MathDB

2

Part of 2019 AIME Problems

Problems(2)

AIME I Problem 2

Source: 2019 AIME I #2

3/14/2019
Jenn randomly chooses a number JJ from 1,2,3,…,19,201, 2, 3,\ldots, 19, 20. Bela then randomly chooses a number BB from 1,2,3,…,19,201, 2, 3,\ldots, 19, 20 distinct from JJ. The value of B−JB - J is at least 22 with a probability that can be expressed in the form mn\frac{m}{n} where mm and nn are relatively prime positive integers. Find m+nm+n.
AMCAIMEAIME Inumber theoryrelatively prime2019 AIME I2019 AMC
Lily pads and a Frog

Source: 2019 AIME II #2

3/22/2019
Lily pads 1,2,3,…1,2,3,\ldots lie in a row on a pond. A frog makes a sequence of jumps starting on pad 11. From any pad kk the frog jumps to either pad k+1k+1 or pad k+2k+2 chosen randomly and independently with probability 12\tfrac12. The probability that the frog visits pad 77 is pq\tfrac pq, where pp and qq are relatively prime positive integers. Find p+qp+q.
AMCAIMEAIME IIprobability