In a convex quadrilateral $PQRS$, the areas of triangles $PQS$, $QRS$ and $PQR$ are in the ratio $3:4:1$. A line through $Q$ cuts $PR$ at $A$ and $RS$ 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 $cd=1$. Prove that there exists at least one positive integer $m$ such that $$ab\le m^2\le (a+c) (b+d). $$
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 $$x_1x_2\le x_2x_3\le x_3x_4\le x_4x_5\le x_5x_6\le x_6x_7\le x_7x_8.$$
In the figure, $BC$ is a diameter of the circle, where $BC=\sqrt{257}$, $BD=1$ and $DA=12$. Find the length of $EC$ and hence find the length of the altitude from $A$ to $BC$. [asy][asy] import cse5; size(200); pair O=(2, 0), B=(0, 0), C=(4, 0), A=(1, 3), D, E; D=MP("D",D(IP(D(CP(O,B)),D(MP("A",D(A),N)--MP("B",D(B),W)))),NW); E=MP("E",D(IP(CP(O,B),D(MP("C",D(C),NE)--A),1)),NE); D(B--C); [/asy][/asy]
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 $15$ questions. Let $O_1, O_2, \dots , O_9$ be the nine objective questions and $F_1, F_2, \dots , F_6$ be the six fill inthe blanks questions. Let $a_{ij}$ be the number of students who attemoted both questions $O_i$ and $F_j$. If the sum of all the $a_{ij}$ for $i=1, 2,\dots , 9$ and $j=1, 2,\dots , 6$ 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 $$x^4+y^4+z^4=2x^2y^2+2y^2z^2+2z^2x^2-63.$$
The perimeter of $\triangle ABC$ is $2$ and it's sides are $BC=a, CA=b,AB=c$. Prove that $$abc+\frac{1}{27}\ge ab+bc+ca-1\ge abc. $$
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 colorings possible.