2019 NMTC Junior

Part of NMTC

Subcontests

(8)

1

Simple Geo

BC

is a diameter of the circle, where

BC=\sqrt{257}

BD=1

DA=12

. Find the length of

EC

and hence find the length of the altitude from

A

BC

. [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]

1

Coloring of disk

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.

1

Simple AM-GM

The perimeter of

\triangle ABC

2

and it's sides are

BC=a, CA=b,AB=c

abc+\frac{1}{27}\ge ab+bc+ca-1\ge abc.

1

Find solution

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.

1

How many students took the math contest?

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

15

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

F_j

. If the sum of all the

a_{ij}

i=1, 2,\dots , 9

j=1, 2,\dots , 6

972

, then find the number of students who took the test in the school.

1

Chain of inequalities

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.

1

There exists an integer

Given positive real numbers

a, b, c, d

cd=1

. Prove that there exists at least one positive integer

m

ab\le m^2\le (a+c) (b+d).

1

Midpoints of two sides passing through a point

In a convex quadrilateral

PQRS

, the areas of triangles

PQS

QRS

PQR

are in the ratio

3:4:1

. A line through

Q

PR

A

RS

B

PA:PR=RB:RS

A

is the midpoint of

PR

B

is the midpoint of

RS