These are the questions of the NMTC final stage.
Hope you people will help me and others by showing how to solve these questions. Different and innovative ways of solving the questions are appreciated:)
a) 28 integers are chosen from the interval [104, 208]. Show that there exists 2 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 of AB. Prove the S is the orthocentre of the triangle APB.
a) a,b,c are distinct real numbers such that . Find the numerical value of abc.
find [a], where [a] denotes the integer part of a.
The arithmetic mean of a number of pair wise distinct prime numbers is 27. Determine the biggest prime among them.
65 bugs are placed at different squares of a 9*9 square board. Abug in each moves to a horizontal or vertical adjacent square. No bug makes 2 horizontal or vertical moves in succession. Show that after some moves, there will be at least 2 bugs in the same square.
f(x) is a fifth degree polynomial. It is given that f(x)+1 is divisible by and f(x)-1 is divisible by . Find f(x).
ABC and DBC are 2 equilateral triangles on the same base BC. A point P is taken on the circle with centre D, radius BD. Show that PA, PB, PC are the sides of a right triangle.
a,b,c are real numbers such that a+b+c=0 and . Prove that . When does the inequality hold?
Please reshare this note so that it can reach to the experienced and other brilliantians who will help us :)