MathDB

2006 Rioplatense Mathematical Olympiad, Level 3

Part of Rioplatense Mathematical Olympiad, Level 3

Subcontests

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Transposing numbers arranged in a circle

The numbers 1,2,…,20061, 2,\ldots, 2006 are written around the circumference of a circle. A move consists of exchanging two adjacent numbers. After a sequence of such moves, each number ends up 1313 positions to the right of its initial position. lf the numbers 1,2,…,20061, 2,\ldots, 2006 are partitioned into 10031003 distinct pairs, then show that in at least one of the moves, the two numbers of one of the pairs were exchanged.

Sequence x_{n+2} = gcd( x_{n+1} , x_{n} ) + 2006

An infinite sequence x1,x2,…x_1,x_2,\ldots of positive integers satisfies xn+2=gcd⁡(xn+1,xn)+2006 x_{n+2}=\gcd(x_{n+1},x_n)+2006 for each positive integer nn. Does there exist such a sequence which contains exactly 10200610^{2006} distinct numbers?
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Recoloring regions bounded by lines in the plane

A given finite number of lines in the plane, no two of which are parallel and no three of which are concurrent, divide the plane into finite and infinite regions. In each finite region we write 11 or −1-1. In one operation, we can choose any triangle made of three of the lines (which may be cut by other lines in the collection) and multiply by −1-1 each of the numbers in the triangle. Determine if it is always possible to obtain 11 in all the finite regions by successively applying this operation, regardless of the initial distribution of 11s and −1-1s.

Find an angle sum in quadrilateral ABCD with AB=AD & CB=CD

Let ABCDABCD be a convex quadrilateral with AB=ADAB = AD and CB=CDCB = CD. The bisector of ∠BDC\angle BDC intersects BCBC at LL, and ALAL intersects BDBD at MM, and it is known that BL=BMBL = BM. Determine the value of 2∠BAD+3∠BCD2\angle BAD + 3\angle BCD.
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Express n as the sum of k distinct divisors of n

(a) For each integer k≥3k\ge 3, find a positive integer nn that can be represented as the sum of exactly kk mutually distinct positive divisors of nn. (b) Suppose that nn can be expressed as the sum of exactly kk mutually distinct positive divisors of nn for some k≥3k\ge 3. Let pp be the smallest prime divisor of nn. Show that 1p+1p+1+⋯+1p+k−1≥1.\frac1p+\frac1{p+1}+\cdots+\frac{1}{p+k-1}\ge1.

Concurrent: line thru orthocenter & circumcenter, BC & other

The acute triangle ABCABC with AB≠ACAB\neq AC has circumcircle Γ\Gamma, circumcenter OO, and orthocenter HH. The midpoint of BCBC is MM, and the extension of the median AMAM intersects Γ\Gamma at NN. The circle of diameter AMAM intersects Γ\Gamma again at AA and PP. Show that the lines APAP, BCBC, and OHOH are concurrent if and only if AH=HNAH = HN.