MathDB

Problem 1. Given a cube

ABCD.EFGH

with an edge with length 4 units and

P

be the midpoint/center of side

EFGH

. If

M

is the midpoint of

PH

, determine the length of segment

AM

.
Problem 2. Find all reals

k

such that the system of equations \begin{align*} a^2 + ab &= kb^2 \\ b^2 + bc &= kc^2 \\ c^2 + ca &= ka^2 \end{align*} have (a) real positive solution(s)

a, b, c

.
Problem 3. Each cell of a checkerboard with size

m \times n

is colored with either black or white, such that: (a) The number of black and white cells in each row are the same. (b) If a row intersects a column at some black cell, then said row and column contain the same number of black tiles. (c) If a row intersects a column at some white cell, then said row and column contain the same number of white tiles. Determine all possible values of

m

and

n

so that the coloring above can be done.
Problem 4. Determine all nonnegative integers

k

such that we can always find a noninteger positive real

x

which satisfies

\lfloor x + k \rfloor^{\lfloor x + k \rfloor} = \lceil x \rceil^{\lfloor x \rfloor} + \lfloor x \rfloor^{\lceil x \rceil}.

Problem 5. On a triangle

ABC

where

AC > BC

, with

O

as its circumcenter. Let

M

be the circumcircle of

ABC

such that

CM

is the bisector of

\angle{ACB}

/ Let

\Gamma

be the circle with diameter

CM

. The bisectors of

BOC

and

AOC

intersect

\Gamma

respectively at points

P

and

Q

. If

K

is the midpoint of

CM

, prove that points

P, Q, O, K

are concyclic.

Problem Statement