TITA} Let $f:\mathbb{R}\to\mathbb{R}$ satisfy $f(x)+2 f(1-x)=x^{2}+3x$ for every real $x$. Find the value of $3 f(5)$.

For positive real numbers x and y,

What is the minimum possible value of $x^2$ + 6x y + 3 + $y^2$ + 6y x + 3?

Let

, defined for x ≠ -1. Denote f(n)(x) the n - fold composition f(f(…f(x)…)) with f applied n times. The value of f(2025)(2) is:

Two cyclists P and Q start simultaneously from towns A and B respectively and ride towards each other on a straight road at constant speeds. They first meet at a point 7.5 km from A. After meeting, each continues to the other town, turns back immediately on reaching it, and they meet for the second time at a point 4 km from B. The distance (in km) between A and B is:

MCQ} A and B together can finish a project in $20$ days. B alone takes $45$ days to finish the project. They start the project together, but A leaves after some days; B alone then takes a further $21.6$ days to complete the remaining work. The number of days A alone would take to complete the project is:

MCQ} In a class, the average mark of the boys is $68$ and that of the girls is $80$. The overall class average is $74$. A new group of students, consisting only of boys with average mark $62$, joins the class, and after they join, the overall class average becomes $71$. The ratio of the original number of boys in the class to the number of boys in the new group is:

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Algebra

Two trains P and Q start simultaneously from stations A and B respectively and move towards each other on the same straight track. After they pass each other, P takes 49 minutes to reach B, and Q takes 64 minutes to reach A. If P is faster than Q by 7.2 km/h, then the distance between A and B (in km) is

A trader marks his goods $50$ above cost. He sells three-fifths of his stock at a discount of $20$ on the marked price, one-fourth of his stock at a discount of $40$ on the marked price, and the remainder at cost price. His overall profit percentage on the entire stock is:

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The sum of the first n terms of an arithmetic progression is denoted by Sn. It is given that S5 = 40 and S10 = 130. The value of S15 -S5 equals

A trader buys two articles A and B for a total cost of Rs. 8100. Article A is marked 50% above its cost price and article B is marked 25% above its cost price. The trader gives the same percentage discount on both marked prices. His total profit is Rs. 600, and the selling price of B is Rs. 1500 more than the selling price of A. What is the cost price of B?

In a batch of 50 students, the average score of the boys is 78 and the average score of the girls is 72. Four boys and six girls are shifted to another section. The average score of these 10 shifted students is 76.2, and the average score of the shifted girls exceeds that of the shifted boys by 12. If the average score of the remaining 40 students is 75, how many boys were originally in the batch?

Two trains start simultaneously from stations P and Q towards each other and meet after 40 minutes. Before this meeting, the train from Q covers 20 km more than the train from P. If the train from P starts 10 minutes late while the train from Q starts on time, the meeting point is 6 km nearer to P than the original meeting point. Find the distance, in km, between P and Q.

A sum of Rs. 15000 is divided into two parts and lent for 2 years at simple interest. One part is lent at r% per annum and the other at (r + 5)% per annum. The total interest earned is Rs. 4200. If the two parts had been interchanged between the two rates, the total interest would have been Rs. 3900. What is the value of r?

Two circles of radii 4 cm and 9 cm respectively are externally tangent to each other, and both are tangent to a common straight line on the same side of the line. A third circle is drawn that is tangent externally to both of these circles and is also tangent to the same straight line (lying in the curvilinear region enclosed by the two circles and the line). The radius of this third circle, in cm, is:

In triangle ABC, AB = 13, AC = 14, and BC = 15. Point D lies on AB such that AD: DB = 1: 2, and point E lies on AC such that AE: EC = 1: 2. Lines BE and CD intersect at P. Find the area of quadrilateral ADPE.