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 […]
ABCD is a quadrilateral and P, and Q are mid-points of CD, and AB respectively. Let AP, DQ meet at X, and BP, CQ meet at Y . Prove that area of ADX + area of BCY = area of quadrilateral PXQY The number of ways in which three non-negative integers \( n_1, n_2, n_3 […]