MathDB

The test this year was held on Monday, 22 August 2022 on 09.10-11.40 (GMT+7) for the essay section and was held on 12.05-13.30 (GMT+7) for the short answers section, which was to be done in an hour using the Moodle Learning Management System. Each problem in this section has a weight of 2 points, with 0 points for incorrect or unanswered problems, whereas in the essay section each problem has a weight of 7 points. As with last year, calculators, protractors and set squares are prohibited. (Of course abacus is also prohibited, but who uses abacus?)
Anyway here are the problems.
Part 1: Speed Round (60 minutes) The problem was presented in no particular order, although here I will order it in a rough order of increasing difficulty.
Problem 1. The number of positive integer solutions

(m,n)

to the equation

m^n = 17^{324}

\ldots.

[url=https://artofproblemsolving.com/community/c6h2909996_7_digits_numbers]Problem 2. Consider the increasing sequence of all 7-digit numbers consisting of all of the following digits:

1,2,3,4,5,6,7

. The 2024th term of the sequence is

\ldots.

Problem 3. Suppose

(a,b)

is a positive integer solution of the equation

\sqrt{a + \frac{15}{b}} = a \sqrt{ \frac{15}{b}}.

The sum of all possible values of

b

\ldots.

Problem 4. It is known that

ABCD

is a trapezoid such that

AB

is parallel to

CD

, with the length of

AB = 6

and

CD = 7

. It is known that points

P

and

Q

are on

AD

and

BC

respectively such that

PQ

is parallel to

AB

. If the perimeter of trapezoid

ABQP

is the same as the perimeter of

PQCD

and

AD + BC = 10

, then the length of

20PQ

\ldots.

Problem 5. Suppose

ABC

is a triangle with side lengths

AB = 16

AC=23

, and

\angle{BAC} = 30^{\circ}

. The maximum possible area of a rectangle whose one of its sides lies on the line

BC

, and the two other vertices each lie on

AB

and

AC

\ldots.

Problem 6. Suppose

a,b,c

are natural numbers such that

a+2b+3c=73

. The minimum possible value of

a^2+b^2+c^2

\ldots.

Problem 7. An equilateral triangle with a side length of

21

is partitioned into

21^2

unit equilateral triangles, and the sides of the small equilateral triangles are all parallel to the original large triangle. The number of paralellograms which are made up of the unit equilateral triangles is

21k

. Then the value of

k = \ldots.

Problem 8. Define the sequence

\{a_n\}

with

a_1 > 3

, and for all

n \geq 1

, the following condition:

2a_{n+1} = a_n(-1 + \sqrt{4a_n - 3})

is satisfied. If

\vert a_1 - a_{2022} \vert = 2023

, then the value of

\sum_{i=1}^{2021} \frac{a_{i+1}^3}{a_i^2 + a_i a_{i+1} + a_{i+1}^2} = \ldots.

Problem 9. Suppose

P(x)

is an integer polynomial such that

P(6)P(38)P(57) + 19

is divisible by

114

. If

P(-13) = 479

, and

P(0) \geq 0

, then the minimum value of

P(0)

\ldots.

Problem 10. The number of nonempty subsets of

S = \{1,2, \ldots, 21\}

such that the sum of its elements is divisble by 4 is

2^k - m

; where

k, m \in \mathbb{Z}

and

0 \leq m < 2022

. The value of

10k + m

\ldots.

Problem Statement