Difference between revisions of "2023 SSMO Speed Round"
(Created page with "==Problem 1== Let <math>S_1 = \{2,0,3\}</math> and <math>S_2 = \{2,20,202,2023\}.</math> Find the last digit of \[ \sum_{a\in S_1,b\in S_2}a^b. \] 2022 SSMO Speed Round...") |
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==Problem 1== | ==Problem 1== | ||
Let <math>S_1 = \{2,0,3\}</math> and <math>S_2 = \{2,20,202,2023\}.</math> Find the last digit of | Let <math>S_1 = \{2,0,3\}</math> and <math>S_2 = \{2,20,202,2023\}.</math> Find the last digit of | ||
− | + | <cmath>\sum_{a\in S_1,b\in S_2}a^b.</cmath> | |
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[[2022 SSMO Speed Round Problems/Problem 1|Solution]] | [[2022 SSMO Speed Round Problems/Problem 1|Solution]] | ||
==Problem 2== | ==Problem 2== |
Revision as of 12:39, 3 July 2023
Contents
Problem 1
Let and Find the last digit of Solution
Problem 2
Let , , be independently chosen vertices lying in the square with coordinates , , , and . The probability that the centroid of triangle lies in the first quadrant is for relatively prime positive integers and Find
Problem 3
Pigs like to eat carrots. Suppose a pig randomly chooses 6 letters from the set Then, the pig randomly arranges the 6 letters to form a "word". If the 6 letters don't spell carrot, the pig gets frustrated and tries to spell it again (by rechoosing the 6 letters and respelling them). What is the expected number of tries it takes for the pig to spell "carrot"?
Problem 4
Let and for all be the Fibonacci numbers. If distinct positive integers satisfies , find the minimum possible value of
Problem 5
In a parallelogram of dimensions a point is choosen such that Find the sum of the maximum, , and minimum values of If you think there is no maximum, let
Problem 6
Find the smallest odd prime that does not divide .
Problem 7
At FenZhu High School, th graders have a 60\% of chance of having a dog and th graders have a 40\% chance of having a dog. Suppose there is a classroom of th grader and th graders. If exactly one person owns a dog, then the probability that a th grader owns the dog is for relatively prime positive integers and Find
Problem 8
Circle has chord of length . Point lies on chord such that Circle with radius and with radius lie on two different sides of Both and are tangent to at and If the sum of the maximum and minimum values of is find .
Problem 9
Find the sum of the maximum and minimum values of under the constraint that
Problem 10
In a circle centered at with radius we have non-intersecting chords and is outisde of quadrilateral and Let and Suppose that . If and , then for and squareless Find