MathDB

Iran number theory

Source: Iranian TST 2018, first exam day 2, problem 5

4/8/2018

Prove that for each positive integer

m

, one can find

m

consecutive positive integers like

n

such that the following phrase doesn't be a perfect power:

\left(1^3+2018^3\right)\left(2^3+2018^3\right)\cdots \left(n^3+2018^3\right)

Proposed by Navid Safaei

number theory proposed

iran tst 2018 geometry

Source: Iranian TST 2018, second exam day 2, problem 5

4/17/2018

Let

\omega

be the circumcircle of isosceles triangle

ABC

(

AB=AC

). Points

P

and

Q

lie on

\omega

and

BC

respectively such that

AP=AQ

.

AP

and

BC

intersect at

R

. Prove that the tangents from

B

and

C

to the incircle of

\triangle AQR

(different from

BC

) are concurrent on

\omega

.
Proposed by Ali Zamani, Hooman Fattahi

geometry

Combinatorics from Iran TST

Source: Iranian TST 2018, third exam day 2, problem 5

4/19/2018

2n-1

distinct positive real numbers with sum

S

are given. Prove that there are at least

\binom {2n-2}{n-1}

different ways to choose

n

numbers among them such that their sum is at least

\frac {S}{2}

.
Proposed by Amirhossein Gorzi

combinatoricsIranIranian TST

5