MathDB

4

Part of 2022 IFYM, Sozopol

Problems(5)

Problem 4 of First round

Source: XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade

9/9/2022
Does there exist a surjective function f:R→Rf:\mathbb{R} \rightarrow \mathbb{R} for which
f(x+y)−f(x)−f(y)f(x+y)-f(x)-f(y)
takes only 0 and 1 for values for random xx and yy?
surjective functionfunction problemalgebra
Problem 4 of Second round

Source: XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade

9/9/2022
a) Prove that for each positive integer nn the number or ordered pairs of integers (x,y)(x,y) for which x2−xy+y2=nx^2-xy+y^2=n is finite and is multiple of 6. b) Find all ordered pairs of integers (x,y)(x,y) for which x2−xy+y2=727x^2-xy+y^2=727.
number theory
Problem 4 of Third round

Source: XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade

9/9/2022
Let x1,…,xnx_1,\dots ,x_n be real numbers. We look at all the 2n−12^{n-1} possible sums between some of the numbers. If the number of different sums is at least 1.8n1.8^n, prove that the number of sums equal to 20222022 is no more than 1.67n1.67^n.
combinatoricsset theory
\prod^n_{i=0} x^{n-1}_i/ \prod_{j \ne i}(x_i - x_j) is fixed

Source: IFYM - XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade, 4th round p4

11/12/2022
Let nn be a natural number. To prove that the value of the expression ∏i=0nxin−1∏j≠i(xi−xj)\prod^n_{i=0}\frac{x^{n-1}_i}{\prod_{j \ne i}(x_i - x_j)} does not depend on the choice of the different real numbers x0,x1,...,xnx_0, x_1, ... , x_n.
algebra
add digits of a naturals, in every move, sometimes on pair only

Source: IFYM - XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade,finals p4

11/13/2022
A natural number xx is written on the board. In one move, we can take the number on the board and between any two of its digits in its decimal notation we can we put a sign ++, or we may not put it, then we calculate the obtained result and we write it on the board in place of xx. For example, from the number 819819. we can get 1818 by 8+1+98 + 1 + 9, 9090 by 81+981 + 9, and 2727 by 8+198 + 19. Prove that no matter what xx is, we can reach a single digit number with at most 44 moves.
number theoryDigitssum of digits