Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2020 CIME I Problems"
Line 47: Line 47:
  
 
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>.
 
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</math>, 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$.
 +
 +
[[2020 CIME I Problems/Problem 9 | Solution]]

Revision as of 16:39, 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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2020_CIME_I_Problems&oldid=132761"