Cono Sur Shortlist - geometry

Part of Cono Sur Olympiad

Subcontests

(41)

1

RS tangent to mixtilinear incircle wanted, isosceles ABC

Given an triangle

ABC

isosceles at the vertex

A

P

Q

be the touchpoints with

AB

AC

, respectively with the circle

T

, which is tangent to both and is internally tangent to the circumcircle of

ABC

R

S

be the points of the circumscribed circle of

ABC

AP = AR = AS

RS

T

1

<CBQ = <PBA wanted, feet of altitudes and circumcircle related

\vartriangle ABC

be a triangle with circumcenter

O

H

, and circumcircle

\omega

AA'

BB'

CC'

are altitudes of

\vartriangle ABC

A'

BC

B'

AC

C'

AB

P

is a point on the segment

AA'

. The perpenicular line to

B'C'

P

BC

K

AA'

\omega

M \ne A

MK

AO

Q

\angle CBQ = \angle PBA

1

max of [ADJE]/AJ^2 wanted, A-excircle related

ABC

be a triangle and

\Gamma

A

- exscribed circle whose center is

J

D

E

be the touchpoints of

\Gamma

AB

AC

, respectively. Let

S

be the area of the quadrilateral

ADJE

, Find the maximum value that

\frac{S}{AJ^2}

has and when equality holds.

1

HD ,LE, BC concurrent iff EO = EC

ABCD

be a parallelogram with vertices in order clockwise and let

E

be the intersection of its diagonals. The circle of diameter

DE

intersects the segment

AD

L

EC

H

. The circumscribed circle of

LEB

intersects the segment

BC

O

. Prove that the lines

HD

LE

BC

are concurrent if and only if

EO = EC

1

concurrency, feets of arc midpoint of circumcircle on 2 triangle sides

ABC

be an acute triangle. Define

A_1

the midpoint of the largest arc

BC

of the circumcircle of

ABC

A_2

A_3

be the feet of the perpendiculars from

A_1

AB

AC

, respectively. Define

B_1

B_2

B_3

C_1

C_2

C_3

analogously. Show that the lines

A_2A_3

B_2B_3

C_2C_3

are concurrent.

1

MA_2 bisects BC, circumrcircle

ABC

be a triangle with circumcircle

\omega

. The bisector of

\angle BAC

\omega

A_1

A_2

be a point on the segment

AA_1

CA_2

AB

\omega

C_1

C_2

, respectively. Similarly,

BA_2

AC

\omega

B_1

B_2

, respectively. Let

M

be the intersection point of

B_1C_2

B_2C_1

MA_2

passes the midpoint of

BC

.
proposed by Jhefferson Lopez, Perú

1

sides of triangle ABC form an arithmetic progression, incircle, circumcircle

ABC

be a triangle whose inscribed circle is

\omega

r_1

be the line parallel to

BC

\omega

r_1 \ne BC

r_2

be the line parallel to

AB

\omega

r_2 \ne AB

. Suppose that the intersection point of

r_1

r_2

lies on the circumscribed circle of triangle

ABC

. Prove that the sidelengths of triangle

ABC

form an arithmetic progression.

1

min sum of squares of distances, centroid - 1993 Cono Sur Shortlist G14

Prove that the sum of the squares of the distances from a point

P

to the vertices of a triangle

ABC

is minimum when

P

is the centroid of the triangle.

1

triangle with sidelengths 5,8,x is obtuse - 1993 Cono Sur Shortlist G13

Determine the real values of

x

such that the triangle with sides

5

8

x

1

lengths of 8 segments by 4 lines, diff. integers - 1993 Cono Sur Shortlist G12

4

lines in the plane such that there are not

2

parallel to each other or no

3

concurrent, we consider the following

8

segments: in each line we have

2

consecutive segments determined by the intersections with the other three lines. Prove that: a) The lengths of the

8

segments cannot be the numbers

1, 2, 3,4, 5, 6, 7, 8

in some order. b) The lengths of the

8

segments can be

8

different integers.

