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2006 Chile Classification / Qualifying NMO Juniors XVIII

Source:

October 9, 2021
algebrageometrycombinatoricsnumber theorychilean NMO

Problem Statement

p1. Let a,ba, b different real numbers such that ab+a+10bb+10a=2\frac{a}{b} +\frac{a + 10b}{b + 10a}= 2. Find ab\frac{a}{b}
[url=https://artofproblemsolving.com/community/c4h1846788p12438105]p2. The vertex EE of a square EFGHEFGH with side 20062006 mm is in the center of the square ABCDABCD with side 1010 mm. Segment EFEF intersects CDCD at point II. Segment EHEH intersects ADAD at JJ. If ∠EID=60o\angle EID = 60^o, find the area of the quadrilateral EIDJEIDJ.
p3. The number 30a0b03‾\overline{30a0b03} in decimal notation is divisible by 1313. Find the possible values of the digits a,ba, b.
p4. In a rectangular board of 20062006 squares, distributed in 3434 rows and 5959 columns, they will be placed three identical buttons in the center of the boxes, determining a triangle. In how many different ways can we can place the buttons forming a ractangle triangle with legs parallel to the edges of the board?
[url=https://artofproblemsolving.com/community/c4h1846785p12438094]p5. Let △ABC\vartriangle ABC be any triangle and △MNP\vartriangle MNP be the triangle formed by the points of tangency of the inscribed circle with the sides of the triangle △ABC\vartriangle ABC, show that if △MNP\vartriangle MNP is equilateral, then △ABC\vartriangle ABC is equilateral.
p6. A natural number is called a palindrome, when the same number is obtained by writing its digits in reverse order, for example 2343223432, 565565, 8 8 are palindromes. Determine all pairs of positive integers m,nm, n such that 1111...11⏟m  times⋅1111...11⏟n  times\underbrace{1111...11} _{m \,\, times} \cdot \underbrace{1111...11}_{n \,\, times} is palindrome.
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