MathDB

4

Part of 1999 IMO Shortlist

Problems(4)

IMO ShortList 1999, geometry problem 4

Source: IMO ShortList 1999, geometry problem 4

11/13/2004
For a triangle T=ABCT = ABC we take the point XX on the side (AB)(AB) such that AX/AB=4/5AX/AB=4/5, the point YY on the segment (CX)(CX) such that CY=2YXCY = 2YX and, if possible, the point ZZ on the ray (CACA such that CXZ^=180−ABC^\widehat{CXZ} = 180 - \widehat{ABC}. We denote by Σ\Sigma the set of all triangles TT for which XYZ^=45\widehat{XYZ} = 45. Prove that all triangles from Σ\Sigma are similar and find the measure of their smallest angle.
geometrytrigonometryTriangleanglessimilarityIMO Shortlist
IMO ShortList 1999, number theory problem 4

Source: IMO ShortList 1999, number theory problem 4

11/13/2004
Denote by S the set of all primes such the decimal representation of 1p\frac{1}{p} has the fundamental period divisible by 3. For every p∈Sp \in S such that 1p\frac{1}{p} has the fundamental period 3r3r one may write 1p=0,a1a2…a3ra1a2…a3r…,\frac{1}{p}=0,a_{1}a_{2}\ldots a_{3r}a_{1}a_{2} \ldots a_{3r} \ldots , where r=r(p)r=r(p); for every p∈Sp \in S and every integer k≥1k \geq 1 define f(k,p)f(k,p) by f(k,p)=ak+ak+r(p)+ak+2.r(p) f(k,p)= a_{k}+a_{k+r(p)}+a_{k+2.r(p)} a) Prove that SS is infinite. b) Find the highest value of f(k,p)f(k,p) for k≥1k \geq 1 and p∈Sp \in S
number theorydecimal representationperiodic functionrationalIMO Shortlist
IMO ShortList 1999, algebra problem 4

Source: IMO ShortList 1999, algebra problem 4

11/14/2004
Prove that the set of positive integers cannot be partitioned into three nonempty subsets such that, for any two integers x,yx,y taken from two different subsets, the number x2−xy+y2x^2-xy+y^2 belongs to the third subset.
algebraColoringpartitionRamsey TheoryIMO Shortlistcombinatorics
IMO ShortList 1999, combinatorics problem 4

Source: IMO ShortList 1999, combinatorics problem 4

11/14/2004
Let AA be a set of NN residues (modN2)\pmod{N^{2}}. Prove that there exists a set BB of of NN residues (modN2)\pmod{N^{2}} such that A+B={a+b∣a∈A,b∈B}A + B = \{a+b|a \in A, b \in B\} contains at least half of all the residues (modN2)\pmod{N^{2}}.
modular arithmeticcombinatoricscountingAdditive combinatoricsAdditive Number TheoryIMO ShortlistHi