MathDB

P6. Determine all integer pairs

(x, y)

satisfying the following system of equations.

\begin{cases} x + y - 6 &= \sqrt{2x + y + 1} \\ x^2 - x &= 3y + 5 \end{cases}

P7. Determine the sum of all (positive) integers

n \leq 2019

such that

1^2 + 2^2 + 3^2 + \cdots + n^2

is an odd number and

1^1 + 2^2 + 3^3 + \cdots + n^n

is also an odd number.
P8. Two quadrilateral-based pyramids where the length of all its edges are the same, have their bases coincide, forming a new 3D figure called "8-plane" (octahedron). If the volume of such "8-plane" (octahedron) is

a^3\sqrt{2}

^3

, determine the volume of the largest sphere that can be fit inside such "8-plane" (octahedron).
P9. Six-digit numbers

\overline{ABCDEF}

with distinct digits are arranged from the digits 1, 2, 3, 4, 5, 6, 7, 8 with the rule that the sum of the first three numbers and the sum of the last three numbers are the same. Determine the probability that such arranged number has the property that either the first or last three digits (might be both) form an arithmetic sequence or a geometric sequence. [hide=Remarks (Answer spoiled)]It's a bit ambiguous whether the first or last three digits mentioned should be in that order, or not. If it should be in that order, the answer to this problem would be

\frac{1}{9}

, whereas if not, it would be

\frac{1}{3}

. Some of us agree that the correct interpretation should be the latter (which means that it's not in order) and the answer should be

\frac{1}{3}

. However since this is an essay problem, your interpretation can be written in your solution as well and it's left to the judges' discretion to accept your interpretation, or not. This problem is very bashy.
P10.

X_n

denotes the number which is arranged by the digit

X

written (concatenated)

n

times. As an example,

2_{(3)} = 222

and

5_{(2)} = 55

. For

A, B, C \in \{1, 2, \ldots, 9\}

and

1 \leq n \leq 2019

, determine the number of ordered quadruples

(A, B, C, n)

satisfying:

A_{(2n)} = 2 \left ( B_{(n)} \right ) + \left ( C_{(n)} \right )^2.

Problem Statement