MathDB

4

Part of 2007 JBMO Shortlist

Problems(3)

n_1 = a,n_k = b, (n_i + n_{i+1}) | n_in_{i+1}

Source: JBMO Shortlist 2007 A4

10/14/2017
Let aa and b b be positive integers bigger than 22. Prove that there exists a positive integer kk and a sequence n1,n2,...,nkn_1, n_2, ..., n_k consisting of positive integers, such that n1=a,nk=bn_1 = a,n_k = b, and (ni+ni+1)∣nini+1(n_i + n_{i+1}) | n_in_{i+1} for all i=1,2,...,k−1i = 1,2,..., k - 1
JBMOalgebrapositive integer
2007 JBMO Shortlist G4

Source: 2007 JBMO Shortlist G4

10/10/2017
Let SS be a point inside ∠pOq\angle pOq, and let kk be a circle which contains SS and touches the legs OpOp and OqOq in points PP and QQ respectively. Straight line ss parallel to OpOp from SS intersects OqOq in a point RR. Let TT be the intersection point of the ray PSPS and circumscribed circle of △SQR\vartriangle SQR and T≠ST \ne S. Prove that OT//SQOT // SQ and OTOT is a tangent of the circumscribed circle of △SQR\vartriangle SQR.
JBMOgeometry
n= ax + by iff n, f(n) = n - n_a - n_b, f(f(n)),.. \in N

Source: JBMO Shortlist 2007 N4

10/14/2017
Let a,ba, b be two co-prime positive integers. A number is called good if it can be written in the form ax+byax + by for non-negative integers x,yx, y. Defi ne the function f:Z→Zf : Z\to Z as f(n)=n−na−nbf(n) = n - n_a - n_b, where sts_t represents the remainder of ss upon division by tt. Show that an integer nn is good if and only if the in finite sequence n,f(n),f(f(n)),...n, f(n), f(f(n)), ... contains only non-negative integers.
JBMOnumber theoryremainder