MathDB

4

Part of 2008 IMO Shortlist

Problems(3)

IMO ShortList 2008, Algebra problem 4

Source: IMO ShortList 2008, Algebra problem 4

7/9/2009
For an integer m m, denote by t(m) t(m) the unique number in {1,2,3} \{1, 2, 3\} such that m \plus{} t(m) is a multiple of 3 3. A function f:Z→Z f: \mathbb{Z}\to\mathbb{Z} satisfies f( \minus{} 1) \equal{} 0, f(0) \equal{} 1, f(1) \equal{} \minus{} 1 and f\left(2^{n} \plus{} m\right) \equal{} f\left(2^n \minus{} t(m)\right) \minus{} f(m) for all integers m m, n≥0 n\ge 0 with 2n>m 2^n > m. Prove that f(3p)≥0 f(3p)\ge 0 holds for all integers p≥0 p\ge 0. Proposed by Gerhard Woeginger, Austria
functionalgebrafunctional equationInequalityIMO Shortlist
IMO ShortList 2008, Number Theory problem 4

Source: IMO ShortList 2008, Number Theory problem 4

7/9/2009
Let n n be a positive integer. Show that the numbers \binom{2^n \minus{} 1}{0},\; \binom{2^n \minus{} 1}{1},\; \binom{2^n \minus{} 1}{2},\; \ldots,\; \binom{2^n \minus{} 1}{2^{n \minus{} 1} \minus{} 1} are congruent modulo 2n 2^n to 1 1, 3 3, 5 5, … \ldots, 2^n \minus{} 1 in some order. Proposed by Duskan Dukic, Serbia
number theorybinomial coefficientsmodular arithmeticIMO ShortlistHi
IMO Shortlist 2008, Geometry problem 4

Source: IMO Shortlist 2008, Geometry problem 4, German TST 5, P3, 2009

7/9/2009
In an acute triangle ABC ABC segments BE BE and CF CF are altitudes. Two circles passing through the point A A and F F and tangent to the line BC BC at the points P P and Q Q so that B B lies between C C and Q Q. Prove that lines PE PE and QF QF intersect on the circumcircle of triangle AEF AEF.
Proposed by Davood Vakili, Iran
geometrycircumcirclereflectionIMO Shortlist