_\text{<math>(2^{i}-1)</math> 0's} 1_2,<math></math> 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>.

Revision as of 19:01, 30 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 $(15,15)$.

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 $$ (Error compiling LaTeX. Unknown error_msg)a_i = \underbrace{1 \ldots 1}_\text{$2^i$ 1's} \underbrace{0 \ldots 0}_\text{$(2^{i}-1)$ 0's} 1_2,$$ (Error compiling LaTeX. Unknown error_msg) 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 $1000$.

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