3. Let a and b are positive real numbers such that a+b = 1. Prove that \( (a^a b^b + a^b b^a \le 1)\) Solution: We use the weighted A.M.-G.M. inequality which states that: \( \frac {w_1 a_1 + w_2 a_2 }{w_1 + w_2} \ge ({a_1}^{w_1} {a_2}^{w_2})^{\frac{1}{w_1 + w_2}} \) First we put \( w_1 […]
2. Let a, b, c be positive integers such that a divides $ (b^5)$ , b divides $(c^5)$ and c divides $ (a^5)$. Prove that abc divides $((a+b+c)^{31})$. Solution: A general term of the expansion of $((a+b+c)^{31})$ is $(\frac {31!}{p!q!r!} a^p b^q c^r)$ where p+q+r = 31 (by multinomial theorem; this may reasoned as following: […]
1. Let ABCD be a unit square. Draw a quadrant of a circle with A as the center and B, D as the end points of the arc. Similarly draw a quadrant of a circle with B as the center and A, C as the end points of the arc. Inscribe a circle Γ touching the […]
Given a triangle ABC, let P and Q be the points on the segments AB and AC, respectively such that AP = AQ. Let S and R be distinct points on segment BC such that S lies between B and R, ∠BPS = ∠PRS, and ∠CQR = ∠QSR. Prove that P, Q, R and S […]
Let ABC be an acute angled scalene triangle with circumcenter O orthocenter H. If M is the midpoint of BC, then show that AO and HM intersect at the circumcircle of ABC. Let n be a positive integer such that 2n + 1 and 3n + 1 are both perfect squares. Show that 5n + […]
1. Let ABC be a triangle. Let D, E, F be points on the segments BC, CA and AB such that AD, BE and CA concur at K. Suppose $latex (\frac{BD}{DC} = \frac{BF}{FA})$ and ∠ADB = ∠AFC. Prove that ∠ABE = ∠CAD. Solution: Diagram Given: ABC be any triangle. AD, BE and CF are drawn […]
Try this TOMATO problem from I.S.I. B.Stat Entrance Objective Problem based on Perfect square and Positive Integer. You may use sequential hints.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Complex roots and equations.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Length and Inequalities.
Try this beautiful problem from the Pre-RMO, 2017 based on Trigonometry & natural numbers. You may use sequential hints to solve the problem.
Try this beautiful problem from TOMATO useful for ISI B.Stat Entrance from Integer based on divisibility. You may use sequential hints.
Try this beautiful problem from Probability: positive factors AMC-10A, 2003. You may use sequential hints to solve the problem
Try this beautiful problem from AMC 10A, 2007 based on Numbers on cube. You may use sequential hints to solve the problem.
Try this beautiful problem from TOMATO useful for ISI B.Stat Entrance based on Pairs of Positive Integer. You may use sequential hints.
Try this beautiful problem from TOMATO useful for ISI B.Stat Entrance based on Quadratic equation You may use sequential hints.
This problem is an application of the non negative integer solution and the symmetry argument. This is from ISI MStat 2015 PSB Problem 4.