NMTC Stage 2 BHASKARA Junior Level 2019

In a convex quadrilateral PQRS, the areas of triangles PQS, QRS and PQR are in the ratio $$3 \colon 4 \colon 1$$. A line through Q cuts PR at A and RS at B such that $$PA \colon PR = RB \colon RS$$. 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 $$ab \leq m^2 \leq (a + c)(b + d)$$.

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 $$x_1x_2 < x_2x_3 < x_3x_4 < x_4x_5 < x_5x_6 < x_6x_7 < x_7x_8$$.

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.

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 $$O_1, O_2, \ldots, O_9$$ be the nine objective type questions and let $$F_1, F_2, \ldots, F_6$$ be the six fill in the blanks questions. Let $$a_{ij}$$ be the number of students who attempted both questions $$O_i$$ and $$F_j$$. If the sum of all the $$a_{ij}$$, for $$i = 1, 2, 3, \ldots, 9$$ and $$j = 1, 2, 3, \ldots, 6$$, 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 $$x^4 + y^4 + z^4 = 2x^2y^2 + 2y^2z^2 + 2z^2x^2 - 63$$.

The perimeter of $$\triangle ABC$$ is 2 and its sides are $$BC = a$$, $$CA = b$$, $$AB = c$$. Prove that $$abc + \frac{1}{27} \geq ab + bc + ca - 1 \geq abc$$.

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 colourings possible.