MathDB

1

Incenter, Excenter and intersections

ABC

be a triangle with incenter

I

, and

A

-excenter

\Gamma

. Let

A_1,B_1,C_1

be the points of tangency of

\Gamma

with

BC,AC

and

AB

, respectively. Suppose

IA_1, IB_1

and

IC_1

intersect

\Gamma

for the second time at points

A_2,B_2,C_2

, respectively.

M

is the midpoint of segment

AA_1

. If the intersection of

A_1B_1

and

A_2B_2

is

X

, and the intersection of

A_1C_1

and

A_2C_2

is

Y

, prove that

MX=MY

.

Problem 5

1

Unlimited candy in PAGMO

Celeste has an unlimited amount of each type of

n

types of candy, numerated type 1, type 2, ... type n. Initially she takes

m>0

candy pieces and places them in a row on a table. Then, she chooses one of the following operations (if available) and executes it:

1.

She eats a candy of type

k

, and in its position in the row she places one candy type

k-1

followed by one candy type

k+1

(we consider type

n+1

to be type 1, and type 0 to be type

n

).

2.

She chooses two consecutive candies which are the same type, and eats them.
Find all positive integers

n

for which Celeste can leave the table empty for any value of

m

and any configuration of candies on the table.

<span class='latex-italic'>Proposed by Federico Bach and Santiago Rodriguez, Colombia</span>

Problem 4

1

Odd digit multiplication

Lucía multiplies some positive one-digit numbers (not necessarily distinct) and obtains a number

n

greater than 10. Then, she multiplies all the digits of

n

and obtains an odd number. Find all possible values of the units digit of

n

.

<span class='latex-italic'>Proposed by Pablo Serrano, Ecuador</span>

Problem 1

1

Coins in a circle

There are

n \geq 2

coins numbered from

1

to

n

. These coins are placed around a circle, not necesarily in order.
In each turn, if we are on the coin numbered

i

, we will jump to the one

i

places from it, always in a clockwise order, beginning with coin number 1. For an example, see the figure below.
Find all values of

n

for which there exists an arrangement of the coins in which every coin will be visited.

Problem 2

1

Isosceles right triangle and tangency

Consider the isosceles right triangle

ABC

with

\angle BAC = 90^\circ

. Let

\ell

be the line passing through

B

and the midpoint of side

AC

. Let

\Gamma

be the circumference with diameter

AB

. The line

\ell

and the circumference

\Gamma

meet at point

P

, different from

B

. Show that the circumference passing through

A,\ C

and

P

is tangent to line

BC

at

C

.

Problem 3

1

Interesting functional equation as P3 in the first-ever PAGMO

Let

\mathbb{R}

be the set of real numbers. Determine all functions

f: \mathbb{R}\longrightarrow \mathbb{R}

so that the equality

f(x+yf(x+y)) +xf(x)= f(xf(x+y+1))+y^2

is true for any real numbers

x,y

.

2021 Pan-American Girls' Math Olympiad

Subcontests

Incenter, Excenter and intersections

Unlimited candy in PAGMO

Odd digit multiplication

Coins in a circle

Isosceles right triangle and tangency

Interesting functional equation as P3 in the first-ever PAGMO