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 […]
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 + […]
Try this beautiful problem from Number Theory based on largest possible value from AMC-10A, 2004. You may use sequential hints to solve the problem.
Try this beautiful problem from AMC 10A, 2004 based on Mensuration: Cylinder. You may use sequential hints to solve the problem.
Try this beautiful problem from AMC-10A, 2004 based on Triangle. You may use sequential hints to solve the problem.
This problem is an easy application in calculus using the basic ideas of curve sketching. This is the problem 1 from ISI MStat 2019 PSB.
This is a problem from ISI MStat Examination,2018.
It involves construction of a most powerful test of size alpha using Neyman Pearson Lemma. The aim is to find its critical region in terms of quantiles of a standard distribution.
This problem is a cute application of joint distribution and conditional probability. This is the problem 5 from ISI MStat 2018 PSB.
Try this TOMATO problem from I.S.I. B.Stat Entrance Objective Problem based on Binomial Expression. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Right angled triangle.
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 Length and Inequalities.