# ALGEBRA MATH TEST

Author: Ngo Kien Khang
The test has 4 highly difficult questions, corresponding to 20 points. Knowledge is within the scope of high school level.

Complete the test and get an award.
Question 1
Find all polynomials P(x) with real coefficients such that:
P($$x^{2}$$+3x+1)=$$P(x)^{2}$$

Question 2
Given 3 real positive numbers a, b, c such that (a+b)(b+c)(c+a)=1.
Find the minimum value of:$$\frac{a}{b(b+{2} c^{2})}$$ + $$\frac{b}{c(c+{2} a^{2})}$$ + $$\frac{c}{a(a+{2} b^{2})}$$

Question 3
Given polynominal f(x)=$$x^{2}$$-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: $$x^{5}$$+$$x^{4}$$+1=$$p^{y}$$. Find the value of p.

Question 6
Given corners a(1), a(2), ..., a(n) with 0°$$\le$$a(i)$$\le$$180° (i=1, 2, ..., n) such that $$\sum_{i=1}^{n }(1+cos a(i))$$ is an odd integer number. Find the minimum value of:$$\sum_{i=1}^{n }sin a(i)$$

Question 7
Find the largest value of n such that the equation $$x^{2}$$+nx+n=0 has integer roots.

Question 8
Given 3 integer numbers a, b, c such that abc=$$2021^{2024}$$. Find the remainder of division A=29$$a^{2}$$+2$$b^{2}$$+2024$$c^{2}$$+2025 by 24.

Question 9
Let a, b, c, d are 4 positive integer numbers such that:
$$b^{2}$$+b+1=ac and $$c^{2}$$+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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