Difference between revisions of "2020 CIME I Problems"
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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} | + | 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]] |
Revision as of 14:07, 30 August 2020
2020 CIME I (Answer Key) | AoPS Contest Collections | ||
Instructions
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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 .
Problem 5
Let be a rectangle with sides and let be the reflection of over . If and the area of is , find the area of .
Problem 6
Find the number of complex numbers satisfying and .