MathDB

1

sum ABCD +BCDA+CDAB+DABC, when each number has least prime divisor 11

A,B,C

and

D

each represent a different digit, so each of the four-digit numbers

ABCD

,

BCDA

,

CDAB

and

DABC

satisfy that its least prime divisor is equal to

11

. Determine all possible values of the sum

ABCD +BCDA+CDAB+DABC

and for each possible value of said sum, give an example of a choice of digits

A,B,C

and

D

with which to obtain that value and which satisfies the conditions established above.

5

1

orthocenter of ABC is also orthocenter of DEF

Let

ABC

be a scalene triangle,

\Gamma

its circumscribed circle and

H

the point where the altitudes of triangle

ABC

meet. The circumference with center at

H

passing through

A

cuts

\Gamma

at a second point

D

. In the same way, the circles with center at

H

and passing through

B

and

C

cut

\Gamma

again at points

E

and

F

, respectively. Prove that

H

is also the point in which the altitudes of the triangle

DEF

meet.

6

1

n bulbs on a line, on off

Let

n > 1

be a natural number. There are

n

bulbs in a line, each of which can be on or off. Every minute, simultaneously, all the lit bulbs turn off and the unlit bulbs that were adjacent to exactly one lit bulb turn on. Determine for what values of

n

there is an initial arrangement such that if this process is followed indefinitely, all the lights will never be off.

3

1

similar triangles wanted, <ABC = <CBD, <ACP=<BCD, (ABC)=(BPCD)

A scalene acute triangle

ABC

is drawn on the plane, in which

BC

is the longest side. Points

P

and

D

are constructed, the first inside

ABC

and the second outside, so that

\angle ABC = \angle CBD

,

\angle ACP = \angle BCD

and that the area of triangle

ABC

is equal to the area of quadrilateral

BPCD

. Prove that triangles

BCD

and

ACP

are similar.

2

1

min and max a, if a+b+c+d+e=8 and a^2+b^2+c^2+d^2+e^2=16

Let

a,b,c,d, e

be real numbers such that they simultaneously satisfy the following equations

a+b+c+d+e=8

a^2+b^2+c^2+d^2+e^2=16

Determine the smallest and largest value that

a

can take.

1

flag with 4 colors

You want to color a flag like the one shown in the following image, for which four different colors are available. Two regions of the flag that share a side (or a segment of a side) must have different colors. The flag cannot be flipped, rotated, or reflected. How many different flags can be colored with these conditions? https://cdn.artofproblemsolving.com/attachments/4/9/879d1e144acdbc63ee2ffe34cf13a920d5d836.png

2018 Regional Olympiad of Mexico West

Subcontests

sum ABCD +BCDA+CDAB+DABC, when each number has least prime divisor 11

orthocenter of ABC is also orthocenter of DEF

n bulbs on a line, on off

similar triangles wanted, <ABC = <CBD, <ACP=<BCD, (ABC)=(BPCD)

min and max a, if a+b+c+d+e=8 and a^2+b^2+c^2+d^2+e^2=16

flag with 4 colors

2018 Regional Olympiad of Mexico West

Subcontests

sum ABCD +BCDA+CDAB+DABC, when each number has least prime divisor 11

orthocenter of ABC is also orthocenter of DEF

n bulbs on a line, on off

similar triangles wanted, &lt;ABC = &lt;CBD, &lt;ACP=&lt;BCD, (ABC)=(BPCD)

min and max a, if a+b+c+d+e=8 and a^2+b^2+c^2+d^2+e^2=16

flag with 4 colors

similar triangles wanted, <ABC = <CBD, <ACP=<BCD, (ABC)=(BPCD)