Difference between revisions of "2020 CIME I Problems"
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==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 \gcd(m, n) = \gcd(p, q) = 1, 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>. |
Revision as of 14:31, 30 August 2020
2020 CIME I (Answer Key) | AoPS Contest Collections | ||
Instructions
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Contents
Problem 1
A knight begins on the point in the coordinate plane. From any point
the knight moves to either
or
. Find the number of ways the knight can reach
.
Problem 2
At the local Blast Store, there are sufficiently many items with a price of for each nonnegative integer
. A sales tax of
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.
Problem 3
In a math competition, all teams must consist of between and
members,
inclusive. Mr. Beluhov has
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
.
Problem 4
There exists a unique positive real number satisfying
. Given that
can be written in the form
for integers
with
, find
.