4. Let X = {1, 2, 3, ... , 10}. Find the number of pairs {A, B} such that A ⊆ X, B ⊆ X, A ≠ B and A∩B = {5, 7, 8}. Solution: First we put 5, 7, 8 in each of A and B. We are left out with 7 elements of […]
4. Let X = {1, 2, 3, ... , 10}. Find the number of pairs {A, B} such that A ⊆ X, B ⊆ X, A ≠ B and A∩B = {5, 7, 8}. Solution: First we put 5, 7, 8 in each of A and B. We are left out with 7 elements of […]
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 […]