MathDB

1

|AB/3- 3ab| -3ab<=1 - ab if |A - 3a| <=1 - a , and |B -3b| <= 1 - b and a,b>0

a, A, b, B

satisfy the inequalities:

|A - 3a| \le 1 - a\,\,\, , \,\,\, |B -3b| \le 1 - b

, and

a, b

are positive. Prove that

\left|\frac{AB}{3}- 3ab\right | - 3ab \le 1 - ab.

8

1

3 lines concurrent with 1 circle

A point

C

lies on a line segment

AB

between

A

and

B

and circles are drawn having

AC

and

CB

as diameters. A common tangent to both circles touches the circle with

AC

as diameter at

P \ne C

and the circle with

CB

as diameter at

Q \ne C

. Prove that

AP, BQ

and the common tangent to both circles at

C

all meet at a single point which lies on the circumference of the circle with

AB

as diameter.

7

1

k blue cells in a blue / red n x n grid , S_B- S_R = 50

Each square of an

n \times n

grid is coloured either blue or red, where

n

is a positive integer. There are

k

blue cells in the grid. Pat adds the sum of the squares of the numbers of blue cells in each row to the sum of the squares of the numbers of blue cells in each column to form

S_B

. He then performs the same calculation on the red cells to compute

S_R

. If

S_B- S_R = 50

, determine (with proof) all possible values of

k

.

6

1

sequence of areas of equilaterals whose perimeter is preceding term

A sequence whose first term is positive has the property that any given term is the area of an equilateral triangle whose perimeter is the preceding term. If the first three terms form an arithmetic progression, determine all possible values of the first term.

5

1

g(g(x)) = 3x/(x + 3) for x>=0, g(x) = (x + 1)/2 for 2<=x<=3

The function

g : [0, \infty) \to [0, \infty)

satisfies the functional equation:

g(g(x)) = \frac{3x}{x + 3}

, for all

x \ge 0

. You are also told that for

2 \le x \le 3

:

g(x) = \frac{x + 1}{2}

. (a) Find

g(2021)

. (b) Find

g(1/2021)

.

4

1

3031 identical dominoes, in 3x 2021 chessboard missing 1 corner square

You have a

3 \times 2021

chessboard from which one corner square has been removed. You also have a set of

3031

identical dominoes, each of which can cover two adjacent chessboard squares. Let

m

be the number of ways in which the chessboard can be covered with the dominoes, without gaps or overlaps. What is the remainder when

m

is divided by

19

?

3

1

n + T(n) = 10n, whereT(n) sends first 2 digits to the end

For each integer

n \ge 100

we define

T(n)

to be the number obtained from

n

by moving the two leading digits to the end. For example,

T(12345) = 34512

and

T(100) = 10

. Find all integers

n \ge 100

for which

n + T(n) = 10n

.

2

1

|CE|x|DG| = |EF| x |CG| for inscribed right isosceles triangle

An isosceles triangle

ABC

is inscribed in a circle with

\angle ACB = 90^o

and

EF

is a chord of the circle such that neither E nor

F

coincide with

C

. Lines

CE

and

CF

meet

AB

at

D

and

G

respectively. Prove that

|CE|\cdot |DG| = |EF| \cdot |CG|

.

1

15! as product of 9 different integers

Let

N = 15! = 15\cdot 14\cdot 13 ... 3\cdot 2\cdot 1

. Prove that

N

can be written as a product of nine different integers all between

16

and

30

inclusive.

10

1

Minimum Distance in a Quarter Plane With Fixed Centroid

Let

P_{1}, P_{2}, \ldots, P_{2021}

be 2021 points in the quarter plane

\{(x, y): x \geq 0, y \geq 0\}

. The centroid of these 2021 points lies at the point

(1,1)

.
Show that there are two distinct points

P_{i}, P_{j}

such that the distance from

P_{i}

to

P_{j}

is no more than

\sqrt{2} / 20

.

2021 Irish Math Olympiad

Subcontests

|AB/3- 3ab| -3ab<=1 - ab if |A - 3a| <=1 - a , and |B -3b| <= 1 - b and a,b>0

3 lines concurrent with 1 circle

k blue cells in a blue / red n x n grid , S_B- S_R = 50

sequence of areas of equilaterals whose perimeter is preceding term

g(g(x)) = 3x/(x + 3) for x>=0, g(x) = (x + 1)/2 for 2<=x<=3

3031 identical dominoes, in 3x 2021 chessboard missing 1 corner square

n + T(n) = 10n, whereT(n) sends first 2 digits to the end

|CE|x|DG| = |EF| x |CG| for inscribed right isosceles triangle

15! as product of 9 different integers

Minimum Distance in a Quarter Plane With Fixed Centroid

2021 Irish Math Olympiad

Subcontests

|AB/3- 3ab| -3ab&lt;=1 - ab if |A - 3a| &lt;=1 - a , and |B -3b| &lt;= 1 - b and a,b&gt;0

3 lines concurrent with 1 circle

k blue cells in a blue / red n x n grid , S_B- S_R = 50

sequence of areas of equilaterals whose perimeter is preceding term

g(g(x)) = 3x/(x + 3) for x&gt;=0, g(x) = (x + 1)/2 for 2&lt;=x&lt;=3

3031 identical dominoes, in 3x 2021 chessboard missing 1 corner square

n + T(n) = 10n, whereT(n) sends first 2 digits to the end

|CE|x|DG| = |EF| x |CG| for inscribed right isosceles triangle

15! as product of 9 different integers

Minimum Distance in a Quarter Plane With Fixed Centroid

|AB/3- 3ab| -3ab<=1 - ab if |A - 3a| <=1 - a , and |B -3b| <= 1 - b and a,b>0

g(g(x)) = 3x/(x + 3) for x>=0, g(x) = (x + 1)/2 for 2<=x<=3