MathDB

2003 IberoAmerican

Part of IberoAmerican

Subcontests

(3)
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2

18th ibmo - argentina 2003/q3

Pablo copied from the blackboard the problem: Consider all the sequences of 20042004 real numbers (x0,x1,x2,…,x2003)(x_0,x_1,x_2,\dots, x_{2003}) such that: x0=1,0≤x1≤2x0,0≤x2≤2x1…,0≤x2003≤2x2002x_0=1, 0\le x_1\le 2x_0,0\le x_2\le 2x_1\ldots ,0\le x_{2003}\le 2x_{2002}. From all these sequences, determine the sequence which minimizes S=⋯S=\cdots
As Pablo was copying the expression, it was erased from the board. The only thing that he could remember was that SS was of the form S=±x1±x2±⋯±x2002+x2003S=\pm x_1\pm x_2\pm\cdots\pm x_{2002}+x_{2003}. Show that, even when Pablo does not have the complete statement, he can determine the solution of the problem.

18th ibmo - argentina 2003/q6

The sequences (an),(bn)(a_n),(b_n) are defined by a0=1,b0=4a_0=1,b_0=4 and for n≥0n\ge 0 an+1=an2001+bn,  bn+1=bn2001+ana_{n+1}=a_n^{2001}+b_n,\ \ b_{n+1}=b_n^{2001}+a_n Show that 20032003 is not divisor of any of the terms in these two sequences.
2
2

18th ibmo - argentina 2003/q2

Let CC and DD be two points on the semicricle with diameter ABAB such that BB and CC are on distinct sides of the line ADAD. Denote by MM, NN and PP the midpoints of ACAC, BDBD and CDCD respectively. Let OAO_A and OBO_B the circumcentres of the triangles ACPACP and BDPBDP. Show that the lines OAOBO_AO_B and MNMN are parallel.

18th ibmo - argentina 2003/q5

In a square ABCDABCD, let PP and QQ be points on the sides BCBC and CDCD respectively, different from its endpoints, such that BP=CQBP=CQ. Consider points XX and YY such that X≠YX\neq Y, in the segments APAP and AQAQ respectively. Show that, for every XX and YY chosen, there exists a triangle whose sides have lengths BXBX, XYXY and DYDY.
1
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18th ibmo - argentina 2003/q1

(a)(a)There are two sequences of numbers, with 20032003 consecutive integers each, and a table of 22 rows and 20032003 columns     ⋯⋯     ⋯⋯  \begin{array}{|c|c|c|c|c|c|} \hline\ \ &\ &\ &\cdots\cdots&\ &\ \\ \hline \ &\ &\ &\cdots\cdots&\ &\ \\ \hline \end{array} Is it always possible to arrange the numbers in the first sequence in the first row and the second sequence in the second row, such that the sequence obtained of the 20032003 column-wise sums form a new sequence of 20032003 consecutive integers? (b)(b) What if 20032003 is replaced with 20042004?

18th ibmo - argentina 2003/q4

Let M={1,2,…,49}M=\{1,2,\dots,49\} be the set of the first 4949 positive integers. Determine the maximum integer kk such that the set MM has a subset of kk elements such that there is no 66 consecutive integers in such subset. For this value of kk, find the number of subsets of MM with kk elements with the given property.