MathDB

7

Part of 2010 IMO Shortlist

Problems(2)

Least common multiple of steps of these progressions

Source: IMO Shortlist 2010, Combinatorics 7

7/17/2011
Let P1,…,PsP_1, \ldots , P_s be arithmetic progressions of integers, the following conditions being satisfied:
(i) each integer belongs to at least one of them; (ii) each progression contains a number which does not belong to other progressions.
Denote by nn the least common multiple of the ratios of these progressions; let n=p1α1⋯pkαkn=p_1^{\alpha_1} \cdots p_k^{\alpha_k} its prime factorization.
Prove that s≥1+∑i=1kαi(pi−1).s \geq 1 + \sum^k_{i=1} \alpha_i (p_i - 1).
Proposed by Dierk Schleicher, Germany
number theorymodular arithmeticcombinatoricsarithmetic sequenceIMO ShortlistSequence
IMO Shortlist 2010 - Problem G7

Source:

7/17/2011
Three circular arcs γ1,γ2,\gamma_1, \gamma_2, and γ3\gamma_3 connect the points AA and C.C. These arcs lie in the same half-plane defined by line ACAC in such a way that arc γ2\gamma_2 lies between the arcs γ1\gamma_1 and γ3.\gamma_3. Point BB lies on the segment AC.AC. Let h1,h2h_1, h_2, and h3h_3 be three rays starting at B,B, lying in the same half-plane, h2h_2 being between h1h_1 and h3.h_3. For i,j=1,2,3,i, j = 1, 2, 3, denote by VijV_{ij} the point of intersection of hih_i and γj\gamma_j (see the Figure below). Denote by VijVkj^VklVil^\widehat{V_{ij}V_{kj}}\widehat{V_{kl}V_{il}} the curved quadrilateral, whose sides are the segments VijVil,V_{ij}V_{il}, VkjVklV_{kj}V_{kl} and arcs VijVkjV_{ij}V_{kj} and VilVkl.V_{il}V_{kl}. We say that this quadrilateral is circumscribedcircumscribed if there exists a circle touching these two segments and two arcs. Prove that if the curved quadrilaterals V11V21^V22V12^,V12V22^V23V13^,V21V31^V32V22^\widehat{V_{11}V_{21}}\widehat{V_{22}V_{12}}, \widehat{V_{12}V_{22}}\widehat{V_{23}V_{13}},\widehat{V_{21}V_{31}}\widehat{V_{32}V_{22}} are circumscribed, then the curved quadrilateral V22V32^V33V23^\widehat{V_{22}V_{32}}\widehat{V_{33}V_{23}} is circumscribed, too.
Proposed by Géza Kós, Hungary
[asy] pathpen=black; size(400); pair A=(0,0), B=(4,0), C=(10,0); draw(L(A,C,0.3)); MP("A",A); MP("B",B); MP("C",C); pair X=(5,-7); path G1=D(arc(X,C,A)); pair Y=(5,7), Z=(9,6); draw(Z--B--Y); struct T {pair C;real r;}; T f(pair X, pair B, pair Y, pair Z) { pair S=unit(Y-B)+unit(Z-B); real s=abs(sin(angle((Y-B)/(Z-B))/2)); real t=10, r=abs(X-A); pair Q; for(int k=0;k<30;++k) { Q=B+t*S; t-=(abs(X-Q)-r)/abs(S)-s*t; }
T T=new T; T.C=Q; T.r=s*t*abs(S); return T; } void g(pair Q, real r) { real t=0; for(int k=0;k<30;++k) { X=(5,t); t+=(abs(X-Q)+r-abs(X-A)); } } pair Z1=(1.07,6); draw(B--Z1); T T=f(X,B,Y,Z1); draw(CR(T.C,T.r)); T T=f(X,B,Y,Z); draw(CR(T.C,T.r)); g(T.C,T.r); path G2=D(arc(X,C,A)); T T=f(X,B,Y,Z1); draw(CR(T.C,T.r)); T=f(X,B,Y,Z); draw(CR(T.C,T.r)); g(T.C,T.r); path G3=D(arc(X,C,A)); pen p=black+fontsize(8); MC("\gamma_1",G1,0.85,p); MC("\gamma_2",G2,0.85,NNW,p); MC("\gamma_3",G3,0.85,WNW,p); MC("h_1",B--Z1,0.95,E,p); MC("h_2",B--Y,0.95,E,p); MC("h_3",B--Z,0.95,E,p); path[] G={G1,G2,G3}; path[] H={B--Z1,B--Y,B--Z}; pair[][] al={{S+SSW,S+SSW,3*S},{SE,NE,NW},{2*SSE,2*SSE,2*E}}; for(int i=0;i<3;++i) for(int j=0;j<3;++j) MP("V_{"+string(i+1)+string(j+1)+"}",IP(H,G[j]),al[j],fontsize(8));[/asy]
geometrygeometric transformationhomothetyincenterIMO Shortlist