p1. Find all real numbers $x$ that satisfy the inequality $$\frac{x^2-3}{x^2-1}+ \frac{x^2 + 5}{x^2 + 3} \ge \frac{x^2-5}{x^2-3}+\frac{x^2 + 3}{x^2 + 1}$$ p2. It is known that $m$ is a four-digit natural number with the same units and thousands digits. If $m$ is a square of an integer, find all possible numbers $m$. p3. In the following figure, $\vartriangle ABP$ is an isosceles triangle, with $AB = BP$ and point $C$ on $BP$. Calculate the volume of the object obtained by rotating $ \vartriangle ABC$ around the line $AP$ p4. A class farewell event is attended by $10$ male students and $ 12$ female students. Homeroom teacher from the class provides six prizes to randomly selected students. Gifts that provided are one school bag, two novels, and three calculators. If the total students The number of male students who received prizes was equal to the total number of female students who received prizes. How many possible arrangements are there of the student who gets the prize? p5. It is known that $S =\{1945, 1946, 1947, ..., 2016, 2017\}$. If $A = \{a, b, c, d, e\}$ a subset of $S$ where $a + b + c + d + e$ is divisible by $5$, find the number of possible $A$'s.

p1. The parabola $y = ax^2 + bx$, $a < 0$, has a vertex $C$ and intersects the $x$-axis at different points $A$ and $B$. The line $y = ax$ intersects the parabola at different points $A$ and $D$. If the area of triangle $ABC$ is equal to $|ab|$ times the area of triangle $ABD$, find the value of $ b$ in terms of $a$ without use the absolute value sign. p2. It is known that $a$ is a prime number and $k$ is a positive integer. If $\sqrt{k^2-ak}$ is a positive integer, find the value of $k$ in terms of $a$. p3. There are five distinct points, $T_1$, $T_2$, $T_3$, $T_4$, and $T$ on a circle $\Omega$. Let $t_{ij}$ be the distance from the point $T$ to the line $T_iT_j$ or its extension. Prove that $\frac{t_{ij}}{t_{jk}}=\frac{TT_i}{TT_k}$ and $\frac{t_{12}}{t_{24}}=\frac{t_{13}}{t_{34}}$ p4. Given a $7$-digit positive integer sequence $a_1, a_2, a_3, ..., a_{2017}$ with $a_1 < a_2 < a_3 < ...<a_{2017}$. Each of these terms has constituent numbers in non-increasing order. Is known that $a_1 = 1000000$ and $a_{n+1}$ is the smallest possible number that is greater than $a_n$. As For example, we get $a_2 = 1100000$ and $a_3 = 1110000$. Determine $a_{2017}$. p5. At the oil refinery in the Duri area, pump-1 and pump-2 are available. Both pumps are used to fill the holding tank with volume $V$. The tank can be fully filled using pump-1 alone within four hours, or using pump-2 only in six hours. Initially both pumps are used simultaneously for $a$ hours. Then, charging continues using only pump-1 for $ b$ hours and continues again using only pump-2 for $c$ hours. If the operating cost of pump-1 is $15(a + b)$ thousand per hour and pump-2 operating cost is $4(a + c)$ thousand per hour, determine $ b$ and $c$ so that the operating costs of all pumps are minimum (express $b$ and $c$ in terms of $a$). Also determine the possible values of $a$.