MathDB

1

Part of 2014 Saudi Arabia IMO TST

Problems(4)

Tarik and Sultan

Source: Saudi Arabia IMO TST Day I Problem 1

7/22/2014
Tarik and Sultan are playing the following game. Tarik thinks of a number that is greater than 100100. Then Sultan is telling a number greater than 11. If Tarik’s number is divisible by Sultan’s number, Sultan wins, otherwise Tarik subtracts Sultan’s number from his number and Sultan tells his next number. Sultan is forbidden to repeat his numbers. If Tarik’s number becomes negative, Sultan loses. Does Sultan have a winning strategy?
number theory unsolvednumber theory
Prove that EI bisects <BEC

Source: Saudi Arabia IMO TST Day II Problem 1

7/22/2014
Let Γ\Gamma be a circle with center OO and AEAE be a diameter. Point DD lies on segment OEOE and point BB is the midpoint of one of the arcs AE^\widehat{AE} of Γ\Gamma. Construct point CC such that ABCDABCD is a parallelogram. Lines EBEB and CDCD meet at FF. Line OFOF meets the minor arc EB^\widehat{EB} at II. Prove that EIEI bisects ∠BEC\angle BEC.
geometry unsolvedgeometry
Perfect Numbers

Source: Saudi Arabia IMO TST Day III Problem 1

7/22/2014
A perfect number is an integer that equals half the sum of its positive divisors. For example, because 2⋅28=1+2+4+7+14+282 \cdot 28 = 1 + 2 + 4 + 7 + 14 + 28, 2828 is a perfect number.
[*] (a) A square-free integer is an integer not divisible by a square of any prime number. Find all square-free integers that are perfect numbers.
[*] (b) Prove that no perfect square is a perfect number.
number theory unsolvednumber theory
Difficult Inequality

Source: Saudi Arabia IMO TST Day IV Problem 1

7/22/2014
Let a1,…,ana_1,\dots,a_n be a non increasing sequence of positive real numbers. Prove that a12+a22+⋯+an2≤a1+a22+1+⋯+ann+n−1.\sqrt{a_1^2+a_2^2+\cdots+a_n^2}\le a_1+\frac{a_2}{\sqrt{2}+1}+\cdots+\frac{a_n}{\sqrt{n}+\sqrt{n-1}}. When does equality hold?
inequalitiesinductioninequalities unsolved