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?
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