online user counteronline user counter

MATHMAA

Only Search Your Site

16th KVJ Math Olympiad

16th KVJ Math Olympiad questions and answers are here. Please send your opinion on this.

1Q) Prove that for every prime P>7, \(P^{6}-1\) is divisible by 504.

2Q) a) Determine the smallest positive integer x, whose last digit is 6 and if we erase this 6 and put it in left most of the number so obtained, the number becomes 4x.

b) For any real numbers a and b, prove that:

    \(3a^{4}-4a^{3}b+b^{4} \geq 0 \)

3Q) If squares of the roots of \(x^{4}+bx^{2}+cx+d=0\) are \(\alpha ,\beta ,\gamma ,\delta\), then prove that:

\( 64 \alpha \beta \gamma \delta -[4\sum\alpha\beta -(\sum\alpha)^{2}]^{2} = 0 \).

4Q) a) Let a, b,c be the length of the sides of a triangle and r be the in-radius. Show that:

         r< \(\frac{a^{2}+b^{2}+c^{2}}{3(a+b+c)}\).

 b) A family consists of a grand father, 6 sons & daughters and 4 grand children. They are to be seated in a row for a dinner. The grand children wish to occupy the two seats at each end and the grand father refuses to have a grand child on either side of him. Determine the number of ways in which they can be seated for the dinner.

5Q) A semi-circle is drawn outwardly on chord AB of the circle with centre O and unit radius. The perpendicular from O to AB, meets the semi-circle on AB at C. Determine the measure of \(\angle AOB \) and length AB so that OC has maximum length.

16th KVJ Math Olympiad

6Q) If \( cos \alpha + cos \beta + cos \gamma = sin \alpha + sin \beta + sin \gamma = 0 \), Prove that

 \( cos (2\alpha) + cos (2\beta) + cos (2\gamma)\) =  \( sin (2\alpha) + sin (2\beta) + sin (2\gamma) =0\).

7Q) A point 'A' is randomly chosen in a square of side length 1 unit. Find the probability that the distance from A to the centre of the square does not exceed x.

8Q) (a) Two parallel chords in a circle have lengths 10cm an 14cm and distance between them is 6 cm. If a chord parallel to these chords and midway between them is length √a, find the value of a.

(b) The line joining two points A(2, 0) and B(3, 1) is rotated about point A in anticlockwise direction through an angle of \( 15^{\circ}\). Find the equation of the line in the new position. IF B goes to C in new position, find the coordinates of C.

9Q) Find the number of numbers \( \leq 10^{8} \) which are neither perfect squares, nor perfect cubes, nor perfect fifth powers .

10Q) Let PQRS be a rectangle such that PQ= a and QR =b. Suppose \(r_{1}\) is the radius of the circle passing through P and Q and touching RS and \(r_{2}\) is the radius of the circle passing through Q and R and touching PS. Show that :

\( 5(a+b) \leq 8(r_{1}+r_{2} )\).

For Solutions of  16th KVJ Math Olympiad click SOLUTIONS link below.


<<KV JMO <<                                                  << Solutions<<




SHARE YOU ENORMOUS EFFORT AND SMART EXAMPLES HERE

!! NEED MORE HELP !!

SBI! Case Studies