MathDB

2022 May Olympiad

Part of May Olympiad

Subcontests

(5)
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winning strategy, vertices of regular n-gon

The vertices of a regular polygon with NN sides are marked on the blackboard. Ana and Beto play alternately, Ana begins. Each player, in turn, must do the following: ∙\bullet join two vertices with a segment, without cutting another already marked segment; or ∙\bullet delete a vertex that does not belong to any marked segment. The player who cannot take any action on his turn loses the game. Determine which of the two players can guarantee victory: a) if N=28N=28 b) if N=29N=29

isosceles triangles cut into 3 triangles, 2/3 isosceles, 2022 May Olympiad L1 P5

Vero had an isosceles triangle made of paper. Using scissors, he divided it into three smaller triangles and painted them blue, red and green. Having done so, he observed that: ∙\bullet with the blue triangle and the red triangle an isosceles triangle can be formed, ∙\bullet with the blue triangle and the green triangle an isosceles triangle can be formed, ∙\bullet with the red triangle and the green triangle an isosceles triangle can be formed. Show what Vero's triangle looked like and how he might have made the cuts to make this situation be possible.
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numbers at vertices of triangle / tetrahedron, consecutive and gcd related

a) A positive integer is written at each vertex of a triangle. Then on each side of the triangle the greatest common divisor of its ends is written. It is possible that the numbers written on the sides be three consecutive integers, in some order? b) A positive integer is written at each vertex of a tetrahedron. Then, on each edge of the tetrahedron is written the greatest common divisor of its ends . It is possible that the numbers written in the edges are six consecutive integers, in some order?

4 squares on a chessboard

Ana and Bruno have an 8×88 \times 8 checkered board. Ana paints each of the 6464 squares with some color. Then Bruno chooses two rows and two columns on the board and looks at the 44 squares where they intersect. Bruno's goal is for these 44 squares to be the same color. How many colors, at least, must Ana use so that Bruno can't fulfill his objective? Show how you can paint the board with this amount of colors and explain because if you use less colors then Bruno can always fulfill his goal.
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min sum by composite / primes numbers by cards 1-9

There are nine cards that have the digits 1,2,3,4,5,6,7,81, 2, 3, 4, 5, 6, 7, 8 and 99 written on them, with one digit on each card. Using all the cards, some numbers are formed (for example, the numbers 88, 213213, 9494, 6565 and 77). a) If all the numbers formed are prime, determine the smallest possible value of their sum. b) If all formed numbers are composite, determine the smallest possible value of their sum.
Note: A number pp is prime if its only divisors are 11 and pp. A number is composite if it has more than two dividers. The number 11 is neither prime nor composite.

listing 3-digit numbers with 6 from 9 digits

Bob chose six of the nine digits from 11 to 99 and wrote the list, ordered from smallest to largest, of all three-digit numbers that can be formed using the digits you chose. At Bob's list, the number 317317 appears at position 2222. What number appears at position 6060 in the list from Bob? Find all possibilities.
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