MathDB

A nice combinatorics from Iranian TST 2017

Source: Iranian TST 2017, first exam day 2, problem 6

4/6/2017

In the unit squares of a transparent

1 \times 100

tape, numbers

1,2,\cdots,100

are written in the ascending order.We fold this tape on it's lines with arbitrary order and arbitrary directions until we reach a

1 \times1

tape with

100

layers.A permutation of the numbers

1,2,\cdots,100

can be seen on the tape, from the top to the bottom. Prove that the number of possible permutations is between

2^{100}

and

4^{100}

. (e.g. We can produce all permutations of numbers

1,2,3

with a

1\times3

tape)
Proposed by Morteza Saghafian

combinatoricsIranIranian TSTHamiltonian pathcatalan

2017 Iran TST2 day2 p6

Source: 2017 Iran TST second exam day2 p6

4/24/2017

Let

k>1

be an integer. The sequence

a_1,a_2, \cdots

is defined as:

a_1=1, a_2=k

and for all

n>1

we have:

a_{n+1}-(k+1)a_n+a_{n-1}=0

Find all positive integers

n

such that

a_n

is a power of

k

.
Proposed by Amirhossein Pooya

algebranumber theoryIranIranian TST

Geometry from Iran TST 2017

Source: 2017 Iran TST third exam day2 p6

4/27/2017

In triangle

ABC

let

O

and

H

be the circumcenter and the orthocenter. The point

P

is the reflection of

A

with respect to

OH

. Assume that

P

is not on the same side of

BC

as

A

. Points

E,F

lie on

AB,AC

respectively such that

BE=PC \ , CF=PB

. Let

K

be the intersection point of

AP,OH

. Prove that

\angle EKF = 90 ^{\circ}

Proposed by Iman Maghsoudi

IranIranian TSTgeometry

6

Problems(3)