2021 AMC A Proposals
by djmathman, Feb 5, 2021, 11:58 PM
hmm
12A 1 (Summer 2020). What is the value of
12A 20 (Winter 2018). Suppose that on a parabola with vertex
and a focus 
there exists a point 
such that 
and 
. What is the sum of all possible values of the length 
12A 21 (Spring 2018?). The five solutions to the equation
may be written in the form 
for 
where 
and 
are real. Let 
be the unique ellipse that passes through the points 
and 
. The eccentricity of can be written in the form 
where 
and 
are relatively prime positive integers. What is 
? (Recall that the eccentricity of an ellipse is the ratio 
, where 
is the length of the major axis of and 
is the is the distence between its two foci.)
10A 22 (Summer 2019). Hiram’s algebra notes are
pages long and are printed on 
sheets of paper; the first sheet contains pages 
and 
, the second sheet contains pages 
and 
, and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly 
. How many sheets were borrowed?
10A 25 (Summer 2019). How many ways are there to place indistinguishable red chips, indistinguishable blue chips, and indistinguishable green chips in the squares of a 
grid so that no two chips of the same color are directly adjacent to each other, either vertically or horizontally?
I can't tell if these are good or bad, or both, or neither. Maybe it doesn't matter. Regardless, five proposals is a lot
12A 1 (Summer 2020). What is the value of
12A 20 (Winter 2018). Suppose that on a parabola with vertex
and a focus 
there exists a point 
such that 
and 
. What is the sum of all possible values of the length 
12A 21 (Spring 2018?). The five solutions to the equation
may be written in the form 
for 
where 
and 
are real. Let 
be the unique ellipse that passes through the points 
and 
. The eccentricity of can be written in the form 
where 
and 
are relatively prime positive integers. What is 
? (Recall that the eccentricity of an ellipse is the ratio 
, where 
is the length of the major axis of and 
is the is the distence between its two foci.)10A 22 (Summer 2019). Hiram’s algebra notes are
pages long and are printed on 
sheets of paper; the first sheet contains pages 
and 
, the second sheet contains pages 
and 
, and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to borrow some pages from the middle of the notes. When Hiram comes back, he discovers that his roommate has taken a consecutive set of sheets from the notes and that the average (mean) of the page numbers on all remaining sheets is exactly 
. How many sheets were borrowed?10A 25 (Summer 2019). How many ways are there to place
indistinguishable red chips, indistinguishable blue chips, and indistinguishable green chips in the squares of a 
grid so that no two chips of the same color are directly adjacent to each other, either vertically or horizontally?I can't tell if these are good or bad, or both, or neither. Maybe it doesn't matter. Regardless, five proposals is a lot
This post has been edited 3 times. Last edited by djmathman, Feb 6, 2021, 12:02 AM
April 2025
November 2024