## Todai Entrance Exam: Subject 2017 – Problem 3

(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 . […]

## 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 […]