MathDB

2016 Chile Junior Math Olympiad

Part of Chile Junior Math Olympiad

Subcontests

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2016 Chile NMO Juniors XXVIII

p1. Consider the sequence of digits obtained from writing the consecutive natural numbers from 11 to 100,000100,000: 1234567891011121314...99998999991000001234567891011121314...9999899999100000 Determine how many times the 20162016 block appears in this sequence.
[url=https://artofproblemsolving.com/community/c4h1845779p12426933]p2. For an equilateral triangle △ABC\triangle ABC, determine whether or not there is a point PP inside △ABC\triangle ABC so that any straight line that passes through PP divides the triangle △ABC\triangle ABC in two polygonal lines of equal length.
p3. On a 1000×10001000 \times 1000 squared board, place domino pieces (2×12\times 1 or 1×21\times 2), so that each piece of domino covers exactly two squares of the board. Two domino pieces are not allowed to be adjacent, and they are allowed to be touch in a vertex. Determine the maximum number of domino pieces that can be put following these rules.
[url=https://artofproblemsolving.com/community/c4h2945587p26369677]p4. The product 12⋅24⋅38⋅416⋅...⋅99299⋅1002100\frac12 \cdot \frac24 \cdot \frac38 \cdot \frac{4}{16} \cdot ... \cdot \frac{99}{2^{99}} \cdot \frac{100}{2^{100}} is written in its most simplified form. What is the last digit of the denominator?
[url=https://artofproblemsolving.com/community/c4h2917781p26063806]p5. Let △ABC\vartriangle ABC be an is osceles triangle with AC=BCAC = BC. Let OO be the center of the circle circumscribed to the triangle and II the center of the inscribed circle. If DD is the point on side BCBC such that ODOD is perpendicular to BIBI. Show that IDID is parallel to ACAC.
p6. Beto plays the following solitaire: initially a machine chooses at random 2626 positive integers between 11 and 20162016, and writes them on a blackboard (there may be numbers repeated). At each step, Beto chooses some of the numbers written on the blackboard, and they subtract from each of them the same non-negative integer number k with the condition that the resulting numbers remain non-negative. The objective of the game is to achieve that in sometime the 2626 numbers are equal to 00, in which case the game ends and Beto win. Determine the fewest steps that guarantee Beto victory, without import the 2626 numbers initially chosen.