Question 1

Find all polynomials P(x) with real coefficients such that:

P([tex]x^{2}[/tex]+3x+1)=[tex]P(x)^{2}[/tex]

Question 2

Given 3 real positive numbers a, b, c such that (a+b)(b+c)(c+a)=1.

Find the minimum value of:[tex]\frac{a}{b(b+{2} c^{2})}[/tex] + [tex]\frac{b}{c(c+{2} a^{2})}[/tex] + [tex]\frac{c}{a(a+{2} b^{2})}[/tex]

Question 3

Given polynominal f(x)=[tex]x^{2}[/tex]-6x+12. Find the smallest value of x in: f(f(f(f(x))))=65539

Question 4

For each positive integer n, let s(n) be the number of positive integer divisors of n. Solve integer equation:s(n)+2023=n.

Question 5

Given 2 positive integer numbers a, b and a prime number p such that: [tex]x^{5}[/tex]+[tex]x^{4}[/tex]+1=[tex]p^{y}[/tex]. Find the value of p.

Question 6

Given corners a(1), a(2), ..., a(n) with 0°[tex]\le[/tex]a(i)[tex]\le[/tex]180° (i=1, 2, ..., n) such that [tex]\sum_{i=1}^{n }(1+cos a(i))[/tex] is an odd integer number. Find the minimum value of:[tex]\sum_{i=1}^{n }sin a(i)[/tex]

Question 7

Find the largest value of n such that the equation [tex]x^{2}[/tex]+nx+n=0 has integer roots.

Question 8

Given 3 integer numbers a, b, c such that abc=[tex]2021^{2024}[/tex]. Find the remainder of division A=29[tex]a^{2}[/tex]+2[tex]b^{2}[/tex]+2024[tex]c^{2}[/tex]+2025 by 24.

Question 9

Let a, b, c, d are 4 positive integer numbers such that:

[tex]b^{2}[/tex]+b+1=ac and [tex]c^{2}[/tex]+c+1=bd. Which of the following statements is true?

Question 10

Let S is the set of the first 2024 positive integers. How many non-empty subsets of S are there such that the sum of all the numbers in each subset is divisible by 256?