Mathematics Times – July 2019

3.Suppose

(c)  ˆ

(d) may or may not be equal to ˆ

a a a 1 , , , , 2 3 a 2012 are integers

arranged on a circle. Each number is equal to

the average of its adjacent numbers. If the

sum of all even indexed numbers is 3018, what

is the sum all numbers? [2012]

(a) 0 (b) 1509 (c) 3018 (d) 6036

Miscellaneous

1. Let

2.Let x,y,z be positive integers such that HCF

1

a ii i for i1,2,.......,20. put

 1 2 3 20 

1

......

20

p a a a   a

and

1 2 20

1 1 1 1

......

20

q

a a a

 

     

 

.

Then

(a)

22

0,

21

p

q

  

 

 

(b)

22 2 22 

,

21 21

p p

q

   

 

 

(c)

2 22  (22 )

,

21 7

p p

q

   

 

 

(d)

22 4 22 

,

7 21

p p

q

   

 

 

x y z, , 1 and x y^2  ^22 z^2. Which of the

following statements are true? [2017]

i. 4 divides x or 4 divides y.

ii. 3 divides x y or 3 divides x y.

iii. 5 divides z x y^2 ^2 
(a) i and ii only (b) ii and iii only
(c) ii only (d) iii only
3.The number of noncongruent integer-sided
triangles whose sides belong to the set

{10, 11, 12,......,22} is [2016]

(a) 283 (b) 446 (c) 448 (d) 449

4.Suppose we have to cover the xy-palne with

identical tiles such that no two tiles overlap

and no gap is left between the tiles. Suppose

that we can choose tiles of the following

shapes; equilateral triangle, square, regular

pentagon, regular hexagon. Then the tiling

can be done with tiles of [2016]

(a) all four shapes

(b) exactly three of the four shapes

(c) exactly two of the four shapes

(d) exactly one of the four shapes

5.The number of ordered pairs (x,y) of real

numbers that satisfy the simultaneous

equations x y x y   ^2212 is [2015]

(a) 0 (b) 1 (c) 2 (d) 4

6.The largest perfect square that divides

2014 2013 2012 2011 .... 2 1^3 ^3 ^3 ^3   ^33

is [2015]

(a) 12 (b) 22 (c) 10072 (d)

7.Suppose a, b, c are postivie integers such that

20142

2 4 8 328a  b c. Then

a b c2 3

abc

 

is

equal to [2015]

(a)

8.Let P be a closed polygon with 10 sides and

10 vertices (assume that the sides do not

interesect except at the vertices). Let k be

the number of interior angles of P that are

greater than

1

2

(b)

5

8

(c)

17

24

(d)

5

6

1800. The maximum possible

value of k is [2013]

(a) 3 (b) 5 (c) 7 (d) 9

9. In the real number system, the equation

x       3 4 1x x 8 6 1 1x has

[2012]

(a) No solution

(b) Exactly two distinct solutions

(c) Exactly four distinct solutions

(d) Infinitely may solutions

3. 
   

[2012]

Miscellaneous

1. 
   

2. 
   

[2017]

3. 
   

[2016]

4.

[2016]

5.

[2015]

6.

[2015]

7.

[2015]

8.

[2013]

9.

[2012]