Given any integer $n \ge 2 $ , we can always find an integer m such that each of the n-1 consecutive integers m + 2, m + 3,..., m + n are composite. True Discussion: Take m=n!. Then the consecutive integers n! + 2 , n! + 3 , ... n! + n are […]
Given any integer $n \ge 2 $ , we can always find an integer m such that each of the n-1 consecutive integers m + 2, m + 3,..., m + n are composite. True Discussion: Take m=n!. Then the consecutive integers n! + 2 , n! + 3 , ... n! + n are […]
Let's discuss a problem based on Least Value of a Sum of Complex Numbers. Try to solve it yourself before reading the solution. Problem: Least Value of a Sum of Complex Numbers If $ z_1 , z_2 , z_3 , z_4 \in \mathbb{C} $ satisfy $ z_1 + z_2 + z_3 + z_4 = 0 […]
Problem: Continuous Functions and open sets Suppose f be a continuous function from X to Y (where X and Y are domain and range). If Y is a closed set (closed interval if we working in $ R^1 $ ) then can we say that the domain is also closed? There is a simple counter […]
(This is a series of discussions on Homothety. It is largely derived from the Math Olympiad Classroom Discussion in Cheenta - cheenta.com) Homothety is a geometric transformation. It has a couple of synonyms: dilation and central similarity. A geometric transformation is a function. It can be thought of as a machine which takes in a […]
A lamp is placed on the ground 100 feet away from a wall. A man six feet tall is walking at a speed of 10 ft/sec from the lamp to the nearest point on the wall. When he is midway between the lamp and the wall, the rate of change in the length of his shadow is (in ft/ sec)?
Problem: Let ABCDE be a regular pentagon inscribed in a circle. P be any point in the minor arc AE. Prove that PA + PC + PE = PB + PD Proof: Suppose length of each side is 's' and each diagonal is 'x'. Apply Ptolemy's Theorem in PABC. We have PA . s + […]
In this post, here are problems from Regional Mathematics Olympiad, RMO 2011 Re-Test Paper. Let ABC be an acute angled scalene triangle with circumcenter O and 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 […]
Let $ABC$ be a triangle. Let $D, E, F$ be points respectively on the segments $BC, CA, AB$ such that $AD, BE, CF$ concur at the point $K$. Suppose $\frac{BD}{DC} = \frac{BF}{FA}$ and $∠ADB = ∠AFC$. Prove that $∠ ABE = ∠ CAD$. Let $ (a_1a_2a_3.....a_{2011}) $ be a permutation (that is arrangement) of the […]
In this post, there are problems from Regional Mathematics Olympiad, RMO 2010. Try out these problems. Let $ABCDEF$ be a convex hexagon in which the diagonals $AD, BE, CF$ are concurrent at $O$. Suppose the area of triangle $OAF$ is the geometric mean of those of $OAB$ and $OEF$; and the area of the triangle […]
In this post, there are problems from Regional Mathematics Olympiad, RMO 2009. Try out these problems. Let $ABC$ be a triangle in which $AB = AC$ and let $I$ be its in-centre. Suppose $BC = AB + AI$. Find $∠BAC$.Discussion Show that there is no integer a such that $ a^2-3a-19 $ is divisible by […]