benchmarks.wiki

benchmarks.wiki / Public workspace

Omni-MATH / There are 8 lily pads in a pond numbered 1,2,…,8. A frog starts on lily pad 1. During the i-th second, the frog jumps from lily pad i to i+1, falling into the water with probabilit…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

There are 8 lily pads in a pond numbered 1,2,…,81,2, \ldots, 8. A frog starts on lily pad 1. During the ii-th second, the frog jumps from lily pad ii to i+1i+1, falling into the water with probability 1i+1\frac{1}{i+1}. The probability that the frog lands safely on lily pad 8 without having fallen into the water at any point can be written as mn\frac{m}{n}, where m,nm, n are positive integers and gcd(m,n)=1\operatorname{gcd}(m, n)=1. Find 100m+n100 m+n.
Plain-text mathematical notation (without MathML)
There are 8 lily pads in a pond numbered 1,2,…,8. A frog starts on lily pad 1. During the i-th second, the frog jumps from lily pad i to i+1, falling into the water with probability (1)/(i+1). The probability that the frog lands safely on lily pad 8 without having fallen into the water at any point can be written as (m)/(n), where m,n are positive integers and gcd(m,n)=1. Find 100m+n.
Original LaTeX notation
There are 8 lily pads in a pond numbered $1,2, \ldots, 8$. A frog starts on lily pad 1. During the $i$-th second, the frog jumps from lily pad $i$ to $i+1$, falling into the water with probability $\frac{1}{i+1}$. The probability that the frog lands safely on lily pad 8 without having fallen into the water at any point can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.

Discussion

Discussion

No discussion posts on this page yet. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

Source and history

task_id
20907cb5-300c-537b-bca1-4de381510bd7
task_key
test--20907cb5-300c-537b-bca1-4de381510bd7
task_revision_id
1
config / split
/ test

Official source

initial import