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 […]
Let's discuss a problem based on Integer Sided Obtuse angled triangles with Perimeter. Find the number of integer-sided isosceles obtuse-angled triangles with perimeter 2008. (Indian RMO 2008) Discussion: Let the three sides be a, a and b. Hence 2a + b = 2008 ... (i) Using the triangular inequality we have 2a > b ...(ii) […]
In this post, there are problems from Regional Mathematics Olympiad, RMO 2008. Try out these problems. Let $ABC$ be an acute-angled triangle, let $D$, $F$ be the mid-points of $BC, AB$ respectively. Let the perpendicular from $F$ to $AC$ and the perpendicular at $B$ to $BC$ meet in $N$. Prove that $ND$ is equal to […]
A square sheet of paper ABCD is so folded that B falls on the mid-point of M of CD. Prove that the crease will divide BC in the ratio 5:3. Discussion: Assuming the side of the square is 's'. Let a part of the crease be 'x' (hence the remaining part is 's-x'). We apply […]