Grossman Math Olympiad, 2017, Juniors
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Question 1 - Enlarging a Rectangle
A rectangle is given in the plane. Is it possible that after each side of the rectangle is increased by 1 cm, the area increases by 1 square meter? Provide an example or prove that it is impossible.
(If the rectangle is 1x5, it becomes 2x6 and no side can be 0)
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Question 2 - Integer Coefficients?
Given real numbers a, b, c such that for every integer x, the number `ax^2+bx+c` is an integer. Does this necessarily imply that a, b, c are all integers? Prove it, or provide a counterexample.
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Question 3 - Equality in Stages
The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are written on the board, and David is supposed to change them in stages. At each stage, David is allowed to choose two numbers and change them by 1, that is, to add 1 to both, subtract 1 from both, or add 1 to one and subtract 1 from the other.
Can David, after a number of stages, reach a situation where all the numbers on the board are equal? If so, show an example, and if not, explain your answer in detail.
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Question 4 - Government Elections
In the parliamentary elections of a certain country, a number of parties participated (the exact number is unknown). Every citizen who participated in the elections voted for one party.
For every positive integer k, let `d_k` denote the number of parties that received k or more votes. Prove that the sum `d_1+d_2+d_3+...` is equal to the number of citizens who voted in the elections.
Topics:Combinatorics -> Induction (Mathematical Induction)