MathDB

8

Part of 2021 MOAA

Problems(4)

Speed P8

Source:

10/16/2021
Andrew chooses three (not necessarily distinct) integers aa, bb, and cc independently and uniformly at random from {1,2,3,4,5,6,7}\{1,2,3,4,5,6,7\}. Let pp be the probability that abc(a+b+c)abc(a+b+c) is divisible by 44. If pp can be written as mn\frac{m}{n} for relatively prime positive integers mm and nn, then compute m+nm+n.
Proposed by Andrew Wen
MOAA 2021speed
Accuracy P8

Source:

10/16/2021
Will has a magic coin that can remember previous flips. If the coin has already turned up heads mm times and tails nn times, the probability that the next flip turns up heads is exactly m+1m+n+2\frac{m+1}{m+n+2}. Suppose that the coin starts at 00 flips. The probability that after 1010 coin flips, heads and tails have both turned up exactly 55 times can be expressed as mn\frac{m}{n} for relatively prime positive integers mm and nn. Compute m+nm+n.
Proposed by Nathan Xiong
MOAA 2021Accuracy
Team Round P8

Source:

10/16/2021
Evaluate 27×30+26×31+25×32+⋯+20×37.2^{7}\times 3^{0}+2^{6}\times 3^{1}+2^{5}\times 3^{2}+\cdots+2^{0}\times 3^{7}.
Proposed by Nathan Xiong
MOAA 2021team
Gunga P8

Source:

10/16/2021
Compute the number of triangles of different sizes which contain the gray triangle in the figure below.
[asy] size(5cm); real n = 4; for (int i = 0; i < n; ++i) { draw((0.5*i,0.866*i)--(n-0.5*i,0.866*i)); } for (int i = 0; i < n; ++i) { draw((n-i,0)--((n-i)/2,(n-i)*0.866)); } for (int i = 0; i < n; ++i) { draw((i,0)--((n+i)/2,(n-i)*0.866)); } filldraw((1.5,0.866)--(2,2*0.866)--(2.5,0.866)--cycle, gray); [/asy]
Proposed by Nathan Xiong
MOAA 2021Gunga