Tower Research Capital LLC Strategist Interview Questions and Answers

Expected length of segment containing (1,0) on a unit circle with three random points.

• Use law of cosines to find length of each segment.

• Calculate expected value using probability density function.

• Answer is (4/pi) + (2/3).

If dist(u,v)>n/2 in an undirected graph, there exists a vertex x such that removing x makes u and v go to different connected components.

• Find the shortest path between u and v

• If the path length is greater than n/2, then there must be a vertex x on the path

• Removing x will separate u and v into different connected components

For any subset of size n+1 of the set S of the first 2n numbers, there exists 2 numbers u and v such that u divides v.

• Divide the set S into two subsets of n numbers each.

• By the pigeonhole principle, at least one of the subsets contains two numbers whose ratio is an integer.

• If the subset contains n+1 numbers, then one of the numbers must be in the subset with the two numbers whose ratio is an integer.

• Therefore, there ex...read more

For an odd-sized grid, there cannot be a Hamiltonian cycle in the graph.

• A Hamiltonian cycle is a path that visits every vertex exactly once and ends at the starting vertex.

• In an n*n grid, there are n^2 vertices and each vertex has degree 4.

• For an odd n, the total degree of all vertices is odd, which means there cannot be a Hamiltonian cycle.

Algorithm to determine visible buildings in a 2D plane with given positions and heights

• Sort the buildings by their x-coordinates

• Traverse the sorted buildings from left to right

• For each building, check if it is visible by comparing its height with the maximum height of previously visited buildings

• If visible, add it to the list of visible buildings

• Return the list of visible buildings

In a group of people with a symmetric relation of knowing each other, there will always be at least two people who know the same number of people.

• Consider the person who knows the maximum number of people in the group.

• If there is no one who knows the same number of people, then everyone else must know a different number of people.

• But this would mean that the total number of people known by everyone else is different fr...read more

Prove that there exists a rectangle with all corners of the same color in a 2D plane with blue and red points.

• Divide the plane into a grid of squares.

• By the pigeonhole principle, there must be at least one row or column with four points of the same color.

• Consider the pairs of points in that row or column and check if any of them form a rectangle.

Prove that F_nk is divisible by F_n where F_i is the ith Fibonacci number. with f_0 = 0

• Use mathematical induction to prove the statement

• Base case: F_n0 = 0, F_n is also 0, so 0 is divisible by 0

• Inductive step: Assume F_nk is divisible by F_n, prove F_n(k+1) is divisible by F_n

• F_n(k+1) = F_nk + F_n(k-1), use the assumption to show that F_nk is divisible by F_n

• Therefore, F_n(k+1) is also divisible by F_n

• Hence, the statem...read more

Maximize tournament rating by optimizing team matchups based on batting and bowling ratings.

• Calculate match rating for each pair of teams using (X1+X2)/(Y1+Y2).

• To maximize the tournament rating, minimize the denominator (Y1+Y2).

• Example: Teams A (X=10, Y=5) and B (X=8, Y=3) yield a match rating of (10+8)/(5+3) = 18/8 = 2.25.

• Consider all possible pairings and select the one with the highest minimum match rating.

• Use sorti...read more

Find the largest substring formed from repetition of a smaller string.

• Identify all possible substrings of the given string.

• Check if each substring can be formed by repeating a smaller string.

• Return the largest substring that can be formed from repetition of a smaller string.