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Explore the Back-StoryIn a convex quadrilateral PQRS, the areas of triangles $P Q S, Q R S$ and $P Q R$ are in the ratio $3: 4: 1$. A line through $Q$ cuts $P R$ at $A$ and $R S$ at $B$ such that $PA : PR=RB: RS$. Prove that $A$ is the midpoint of $PR$ and $B$ is the midpoint of $RS$.

Given positive real numbers $a, b, c, d$ such that $c d=1$. Prove that there exist at least one positive integer $m$ such that $a b \leq m^2 \leq(a+c)(b+d)$.

Find the number of permutations $x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8$ of the integers $-3,-2,1,0,1,2,3,4$ that satisfy the chain of inequalities.

$$

\mathrm{x}_1 \mathrm{x}_2 \leq \mathrm{x}_2 \mathrm{X}_3 \leq \mathrm{x}_3 \mathrm{X}_4 \leq \mathrm{x}_4 \mathrm{X}_5 \leq \mathrm{x}_5 \mathrm{X}_6 \leq \mathrm{X}_6 \mathrm{x}_7 \leq \mathrm{x}_7 \mathrm{X}_8

$$

In the figure, $\mathrm{BC}$ is a diameter of the circle, where $\mathrm{BC}=\sqrt{257}, \mathrm{BD}=1$, and $\mathrm{DA}=12$. Find the length of $\mathrm{EC}$ and hence find the length of the altitude from $\mathrm{A}$ to $\mathrm{BC}$.

A math contest consists of $9$ objective type questions and $6$ fill in the blanks questions. From a school some number of students took the test and it was noticed that all students had attempted exactly $14$ out of the $15$ questions. Let $O_1, O_2$, $F_6$ be the six fill in the blanks questions. Let $a_{i j}$ be the number of students who attempted both questions $\mathrm{O}_i$ and $F_j$. If the sum of all the $a_{ij}, i=1,2,3, \ldots \ldots, 9$ and $j=1,2,3, \ldots \ldots ., 6$ is $972$ , then find the number of students who took the test in the school.

Find all positive integer triples $(x, y, z)$ that satisfy the equation $x^4+y^4+z^4=2 x^2 y^2+2 y^2 z^2+2 z^2 x^2-63$

The perimeter of $\triangle A B C$ is 2 and its sides are $B C=a, C A=b, A B=c$. Prove that $a b c+\frac{1}{27} \geq a b+b c+c a-1 \geq a b c$.

A circular disc is divided into $12$ equal sectors and one of $6$ different colours is used to colour each sector. No two adjacent sectors can have the same colour. Find the number of such distinct colourings possible.

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