1. For every real number x, let ⌊x⌋ denote the greatest integer not exceeding x, and let$f(x)=$⌊x⌋(2021^$x$-⌊x⌋-1) .The set of all numbers x such that 1 ≤x ≤ 2021 and let$f(x)\le 1$ is a union of disjoint intervals. What is the sum of the lengths of those intervals?

2. The parabola P has focus (0, 0) and goes through the points (4, 3) and (-4, -3). For how many points (x, y)∈ P, with integer coordinates is it true that |4x + 3y| ≤1000?

3. Consider all polynomials of a complex variable P(z)=4z⁴+az³+bz²+cz+d,  where a, b, c and d are integers, 0≤d≤c≤b≤a≤4, and the polynomial has a zero z₀ with |z| = 1. What is the sum of all values P(1) over all the polynomials with these properties?

4. How many ordered triples （x, y, z） of positive integers satisfy lcm(x, y) = 72, lcm(x, z) = 600, and lcm(y, z) = 900?

5. The arithmetic mean of two distinct positive integers x and y is a two-digit integer. The geometric mean of x and y is obtained by reversing the digits of the arithmetic mean. What is |x-y|?

6. Find the sum of all the positive solutions of $2cos2x$($cos2x-cos$($\frac{2021{\pi }^2}{\frac{1}{2}x}$))=$cos4x-1$

7. Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly 7.  Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?

8. For a positive integer n and nonzero digits a, b, and c, let A_$n$ be the n-digit integer each of whose digits is equal to a; let B_$n$ be the n-digit integer each of whose digits is equal to b, and let C_$n$ be the 2n-digit (not n-digit) integer each of whose digits is equal to _$C$. What is the greatest possible value of a + b + c for which there are at least two values of n such that C_$n$-B_$n$=A_$n$^$2$ ?

9. Suppose f is a function that satisfies the equation$f(x+y)=f(x)+f(y)+csy+x{y}^2$For all read numbers x and y, where c is a constant. Suppose also that $$\underset{n\to 0}{lim}\frac{f(x)}{x}=1$$ Find $$\underset{n\to 0}{lim}\frac{\sqrt[3]{f^{\prime }(x)}-1}{x}=1$$

10. Find the maximum area of a rectangle with two sides on the 𝑥 and 𝑦 axes, and one vertex on the tangent line of the function $f(x)=(2x-1{)}^{5}ln|x|+\sqrt{x}$ that passes point P (1, 1).