MathDB

Problem 1. Determine all triples

(x, y, z)

where

x, y, z

are reals, which satisfy the following simultaneous system of equations: \begin{align*} x^2 + 4 &= y^3 + 4x - z^3 \\ y^2 + 4&= z^3 + 4y - x^3 \\ z^2 + 4 &= x^3 + 4z - y^3. \end{align*}
Problem 2. On triangle

ABC

, it is given points

D

E

, and

F

on sides

BC, CA

, and

AB

such that

AD, BE,

and

CF

concur at a point

O

. Prove that

\frac{AO}{AD} + \frac{BO}{BE} + \frac{CO}{CF} = 2.

Problem 3. Beni, Coki, and Dodi are living in the same house (cohabitating, don't ask me why :v) and are going to the same school. It takes Beni, Coki, and Dodi 2, 4, and 8 minutes to travel from the house to school, respectively. With a bike, it takes each person 1 minute to travel. Show that it is possible for the three of them to travel to school in no more than

2 \frac{3}{4}

minutes.
Problem 4. Prove that there is no natural number

m

so that there exists

k, e

where

e \geq 2

, which satisfy

m(m^2 + 1) = k^e.

Problem 5. A lattice point is a point whose coordinates are a pair of integers. Let

P_1, P_2, P_3, P_4, P_5

are five lattice points on a plane. Prove that there exists (at least) a pair

(P_i, P_j)

, where

i \neq j

, such that the segment

P_iP_j

crosses another lattice point besides

P_i

and

P_j

Problem Statement