1

BP=1/3 AB wanted, semicircle - 1993 Cono Sur Shortlist G11

\Gamma

be a semicircle with center

O

AB

D

is the midpoint of arc

AB

OD

E

OE = BD

BE

intersects the semicircle at

F

P

is the point on

AB

FP

is perpendicular to

AB

BP=\frac13 AB

1

max inradii of OAP, unit circle related - 1993 Cono Sur Shortlist G10

\omega

be the unit circle centered at the origin of

R^2

. Determine the largest possible value for the radius of the circle inscribed to the triangle

OAP

P

lies the circle and

A

is the projection of

P

OX

1

line bisecting area and perimeter, incenter - 1993 Cono Sur Shortlist G9

Prove that a line that divides a triangle into two polygons of equal area and equal perimeter passes through the center of the circle inscribed in the triangle. Prove an analogous property for a polygon that has an inscribed circle.

1

DEF, ABC similar iff ABC equilateral, incircle - 1993 Cono Sur Shortlist G8

ABC

D

E

F

be the touchpoints of the inscribed circle and the sides

AB

BC

CA

. Show that the triangles

DEF

ABC

are similar if and only if

ABC

is equilateral.

1

O is centroid iff ABCD # - 1993 Cono Sur Shortlist G7

ABCD

be a convex quadrilateral, where

M

is the midpoint of

DC

N

is the midpoint of

BC

O

is the intersection of diagonals

AC

BD

O

is the centroid of the triangle

AMN

ABCD

is a parallelogram.

1

DEF equilateral if AD=BE = CF - 1993 Cono Sur Shortlist G6

Consider in the interior of an equilateral triangle

ABC

D, E

F

D

belongs to segment

BE

E

belongs to segment

CF

F

AD

AD=BE = CF

DEF

is equilateral.

1

angle bisector construction - 1993 Cono Sur Shortlist G3

Justify the following construction of the bisector of an angle with an inaccessible vertex: https://cdn.artofproblemsolving.com/attachments/9/d/be4f7799d58a28cab3b4c515633b0e021c1502.png

M \in a

N \in b

4

bisectors of the

4

internal angles formed by

MN

are traced with

a

b

. Said bisectors intersect at

P

Q

PQ

is the bisector sought.

1

distances inside a square - 1993 Cono Sur Shortlist G5

A block of houses is a square. There is a courtyard there in which a gold medal has fallen. Whoever calculates how long the side of said apple is, knowing that the distances from the medal to three consecutive corners of the apple are, respectively,

40

60

80

m, will win the medal.

1

51 circles of diameter 1 in 5x8 - 1993 Cono Sur Shortlist G4

Is it possible to locate in a rectangle of

5

8

51

circles of diameter

1

cm, so that they don't overlap? Could it be possible for more than

40

1

trapezoid and areas wanted - 1993 Cono Sur Shortlist G2

ABCD

be a quadrilateral and let

O

be the point of intersection of diagonals

AC

BD

. Knowing that the area of triangle

AOB

1

, the area of triangle

BOC

2

, and the area of triangle

COD

4

, calculate the area of triangle

AOD

ABCD

is a trapezoid.

1

circle tangent to concentric circles - 1993 Cono Sur Shortlist G1

C_1

C_2

be two concentric circles and

C_3

an outer circle to

C_1

C_2

and tangent to both. If the radius of

C_2

1

, how much must the radius of

C_1

be worth, so that the area of is twice that of

C_3

1

concurrent wanted, circles passing through a vertice, H , tangent to (ABC)

ABC

be an acute triangle with circumcenter

O

and orthocenter

H

. The circle with center

X_A

passes through the points

A

H

and is tangent to the circumcircle of the triangle

ABC

. Similarly, define the points

X_B

X_C

O_A

O_B

O_C

be the reflections of

O

with respect to sides

BC

CA

AB

, respectively. Prove that the lines

O_AX_A

O_BX_B

O_CX_C

are concurrent.

