Quant Boosters  Hemant Yadav  Set 1

If h1, h2, h3 are the heights corresponding to the sides a, b, c, then
a * h1 = b * h2 = c * h3 = 2A (where A is area of the triangle)
=> a = 2A/h1, b = 2A/h2 and c = 2A/h3
Now, a + b > c
=> 2A/h1 + 2A/h2 > 2A/h3
=> 1/h1 + 1/h2 > 1/h3
=> 1/2 + 1/3 > 1/h3
=> h3 > 6/5Also, b + c > a
=> 2A/h2 + 2A/h3 > 2A/h1
=> 1/h2 + 1/h3 > 1/h1
=> 1/3 + 1/h3 > 1/2
=> h3 < 6So, 6/5 < h3 < 6
=> Integral values that h3 can take are 2, 3, 4, 5

minimum of x+x17...+x440
median of 0,17,40,85,173,440 : (40+85)/2 = 62.5
put x=62.5..so 641
[swetabh kumar]

for 1st meet: ad/(a+b)= 400
distance by A til 2nd meet: a/(a+b) * 3d = d + 300
so 1200 = d + 300, d = 900
[swetabh kumar]

ab  a  b = 35
=> (a  1)(b  1) = 36
(a, b) = (2, 36), (3, 19), (4, 13), (5, 10), (7, 7)
In last two cases a and b are not coprime

case 1: when there is no power of 2
so 3 * 3 * 3=27 cases
case 2: when there is only 1(2) in it
one of the dice will have 2 or 6... and other two dices will have odd numbers.. so
2 * 3 * 3 * 3c1=54
case 3:
when there are exactly two (2's)
two dices will have (2,6) and one dice will have (1,3,5)
so 2 * 2 * 3 * 3c1=36
part2:
exactly one dice will have 4 in it.. other two will have (1,3,5)
so 3 * 3 * 3c1=27adding all 144 cases which will not have a multiple of 8.. remaining 72..
so 72/216=1/3[jasneet dua]

17 * 20 * 23
= (203) * 20(20+3)
= 20 * (20²  3²)
= 8000  180
(90  1)² = 8100  2 * 90 + 1 = 8000  180 + 101 is a perfect square
=> 101 is least such number

h(k1 + k2) = 1001
1001 = 7 * 11 * 13
Possible values of h : 2 * 2 * 2 = 8
But if h = 1001 ; Sum = 1 => Not possible.
Hence 8  1 = 7 possible values
[Anubhav Sehgal]

d = x
c = d + y = x + y
b = c + z = x + y + z
a = b + w = w + z + y + x, where x, y, z, w are natural numbersAdd them to get 4x + 3y + 2z + w = 24
x, y, z, w are natural numbers
=> 4x' + 3y' + 2z' + w' = 14
(8 + 6 + 5 + 3 + 2) + (6 + 4 + 3 + 1) + (4 + 2 + 1) + (2) = 47
So, 24 * 47 = 1128 solutions

Total number of 3 digit numbers = 900
3digit numbers having consecutive digits in first 2 places = (2*9  1)*10 = 170 ..as we can choose consecutive digits in 9 ways, but 0, is not allowed at first place)
3digit numbers having consecutive digits in last 2 places = (2*9)*9 = 162
But above 2 cases have some number common. When 0 is middle digit, then we will have 1 common term(101), 1 as middle digit  2 common terms (210, 212), 2 as middle digit  4 common terms, when 3, 4, 5, 6, 7, 8 are middle digit then 4 common terms for each. For 9  one common term
SO total number of 3digit number such that no adjacent digits are consecutive = 900  170  162 + 32 = 600

1/x + 1/y = 1/n
=> xy = nx + ny
=> (x  n)(y  n) = n²Now, n² must have 5 factors to get 5 ordered pairs
=> n must be of form p², where p is prime numberHence all primes less than 20