MathDB

2007 JBMO Shortlist

Part of JBMO ShortLists

Subcontests

(5)
5
1
4
3

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

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

2007 JBMO Shortlist G4

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.

n= ax + by iff n, f(n) = n - n_a - n_b, f(f(n)),.. \in N

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.
3
4
1
3

İnterior point

G1\boxed{\text{G1}} Let MM be interior point of the triangle ABCABC with AB=MCAB=MC

tilings with m x n rectangles with corners

We call a tiling of an m×nm \times n rectangle with corners (see figure below) "regular" if there is no sub-rectangle which is tiled with corners. Prove that if for some mm and nn there exists a "regular" tiling of the m×nm \times n rectangular then there exists a "regular" tiling also for the 2m×2n2m \times 2n rectangle.

1/x+1/y+1/[x, y]+1/(x, y)=1/2

Find all the pairs positive integers (x,y)(x, y) such that 1x+1y+1[x,y]+1(x,y)=12\frac{1}{x}+\frac{1}{y}+\frac{1}{[x, y]}+\frac{1}{(x, y)}=\frac{1}{2} , where (x,y)(x, y) is the greatest common divisor of x,yx, y and [x,y][x, y] is the least common multiple of x,yx, y.
2
2