6. Find all positive integers n such that $latex (3^{2n} + 3 n^2 + 7 )$ is a perfect square. Solution: We use the fact that between square of two consecutive numbers there exist no perfect square. That is between $(k^2 )$ and $((k+1)^2 )$ there is no square. Note that $(3^{2n} = (9^n)^2 )$ […]
5. Let ABC be a triangle. Let D, E be points on the segment BC such that BD = DE = EC. Let F be the mid point of AC. Let BF intersect AD in P and AE in Q respectively. Determine the ratio of triangle APQ to that of the quadrilateral PDEQ. Solution: Applying Menelaus' […]
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 […]
Try this beautiful problem from Geometry: Area of Trapezoid from AMC-10A, 2002. You may use sequential hints to solve the problem.
Try this beautiful problem from AMC-10A, 2004 based on ratio of two triangles.You may use sequential hints to solve the problem.
Try this beautiful problem from Algebra on Quadratic equation from AMC-10A, 2002. You may use sequential hints to solve the problem.
Try this TOMATO problem from I.S.I. B.Stat Entrance Objective Problem based on Arbitrary Arrangement. You may use sequential hints.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Remainders and Functions.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1992 based on Rational Numbers.
This problem is a simple application of the sequential definition of continuity from ISI MStat 2015 PSB Problem 1.
This is a problem from ISI MStat Examination,2019. This tests one's familiarity with the simple and multiple linear regression model and estimation of model parameters and is based on the Invariant Regression Coefficient.
This is a problem from the ISI MStat Entrance Examination, 2019. This primarily tests one's familiarity with size, power of a test and whether he/she is able to condition an event properly.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Points of Equilateral triangle.