MathDB

5

Part of 2015 Tournament of Towns

Problems(2)

Coefficients of Products of 2 Polynomials

Source: Tournament of Towns Spring 2015 Senior A-level

2/24/2017
Do there exist two polynomials with integer coefficients such that each polynomial has a coefficient with an absolute value exceeding 20152015 but all coefficients of their product have absolute values not exceeding 11? (1010 points)
absolute valuealgebrapolynomial
Symbol Combinations on Real Numbers

Source: Tournament of Town Fall 2015 Senior A-level

2/23/2017
Several distinct real numbers are written on a blackboard. Peter wants to create an algebraic expression such that among its values there would be these and only these numbers. He may use any real numbers, brackets, signs +,−,×+, -, \times and a special sign ±\pm. Usage of ±\pm is equivalent to usage of ++ and −- in all possible combinations. For instance, the expression 5±15 \pm 1 results in {4,6}\{4, 6\}, while (2±0.5)±0.5(2 \pm 0.5) \pm 0.5 results in {1,2,3}\{1, 2, 3\}. Can Peter construct an expression if the numbers on the blackboard are : (a) 1,2,41, 2, 4 ? (22 points) (b) any 100100 distinct real numbers ? (66 points)
combinatoricsinvariant