How many solutions (x, y, z) of the equation x + y² + z³ = 50 exist, where x, y and z are positive integers?

Rajan is a fruit seller. On any day, he sells only one kind of fruit. On the first day, he buys 9 kg of blueberries. On the second day, he buys 22 kg of kiwis. On the third day, he buys 50 kg of peaches. The per kg purchase price of each fruit is an integer. Further, on each of t…

A triangular plot is such that two of its sides, of lengths 90m (meter) and 60m, are perpendicular to each other. There is a housing complex in a rectangular region within the plot. The area of the rectangular region is 4/9th of the area of the triangular plot. Additionally, two…

Consider two circles, each having radius of 5cm (centimeters), touching each other at a point P. A direct tangent QR is drawn touching one circle at a point Q and the other circle at a point R. Inside the region PQR inscribed by the two circles and the tangent, a square ABCD is i…

During Durga Puja, for the purpose of lighting, one puja pandal in Kolkata used many identical structures made of wooden sticks. The design of the structures was as follows: each structure was constructed with the help of six wooden sticks by combining an isosceles triangular str…

Let a₁ < a₂ < .... < aₙ be the list of all prime numbers less than 25. Define Xᵢ = bᵢ/aᵢ, where bᵢ is the sum of all aₖ where k ranges from 1 to n, k ≠ i. Let B be the set of all integer-valued Xᵢ. What is the Smallest element of B?

A park has two gates, Gate 1 and Gate 2. These two gates are connected via two alternate paths. If one takes the first path from Gate 1, they need to walk 80m (meters) towards east, then 80m towards south, and finally 20m towards west to arrive at Gate 2. The second path is a sem…

Let f : R² → R be a real-valued function defined as f(0, y) = y + 1 and f(x + 1, y) = f(x, f(x, y)) + x. What is the value of f(2, 2)?

Ronny uses a 5-digit key for a combination lock, where 5 digits need to be entered in a fixed sequence. While he remembers that the 5 digits are 9, 8, 7, 5 and 4, he has forgotten the sequence he uses. He also remembers that the sum of the first three digits is a multiple of 3, a…