MathDB

2007 CentroAmerican

Part of CentroAmerican

Subcontests

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Quadratic polynomials and a set of integers

Let SS be a finite set of integers. Suppose that for every two different elements of SS, pp and qq, there exist not necessarily distinct integers a≠0a \neq 0, bb, cc belonging to SS, such that pp and qq are the roots of the polynomial ax2+bx+cax^{2}+bx+c. Determine the maximum number of elements that SS can have.

Nice problem from a friend: prove G is the centroid of ABP

Consider a circle SS, and a point PP outside it. The tangent lines from PP meet SS at AA and BB, respectively. Let MM be the midpoint of ABAB. The perpendicular bisector of AMAM meets SS in a point CC lying inside the triangle ABPABP. ACAC intersects PMPM at GG, and PMPM meets SS in a point DD lying outside the triangle ABPABP. If BDBD is parallel to ACAC, show that GG is the centroid of the triangle ABPABP. Arnoldo Aguilar (El Salvador)
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n divides the year of the nth Olympiad

The Central American Olympiad is an annual competition. The ninth Olympiad is held in 2007. Find all the positive integers nn such that nn divides the number of the year in which the nn-th Olympiad takes place.

Synonymous words

In a remote island, a language in which every word can be written using only the letters aa, bb, cc, dd, ee, ff, gg is spoken. Let's say two words are synonymous if we can transform one into the other according to the following rules: i) Change a letter by another two in the following way: a→bc, b→cd, c→de, d→ef, e→fg, f→ga, g→aba \rightarrow bc,\ b \rightarrow cd,\ c \rightarrow de,\ d \rightarrow ef,\ e \rightarrow fg,\ f\rightarrow ga,\ g\rightarrow ab ii) If a letter is between other two equal letters, these can be removed. For example, dfd→fdfd \rightarrow f. Show that all words in this language are synonymous.