(1) Let be the values of after processing . (2) We use induction proof: If a list has the number of varieties of user IDs being at most one then . Base case: , so when . […]

# Month: June 2019

## Todai Entrance Exam: Math 2017 – Problem 3

(1) (2) (3) (4) (5) From (1), we have:

## Todai Entrance Exam: Math 2017 – Problem 2

(1) (2) Let and . So, thanks to the given fact: (3) We solve the characteristic equation of the above homogenous second order linear differential equations: So: If : which is the excluded solution. If […]

## Todai Entrance Exam: Math 2017 – Problem 1

(1) Therefore: (2) Then we solve the equation for each , we then arrive: (3) (4) As: Therefore: So: (5) Notice: Therefore and . (6) Rayleigh quotient? […]

## Todai Entrance Exam: Math 2019 – Problem 3

(1) (2) The easy way: The projection of on is . So . The hard way: Let be the intersection between and . As the triangle ABC is symmetric with respect to the line , every point on has a corresponding point on . So: Note that as . (3) We […]

## Todai Entrance Exam: Math 2019 – Problem 2

(1) Let , so : (2) (i) From Eq. (2.1), we have: Remember that: So: Notice that: So: Working… (ii) I speculate the general form of the solution is: . Eq. (2.3) implies that: […]

## Todai Entrance Exam: Math 2019 – Problem 1

(1) (2) Prove that if is unitary then is orgthogonal: As , if is unitary: Then, (because they are the real part) and (because they are the imaginary part). Therefore: So the statement follows. Prove that if is orthogonal then is unitary: If then as shown above, it must […]