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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 <math>(15,15)</math>.
 +
 +
[[2020 CIME I Problems/Problem 1 | Solution]]
  
 
==Problem 2==
 
==Problem 2==
 
At the local Blast Store, there are sufficiently many items with a price of <math>\$n.99</math> for each nonnegative integer <math>n</math>. A sales tax of <math>7.5\%</math> 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.
 
At the local Blast Store, there are sufficiently many items with a price of <math>\$n.99</math> for each nonnegative integer <math>n</math>. A sales tax of <math>7.5\%</math> 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.
 +
 +
[[2020 CIME I Problems/Problem 2 | Solution]]
  
 
==Problem 3==
 
==Problem 3==
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teams so that each of his students is on exactly one team. Find the sum of all
 
teams so that each of his students is on exactly one team. Find the sum of all
 
possible values of <math>n</math>.
 
possible values of <math>n</math>.
 +
 +
[[2020 CIME I Problems/Problem 3 | Solution]]
  
 
==Problem 4==
 
==Problem 4==
−
There exists a unique positive real number <math>x</math> satisfying <cmath>x=\sqrt{x^2+\frac{1}{x^2}} - \sqrt{x^2-\frac{1}{x^2}}</cmath>. Given that <math>x</math> can be written in the form <math>x=2^\frac{m}{n} \cdot 3^\frac{-p}{q}</math> for integers <math>m, n, p, q</math> with <math>\gcd(m, n) = \gcd(p, q) = 1</math>, find <math>m+n+p+q</math>.
+
There exists a unique positive real number <math>x</math> satisfying <cmath>x=\sqrt{x^2+\frac{1}{x^2}} - \sqrt{x^2-\frac{1}{x^2}}.</cmath> Given that <math>x</math> can be written in the form <math>x=2^\frac{m}{n} \cdot 3^\frac{-p}{q}</math> for integers <math>m, n, p, q</math> with <math>\gcd(m, n) = \gcd(p, q) = 1</math>, find <math>m+n+p+q</math>.
 +
 
 +
[[2020 CIME I Problems/Problem 4 | Solution]]
 +
 
 +
==Problem 5==
 +
Let <math>ABCD</math> be a rectangle with sides <math>AB>BC</math> and let <math>E</math> be the reflection of <math>A</math> over <math>\overline{BD}</math>. If <math>EC=AD</math> and the area of <math>ECBD</math> is <math>144</math>, find the area of <math>ABCD</math>.
 +
 
 +
==Problem 6==
 +
Find the number of complex numbers <math>z</math> satisfying <math>|z|=1</math> and <math>z^850+z^350+1=0</math>.
 +
 
 +
==Problem 7==
 +
For every positive integer <math>n</math> define <cmath>f(n)=\frac{n}{1 \cdot 3 \cdot 5 \cdot \cdot \cdot (2n+1).</cmath> Suppose that the sum <math>f(1)+f(2)+\cdot \cdot \cdot+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>.

Revision as of 13:38, 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 $ABCD$.

Problem 6

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

Problem 7

For every positive integer $n$ define 

\[f(n)=\frac{n}{1 \cdot 3 \cdot 5 \cdot \cdot \cdot (2n+1).\] (Error compiling LaTeX. Unknown error_msg)

Suppose that the sum $f(1)+f(2)+\cdot \cdot \cdot+f(2020)$ can be expressed as $\frac{p}{q}$ for relatively prime integers $p$ and $q$. Find the remainder when $p$ is divided by $1000$.

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