IPM Indore 2023

Let a, b, c, d be positive integers such that a + b + c + d = 2023. If a : b = 2 : 5 and c : d = 5 : 2 then the maximum possible value of a + c is________.

In the xy-plane let A = (-2,0), B = (2,0). Define the set S as the collection of all points C on the circle $$x^{2} + y^{2} = 4$$ such that the area of the triangle ABC is an integer. The number of points in the set S is ___________.

The length of the line segment joining the two intersection points of the curves $$y = 4970 - |x|$$ and $$y = x^{2}$$ is_________.

Assume it is the beginning of the year today. Ankita will earn INR 10,000 at the end of the year, which she plans to invest in a bank deposit immediately at a fixed simple interest of 0.5% per annum. Her yearly income will increase by INR 10,000 every year, and the fixed simple interest offered by the bank on new deposits will also increase by 0.5% per annum every year. If Ankita continues to invest all her yearly income in new bank deposits at the end of each year, the total interest earned by her, in INR, in five years from today will be__________.

In a chess tournament, there are four groups, each containing an equal number of players. Each player plays
• against every other player belonging to one's own group exactly once;
• against each player belonging to one of the remaining three groups exactly twice;
• against each player belonging to one of the remaining two groups exactly three times; and
• against each player belonging to the remaining group exactly four times.
If there are more than 1000 matches being played in the tournament, the minimum possible number of players in each group is_______.

If $$\log_{cos  x} (\sin  x) + \log_{\sin  x}(cos  x) = 2$$, then the value of x is

A person standing at the center of an open ground first walks 32 meters towards the east, takes a right turn and walks 16 meters, takes another right turn and walks 8 meters, and so on. How far will the person be from the original starting point after an infinite number of such walks in this pattern?

The equation $$x^{2} + y^{2} − 2x − 4y + 5 = 0$$ represents

A goldsmith bought a large solid golden ball at INR 1,000,000 and melted it to make a certain number of solid spherical beads such that the radius of each bead was one-fifth of the radius of the original ball. Assume that the cost of making golden beads is negligible. If the goldsmith sold all the beads at 20% discount on the listed price and made a total profit of 20%, then the listed price of each golden bead, in INR, was

A helicopter flies along the sides of a square field of side length 100 kms. The first side is covered at a speed of 100 kmph, and for each subsequent side the speed is increased by 100 kmph till it covers all the sides. The average speed of the helicopter is