MathDB

[url=https://artofproblemsolving.com/community/c4h2372772p19406382]p1. In the figure, point

P, Q,R,S

lies on the side of the rectangle

ABCD

. https://1.bp.blogspot.com/-Ff9rMibTuHA/X9PRPbGVy-I/AAAAAAAAMzA/2ytG0aqe-k0fPL3hbSp_zHrMYAfU-1Y_ACLcBGAsYHQ/s426/2020%2BIndonedia%2BMO%2BProvince%2BP2%2Bq1.png If it is known that the area of the small square is

1

unit, determine the area of the rectangle

ABCD

.
p2. Given a quadratic function

f(x) = x^2 + px + q

where

p

and

q

are integers. Suppose

a, b

, and

c

are distinct integers so that

2^{2020}

divides

f(a)

f(b)

and

f(c)

, but

2^{1000}

doesn't divide

b-a

nor

c-a

. Prove that

2^{1021}

divides

b-c

.
p3. Find all the irrational numbers

x

such that

x^2 +20x+20

and

x^3-2020x+1

both are rational numbers.
[url=https://artofproblemsolving.com/community/c6h2372767p19406215]p4. It is known that triangle

ABC

is not isosceles with altitudes of

AA_1, BB_1

, and

CC_1

. Suppose

B_A

and

C_A

respectively points on

BB_1

and

CC_1

so that

A_1B_A

is perpendicular on

BB_1

and

A_1C_A

is perpendicular on

CC_1

. Lines

B_AC_A

and

BC

intersect at the point

T_A

. Define in the same way the points

T_B

and

T_C

. Prove that points

T_A, T_B

, and

T_C

are collinear.
p5. In a city,

n

children take part in a math competition with a total score of non-negative round. Let

k < n

be a positive integer. Each child

s

: (i) gets

k

candies for each score point he gets, and (ii) for every other child

t

whose score is higher than

s

, then

s

gets

1

candy for each point the difference between the values of

t

and

s

. After all the candy is distributed, it turns out that no child gets less candy from Badu, and there are

i

children who get higher scores than Badu. Determine all values of

i

which may take.

Problem Statement