1

min of L/l, max side inscribed in triangle, min square circumscribed in triangle

We say that a polygon

P

is inscribed in another polygon

Q

when all the vertices of

P

belong to the perimeter of

Q

. We also say in this case that

Q

is circumscribed to

P

. Given a triangle

T

\ell

be the largest side of a square inscribed in

T

L

is the shortest side of a square circumscribed to

T

. Find the smallest possible value of the ratio

L/\ell

1

JK bisects BC, incenter, arc midpoint of circumcircle, AE = AF

ABC

be an acute triangle with

AC > AB

\Gamma

be the circle circumscribed to the triangle

ABC

D

the midpoint of the smaller arc

BC

of this circle. Let

I

be the incenter of

ABC

E

F

be points on sides

AB

AC

, respectively, such that

AE = AF

I

lies on the segment

EF

P

be the second intersection point of the circumcircle of the triangle

AEF

\Gamma

P \ne A

G

H

be the intersection points of the lines

PE

PF

\Gamma

P

, respectively. Let

J

K

be the intersection points of lines

DG

DH

with lines AB and

AC

, respectively. Show that the line

JK

passes through the midpoint of

BC

1

computational in a pentagon ABCDE, AB=AE, AC=AD, <BAE=90^o , <ACB=<ADE=135^o

Consider the pentagon

ABCDE

AB = AE = x

AC = AD = y

\angle BAE = 90^o

\angle ACB = \angle ADE = 135^o

. It is known that

C

D

are inside the triangle

BAE

. Determine the length of

CD

x

y

1

computational in a square, a^2 = b^2 + c^2, right angle wanted

P

point to a square

ABCD

PA = a, PB = b

PC = b + c

, where the numbers

a, b

c

satisfy the relationship

a^2 = b^2 + c^2

. Prove that the angle

BPC

1

segments intersect at their midpoint, 4 orthocenters related

ABC

P

be a point on its circumscribed circle (on the arc

AC

that does not contain

B

H,H_1,H_2

H_3

be the orthocenters of triangles

ABC, BCP, ACP

ABP

, respectively. Let

L = PB \cap AC

J = HH_2 \cap H_1H_3

M

N

are the midpoints of

JH

LP

, respectively, prove that

MN

JL

intersect at their midpoint.

1

locus of lines' intersection from Cono Sur, condition 1/TH= 1/d -1/PH

L_1

L_2

be two parallel lines and

L_3

a line perpendicular to

L_1

L_2

H

P

, respectively. Points

Q

R

L_1

QR = PR

Q \ne H

d

be the diameter of the circle inscribed in the triangle

PQR

T

L_2

in the same semiplane as

Q

with respect to line

L_3

\frac{1}{TH}= \frac{1}{d}- \frac{1}{PH}

X

be the intersection point of

PQ

TH

. Find the locus of the points

X

Q

L_1

1

externally tangent circles in pairs , diameter wanted

C_1, C_2

C_3

are externally tangent in pairs (each tangent to other two externally). Let

M

the common point of

C_1

C_2, N

the common point of

C_2

C_3

P

the common point of

C_3

C_1

A

be an arbitrary point of

C_1

AM

C_2

B

BN

C_3

C

CP

C_1

D

AD

C_1

1

angle chasing from Cono Sur, isosceles, circumcircle, equal angles wanted

O

be the circumcenter of the isosceles triangle

ABC

AB = AC

P

be a point of the segment

AO

Q

the symmetric of

P

with respect to the midpoint of

AB

OQ

AB

K

and the circle that passes through

A, K

O

AC

L

\angle ALP = \angle CLO

1

2 pairs of equal angles and points of circumcircle lead to concurrency

O

be the circumcenter of an acute triangle

ABC

A_1

a point of the minor arc

BC

ABC

A_2

A_3

be points on sides

AB

AC

respectively such that

\angle BA_1A_2=\angle OAC

\angle CA_1A_3=\angle OAB

B_2, B_3, C_2

C_3

are similarly constructed, with

B_2

BC, B_3

AB, C_2

AC

C_3

BC

. Prove that lines

A_2A_3, B_2B_3

C_2C_3

are concurrent.

