MathDB

1.5

Part of 2021 CMIMC

Problems(3)

2021 Alg/NT Div 1 P5

Source:

3/2/2021
Suppose ff is a degree 42 polynomial such that for all integers 0≤i≤420\le i\le 42,
f(i)+f(43+i)+f(2⋅43+i)+⋯+f(46⋅43+i)=(−2)if(i)+f(43+i)+f(2\cdot43+i)+\cdots+f(46\cdot43+i)=(-2)^i
Find f(2021)−f(0)f(2021)-f(0).
Proposed by Adam Bertelli
algebra
2021 Geo Div 1 P5

Source:

3/2/2021
Let γ1,γ2,γ3\gamma_1, \gamma_2, \gamma_3 be three circles with radii 3,4,9,3, 4, 9, respectively, such that γ1\gamma_1 and γ2\gamma_2 are externally tangent at C,C, and γ3\gamma_3 is internally tangent to γ1\gamma_1 and γ2\gamma_2 at AA and B,B, respectively. Suppose the tangents to γ3\gamma_3 at AA and BB intersect at X.X. The line through XX and CC intersect γ3\gamma_3 at two points, PP and Q.Q. Compute the length of PQ.PQ.
Proposed by Kyle Lee
geometry
2021 Combo Div 1 P5

Source:

3/2/2021
There are exactly 7 possible tetrominos (groups of 4 connected squares in a grid):
https://cdn.discordapp.com/attachments/813077401265242143/816189385859006474/tetris.png
Daniel has a 2×202102 \times 20210 rectangle and wants to tile the interior with tetrominos without overlaps, pieces sticking out, or extra pieces left over. Note that you are allowed to rotate tetrominos but not reflect them.
For how many multisets of tetrominos (ie. an ordered tuple of how many of each tile he has) is it possible to exactly tile his 2×202102\times20210 rectangle?
Proposed by Dilhan Salgado
combinatorics