2017 NMTC Junior

Part of NMTC

Subcontests

(6)

1

Algebra problem

a,b,c,d

are positive reals such that

a^2+b^2=c^2+d^2

a^2+d^2-ad=b^2+c^2+bc

, find the value of

\frac{ab+cd}{ad+bc}

1

Eisenstein? And Combi

x^4+3x^3+6x^2+9x+12

cannot be expressed as product of two polynomials of degree 2 with integers coefficients.
(b)

2n+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.

1

Inequality and Geometry

a,b,c,d

are positive reals such that

abcd=1

\sum_{cyc} \frac{1+ab}{1+a}\geq 4.

(b)In a scalene triangle

ABC

\angle BAC =120^{\circ}

. The bisectors of angles

A,B,C

meets the opposite sides in

P,Q,R

respectively. Prove that the circle on

QR

as diameter passes through the point

P

1

Geometry without trigonometry

ADC

ABC

are triangles such that

AD=DC

AC=AB

\angle CAB=20^{\circ}

\angle ADC =100^{\circ}

, without using Trigonometry, prove that

AB=BC+CD

1

Polynomial?

x,y,z,p,q,r

are real numbers such that

\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}.

Find the numerical value of

\frac{1}{p}+\frac{1}{q}+\frac{1}{r}

1

NT and algebra

(a) Find all prime numbers

p

4p^2+1

6p^2+1

are also primes.
(b)Find real numbers

x,y,z,u

xyz+xy+yz+zx+x+y+z=7

yzu+yz+zu+uy+y+z+u=10

zux+zu+ux+xz+z+u+x=10

uxy+ux+xy+yu+u+x+y=10