NMTC 2015 Stage II - BHASKARA (Class 9, 10) - Problems and Solutions

Problem 1

a) $28$ integers are chosen from the interval $[104,208]$. Show that there exit two of them having a common prime divisor.
b) $AB$ is a line segment .$C$ is a point on $AB$. $ACPQ$ and $CBRS$ are squares drawn on the same side $AB$, Prove the $S$ is the orthocentre of the triangle $APB$.

Problem 2

a) $a,b,c$ are distinct real numbers such that $a^3=3(b^2+c^2)-25$, $b^3=3(c^2+a^2)-25$, $c^3=3(a^2+b^2)-25.$ Find the numerical value of $abc$.
b) $$a=1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\ldots \ldots \ldots \ldots .+\frac{1}{2015^2}$$
find $[a]$, where $[a]$ denotes the integer part of $a$.

Problem 3

The arithmetic mean of a number of pair wise distinct prime numbers is $27$ . Determine the biggest prime among them.

Problem 4

$65$ bugs are placed at different squares of a $9 \times 9$ square board. A bug in each moves to a horizontal or vertical adjacent square. No bug makes two horizontal or two vertical moves in succession. Show that after some moves, there will be at least two bugs in the same square.

Problem 5

$f(x)$ is a fifth degree polynomial. It is given that $f(x)+1$ in divisible by $(x-1)^3$ and $f(x)-1$ is divisible by $(x+1)^3$. Find $f(x)$.

Problem 6

$\mathrm{ABC}$ and $\mathrm{DBC}$ are two equilateral triangles on the same base $\mathrm{BC}$ . $\mathrm{A}$ point $\mathrm{P}$ is taken on the circle with centre $\mathrm{D}$, radius $\mathrm{BD}$. Show that $\mathrm{PA}, \mathrm{PB}, \mathrm{PC}$ are the sides of a right triangle.

Problem 7

$a,b,c$ are real numbers such that $a+b+c=0$ and $a^2+b^2+c^2=1$. Prove that $a^{2}b^{2}c^{2}\leq \frac{1}{54}$. When does the equality hold?