MathDB

1

2^n + 2021n is a perfect square

n

for which

2^n + 2021n

is a perfect square.

P4

1

One more card game

Let

n

be a positive integer. Andrés has

n+1

cards and each of them has a positive integer written, in such a way that the sum of the

n+1

numbers is

3n

. Show that Andrés can place one or more cards in a red box and one or more cards in a blue box in such a way that the sum of the numbers of the cards in the red box is equal to twice the sum of the numbers of the cards in the blue box. Clarification: Some of Andrés's letters can be left out of the boxes.

P3

1

10 students in an exam

Ten students are seated around a circular table. The teacher has a list of fifteen problems and each student is given six problems, in such a way that each problem is given exactly four times and any two students they have at most three problems in common. Prove that no matter how the teacher distributes the problems, there will always be two students sitting next to each other who have at least one problem in common.

P1

1

A manual count doesn&#039;t seem abusive... right?

How many multiples of

11

of four digits, of the form

\overline{abcd}

, satisfy that

a\neq b, b\neq c

and

c\neq a

?

P6

1

length of segment is constant, independent of choice of line (2 circles)

Two circles

\omega_1

and

\omega_2

, which have centers

O_1

and

O_2

, respectively, intersect at

A

and

B

. A line

\ell

that passes through

B

cuts to

\omega_1

again at

C

and cuts to

\omega_2

again in

D

, so that points

C, B, D

appear in that order. The tangents of

\omega_1

and

\omega_2

in

C

and

D

, respectively, intersect in

E

. Line

AE

intersects again to the circumscribed circumference of the triangle

AO_1O_2

in

F

. Try that the length of the

EF

segment is constant, that is, it does not depend on the choice of

\ell

.

P2

1

3A_{2015} + B_{2015}> 4A_{4030}, area regular n-gons inequality

Let

\omega

be a circle. For each

n

, let

A_n

be the area of a regular

n

-sided polygon circumscribed to

\omega

and

B_n

the area of a regular

n

-sided polygon inscribed in

\omega

. Try that

3A_{2015} + B_{2015}> 4A_{4030}

2016 Peru Cono Sur TST

Subcontests

2^n + 2021n is a perfect square

One more card game

10 students in an exam

A manual count doesn&#039;t seem abusive... right?

length of segment is constant, independent of choice of line (2 circles)

3A_{2015} + B_{2015}> 4A_{4030}, area regular n-gons inequality

2016 Peru Cono Sur TST

Subcontests

2^n + 2021n is a perfect square

One more card game

10 students in an exam

A manual count doesn&amp;#039;t seem abusive... right?

length of segment is constant, independent of choice of line (2 circles)

3A_{2015} + B_{2015}&gt; 4A_{4030}, area regular n-gons inequality

A manual count doesn't seem abusive... right?

3A_{2015} + B_{2015}> 4A_{4030}, area regular n-gons inequality