1

2 tangents. a secant, a ratio segment given, a ratio segment wanted

AM

AN

be the tangents to a circle

\Gamma

drawn from a point

A

M

N

lie on the circle). A line passing through

A

\Gamma

B

C

, with B between

A

C

AB: BC = 2: 3

P

is the intersection point of

AB

MN

, calculate the ratio

AP: CP

1

ratio of sum of areas between 4 mutually internally tangent circles

Find the ratio between the sum of the areas of the circles and the area of the fourth circle that are shown in the figure Each circle passes through the center of the previous one and they are internally tangent. https://cdn.artofproblemsolving.com/attachments/d/2/29d2be270f7bcf9aee793b0b01c2ef10131e06.jpg

1

altitudes and incenters lead to perpendicularity

AA _1

CC_1

be altitudes of an acute triangle

ABC

I

J

be the incenters of the triangles

AA_1C

AC_1C

respectively. The

C_1J

A_1 I

T

. Prove that lines

AT

TC

are perpendicular.

1

angle independent of the point X, starting with cyclic trapezoid

ABCD

AB

CD

, is inscribed in a circumference

\Gamma

X

a variable point of the arc

AB

\Gamma

that does not contain

C

D

Y

to the point of intersection of

AB

DX

, and let Z be the point of the segment

CX

\frac{XZ}{XC}=\frac{AY}{AB}

. Prove that the measure of

\angle AZX

does not depend on the choice of

X.

1

Convex polygon distances

We have a convex polygon

P

in the plane and two points

S,T

in the boundary of

P

, dividing the perimeter in a proportion

1:2

. Three distinct points in the boundary, denoted by

A,B,C

start to move simultaneously along the boundary, in the same direction and with the same speed. Prove that there will be a moment in which one of the segments

AB, BC, CA

will have a length smaller or equal than

ST

1

Construct triangle given all lenght of it altitudes.

Construct triangle given all lenght of it altitudes.
Please, do it elementary with Euclidian geometry (no trigonometry or coordinate geometry).

1

Cono Sur Shortlist 2012, Problem G5

ABC

be an acute triangle, and let

H_A

H_B

H_C

be the feet of the altitudes relative to vertices

A

B

C

, respectively. Define

I_A

I_B

I_C

as the incenters of triangles

AH_B H_C

BH_C H_A

CH_A H_B

, respectively. Let

T_A

T_B

T_C

be the intersection of the incircle of triangle

ABC

BC

CA

AB

, respectively. Prove that the triangles

I_A I_B I_C

T_A T_B T_C

1

Cono Sur Shortlist 2012, Problem G3

ABC

be a triangle, and

M

N

P

be the midpoints of

AB

BC

CA

respectively, such that

MBNP

is a parallelogram. Let

R

S

be the points in which the line

MN

intersects the circumcircle of

ABC

AC

is tangent to the circumcircle of triangle

RPS

1

Cono Sur Shortlist 2012, Problem G2

ABC

be a triangle, and

M

N

variable points on

AB

AC

respectively, such that both

M

N

do not lie on the vertices, and also,

AM \times MB = AN \times NC

. Prove that the perpendicular bisector of

MN

passes through a fixed point.

1

Cono Sur Shortlist 2012, Problem G1

ABCD

be a cyclic quadrilateral. Let

P

be the intersection of

BC

AD

AC

intersects the circumcircle of triangle

BDP

S

T

S

A

C

BD

intersects the circumcircle of triangle

ACP

U

V

U

B

D

PS

PT

PU

PV