MathDB

1

K is incenter of ABC if PQ _|_ BL

ABC

be an acute scalene triangle and

K

be a point inside it that belongs to the bisector of the angle

\angle ABC

. Let

P

be the point where the line

AK

intersects the line perpendicular to

AB

that passes through

B

, and let

Q

be the point where the line

CK

intersects the line perpendicular to

CB

that passes through

B

. Let

L

be the foot of the perpendicular drawn from

K

on the line

AC

. Prove that if

P Q

is perpendicular to

BL

, then

K

is the incenter of

ABC

.

3

1

cut a regular hexagon into n similar quads

Prove that, for every integer

n \ge 2

, it is possible to divide a regular hexagon into

n

quadrilaterals such that any two of them are similar.
Clarification: Two quadrilaterals are similar if they have their corresponding sides proportional and their corresponding angles are equal, that is, the quadrilaterals

ABCD

and

EFGH

are similar if

\frac{AB}{EF}= \frac{BC}{FG}= \frac{CD}{GH} = \frac{DA}{HE}

,

\angle ABC = \angle EFG

,

\angle BCD = \angle FGH

,

\angle CDA = \angle GHE

and

\angle DAB = \angle HEF

.

2

1

f(a),f(b),f(c) sides of triangle if a,b,c sides of triangle, f(x)=x/(1+x)

For each positive real number

x

, let

f(x)=\frac{x}{1+x}

. Prove that if

a

,

b,

c

are the sidelengths of a triangle, then

f(a)

,

f(b),

f(c)

are sidelengths of a triangle.

1

min, max pos. integer that divides exactly two elements of {1^2, .., 100^2}

We define the set

M = \{1^2,2^2,3^2,..., 99^2, 100^2\}

. a) What is the smallest positive integer that divides exactly two elements of

M

? b) What is the largest positive integer that divides exactly two elements of

M

?

2023 Peru MO (ONEM)

Subcontests

K is incenter of ABC if PQ _|_ BL

cut a regular hexagon into n similar quads

f(a),f(b),f(c) sides of triangle if a,b,c sides of triangle, f(x)=x/(1+x)

min, max pos. integer that divides exactly two elements of {1^2, .., 100^2}