Let p and q be positive integers such that the equation $x^2$ -p

has two distinct positive integer roots, and the equation $x^2$ -q

also has two distinct positive integer roots. The value of p + q equals

Logarithm system with reciprocal relation. Let x > $y > 1$, xy = 64, and $\log_x y + \log_y x$ = 5 2. What is x -y?

Find the number of integer values of a for which $x^2$ -ax + 16 $x^2$ -6x + 8 ≥0 holds for every real $x$ < 2.

A function f is defined on all non - zero real numbers and satisfies $2f(x)$ + 3 f 1

x for every x ≠ 0. Find the value of f(2) + f 1.

For each natural number k, let ak = 2k. Find the smallest natural number m for which (a1)2($a^2$)3($a^3$)4 · · · (a15)16 < a16a17 · · · a15+m.

If x is a real number greater than 1 satisfying (log₂ x)2 -log₂

, then the product of all possible values of x equals.

A function f is defined on the positive integers and satisfies f(1) = 2,

for every n ≥1. The number of positive integers n with 1 ≤n ≤100 for which f(n) is divisible by 5 is:

Factorisation hidden inside a two-variable Diophantine equation Positive integers m and n, with m < n, satisfy

Find the minimum possible value of m + n.

Let x, y, z be powers of 2 greater than 1, not necessarily distinct. They satisfy $\log_x 64 + \log_y 64 + \log_z 64 = \frac{11}{2}$. If xyz is as small as possible, what is x + y + z?

A vessel is completely filled with a mixture of milk and water in the ratio 7: 3. From this vessel, 20 litres of the mixture is drawn out and replaced by an equal quantity of water. As a result, the ratio of milk to water in the vessel becomes 21: 29. What is the capacity (in litres) of the vessel?

A trader, while purchasing goods from his supplier, uses a balance whose pan reads “1000 g” when the actual weight placed is 1200g(he gains the extra weight). While selling those same goods to a customer, he uses a different balance whose pan reads “1000 g” when the actual weight given is 800 g. He pays his supplier and charges his customer at the same rate per kg (as displayed by the balance in each case). What is his overall profit percentage?

The average age of a certain group of people is 32 years. When a person aged 60 years joins the group, the average becomes 34 years. Subsequently, two more persons aged x years and 2x years respectively also join the group (in addition to the 60 - year - old), and the average of the resulting group of all members becomes 38 years. What is the value of x?

A shop sold a product in four months. The numbers of units sold in January, February, March and April were 80, 100, 120 and 150, respectively. The cost price was Rs. 320 per unit. A fixed marked price was used in all four months. Discounts of 25%, 15% and 10% were given in January, February and March, respectively, while no discount was given in April. If the total profit over the four months was Rs. 117950, then the marked price per unit, in rupees, was

A car was scheduled to cover a distance of 480 km at a constant speed. After travelling for 2 hours at the scheduled speed, it stopped for 30 minutes. It then increased its speed by 20% and reached the destination 10 minutes before the scheduled arrival time. What was the scheduled speed of the car, in km per hour?

A shopkeeper buys two articles X and Y. He sells X at a profit of 25% on its cost and sells Y at a loss of 20% on its cost. The cost price of Y is 40% more than the cost price of X. If the total selling price of the two articles is Rs. 2370, the cost price of X (in Rs.) is