Try this beautiful problem from TOMATO Objective no. 258 based on Real Roots of a Cubic Polynomial. Problem: Real Roots of a Cubic Polynomial Let a,b,c be distinct real numbers. Then the number of real solution of [latex](x-a)^3+(x-b)^3+(x-c)^3=0[/latex] is (A) 1 (B) 2 (C) 3 (D) depends on a,b,c Solution: Ans: (A) Let [latex]f(x)=(x-a)^3+(x-b)^3+(x-c)^3[/latex] [latex]=> f'(x)=3(x-a)^2+3(x-b)^2+3(x-c)^2=0[/latex] [latex]=> […]
Try this beautiful problem from TOMATO Objective no. 257 based on Roots of a Quintic Polynomial. Problem: Roots of a Quintic Polynomial The number of real roots of [latex] x^5+2x^3+x^2+2=0[/latex] is (A) 0 (B) 3 (C) 5 (D) 1 Solution: Answer: (D) [latex] x^5+2x^3+x^2+2=0[/latex] [latex] \implies x^3(x^2+2)+(x^2+2)=0[/latex] [latex] \implies (x^3+1)(x^2+2)=0[/latex] [latex] \implies (x+1)\bold{\underline{(x^2-x+1)(x^2+2)}}=0[/latex] The expression in underline doesn't have any […]
Problem: The number of terms in the expression of $latex [(a+3b)^2 (a-3b)^2]^2 $ A) 4; B) 5; C) 6; D) 7; Solution: $latex [(a+3b)^2 (a-3b)^2]^2 $ $latex = [\{(a+3b)(a-3b)\}^2]^2 $ $latex = \{ (a^2 -9b^2)^2\}^2 = (a^2 - 9b^2)^4 $ By Binomial Theorem, the given expression contains 5 terms (since $latex (x +y)^n $ has […]
Try this beautiful problem from TOMATO Objective no. 27 based on Closure of a set of even numbers. Problem: Closure of a set of even numbers S is the set whose elements are zero and all even integers, positive and negative. Consider the 5 operations- [1] addition; [2] subtraction; [3] multiplication; [4] division; and […]
Here is the post for the Regional Mathematics Olympiad (India) RMO Number Theory Problems. These are problems from previous year papers. (This is a work in progress. More problems will be added soon). RMO Number Theory Problems: Find all triples (p, q, r) of primes such that pq = r + 1 and 2(p 2 […]
Try this beautiful problem from TOMATO Objective no. 13 based on Calendar Problem. This problem is useful for BSc Maths and Stats Entrance Exams. Problem: June 10, 1979, was a SUNDAY. Then May 10, 1972, was a (A) Wednesday; (B) Friday; (C) Sunday; (D) Tuesday; Solution: In a (non-leap) year there are 365 days. $365 […]
Try this beautiful problem from TOMATO Subjective Problem no. 173 based on the Sum of Polynomials. Problem : Sum of polynomials Let [latex] {{P_1},{P_2},...{P_n}}[/latex] be polynomials in [latex] {x}[/latex], each having all integer coefficients, such that [latex] {{P_1}={{P_1}^{2}+{P_2}^{2}+...+{P_n}^{2}}}[/latex]. Assume that [latex] {P_1}[/latex] is not the zero polynomial. Show that [latex] {{P_1}=1}[/latex] and [latex] {{P_2}={P_3}=...={P_n}=0}[/latex] Solution : As [latex] […]
This problem is from the Test of Mathematics, TOMATO Subjective Problem no. 172 based on the Round Robin tournament. Problem : Suppose there are [latex] {k}[/latex] teams playing a round robin tournament; that is, each team plays against all the other teams and no game ends in a draw.Suppose the [latex] {i^{th}}[/latex] team loses [latex] […]