MathDB

1

Part of 2008 Iran MO (3rd Round)

Problems(6)

n|1^n+2^n+...+k^n

Source: Iranian National Olympiad (3rd Round) 2008

8/30/2008
Let k>1 k>1 be an integer. Prove that there exists infinitely many natural numbers such as n n such that: n|1^n\plus{}2^n\plus{}\dots\plus{}k^n
inductionnumber theory proposednumber theory
a^3+1 is not a root

Source: Iranian National Olympiad (3rd Round) 2008

8/30/2008
Suppose that f(x)∈Z[x] f(x)\in\mathbb Z[x] be an irreducible polynomial. It is known that f f has a root of norm larger than 32 \frac32. Prove that if α \alpha is a root of f f then f(\alpha^3\plus{}1)\neq0.
algebrapolynomialalgebra proposed
A counting problem

Source: Iranian National Olympiad (3rd Round) 2008

8/31/2008
Prove that the number of pairs (α,S) \left(\alpha,S\right) of a permutation α \alpha of {1,2,…,n} \{1,2,\dots,n\} and a subset S S of {1,2,…,n} \{1,2,\dots,n\} such that ∀x∈S:α(x)∉S \forall x\in S: \alpha(x)\not\in S is equal to n!F_{n \plus{} 1} in which Fn F_n is the Fibonacci sequence such that F_1 \equal{} F_2 \equal{} 1
combinatorics proposedcombinatorics
A root on unit circle

Source: Iranian National Olympiad (3rd Round) 2008

9/12/2008
Prove that for n>0 n > 0 and a≠0 a\neq0 the polynomial p(z) \equal{} az^{2n \plus{} 1} \plus{} bz^{2n} \plus{} \bar bz \plus{} \bar a has a root on unit circle
algebrapolynomialtrigonometryfunctioncalculusintegrationratio
Equal perpendiculars

Source: Iranian National Olympiad (3rd Round) 2008

9/12/2008
Let ABC ABC be a triangle with BC>AC>AB BC > AC > AB. Let A′,B′,C′ A',B',C' be feet of perpendiculars from A,B,C A,B,C to BC,AC,AB BC,AC,AB, such that AA' \equal{} BB' \equal{} CC' \equal{} x. Prove that: a) If ABC∼A′B′C′ ABC\sim A'B'C' then x \equal{} 2r b) Prove that if A′,B′ A',B' and C′ C' are collinear, then x \equal{} R \plus{} d or x \equal{} R \minus{} d. (In this problem R R is the radius of circumcircle, r r is radius of incircle and d \equal{} OI)
geometrycircumcirclegeometric transformationreflectionincentercyclic quadrilateralgeometry proposed
Arresting Kaiser

Source: Iranian National Olympiad (3rd Round) 2008

9/20/2008
Police want to arrest on the famous criminals of the country whose name is Kaiser. Kaiser is in one of the streets of a square shaped city with n n vertical streets and n n horizontal streets. In the following cases how many police officers are needed to arrest Kaiser? http://i38.tinypic.com/2i1icec_th.png http://i34.tinypic.com/28rk4s3_th.png a) Each police officer has the same speed as Kaiser and every police officer knows the location of Kaiser anytime. b) Kaiser has an infinite speed (finite but with no bound) and police officers can only know where he is only when one of them see Kaiser. Everybody in this problem (including police officers and Kaiser) move continuously and can stop or change his path.
searchcombinatorics proposedcombinatorics