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NMTC 2017 Stage II - BHASKARA (Class 9, 10) - Problems and Solutions

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Problem 1

(a) Find all prime numbers $p$ such that $4 p^2+1$ and $6 p^2+1$ are also primes.
(b) Determine real numbers $x, y, z, u$ such that
$$
\begin{aligned}
& x y z+x y+y z+z x+x+y+z=7 \\
& y z u+y z+z u+u y+y+z+u=9 \\
& z u x+z u+u x+x z+z+u+x=9 \\
& u x y+u x+x y+y u+u+x+y=9
\end{aligned}
$$

Problem 2

If $x, y, z, p, q, r$ are distinct real numbers such that
$$
\begin{aligned}
& \frac{1}{x+p}+\frac{1}{y+p}+\frac{1}{z+p}=\frac{1}{p} \\
& \frac{1}{x+q}+\frac{1}{y+q}+\frac{1}{z+q}=\frac{1}{q} \\
& \frac{1}{x+r}+\frac{1}{y+r}+\frac{1}{z+r}=\frac{1}{r}
\end{aligned}
$$
find the numerical value of $\left(\frac{1}{p}+\frac{1}{q}+\frac{1}{r}\right)$.

Problem 3

$\mathrm{ADC}$ and $\mathrm{ABC}$ are triangles such that $\mathrm{AD}=\mathrm{DC}$ and $\mathrm{CA}=\mathrm{AB}$. If $\angle \mathrm{CAB}=20^{\circ}$ and $\angle \mathrm{ADC}=100^{\circ}$, without using Trigonometry, prove that $\mathrm{AB}=\mathrm{BC}+\mathrm{CD}$.

Problem 4

(a) a, b, c, d are positive real numbers such that $a b c d=1$. Prove that $$\frac{1+a b}{1+a}+\frac{1+b c}{1+b}+\frac{1+c d}{1+c}+\frac{1+d a}{1+d} \geq 4.$$
(b) In a scalene triangle $\mathrm{ABC}, \angle \mathrm{BAC}=120^{\circ}$. The bisectors of the angles $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ meet the opposite sides in $\mathrm{P}, \mathrm{Q}$ and $\mathrm{R}$ respectively. Prove that the circle on $\mathrm{QR}$ as diameter passes through the point $P$.

Problem 5

(a) Prove that $x^4+3 x^3+6 x^2+9 x+12$ cannot be expressed as a product of two polynomials of degree 2 with integer coefficients.
(b) $2 n+1$ segments are marked on a line. Each of these segments intersects at least $n$ other segments. Prove that one of these segments intersects all other segments.

Problem 6

If $a, b, c, d$ are positive real numbers such that $a^2+b^2=c^2+d^2$ and $a^2+d^2-a d=b^2+c^2+bc$, find the value $$\frac{a b+c d}{a d+b c}$$

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