My Math Forum > Math Not-so-tough questions for experts, not-so-simple for me

 Math General Math Forum - For general math related discussion and news

 October 6th, 2016, 08:57 AM #1 Member     Joined: Sep 2013 From: New Delhi Posts: 38 Thanks: 0 Math Focus: Calculus Not-so-tough questions for experts, not-so-simple for me Problem 1. Let p, q, r be integers such that $\displaystyle \frac {pq}{r} + \frac {qr}{p} +\frac {rp}{q}$ is an integer. Prove that each of the numbers $\displaystyle \frac {pq}{r}, \frac {qr}{p}, \frac {rp}{q}$ is an integer. Problem 2 . Two circles $\displaystyle \Gamma_1$ and $\displaystyle \Gamma_2$ of radii $\displaystyle r_1$ and $\displaystyle r_2$ respectively touch each other externally. The points A, B on $\displaystyle \Gamma_1$ and C, D on $\displaystyle \Gamma_2$ are such that AD and BC are common external tangents to the two circles . Prove that a circle can be inscribed in the quadrilateral ABCD and determine the radius of this circle. Problem 3. Let $\displaystyle a, b, c \geq -1$ be real numbers with $\displaystyle a^3+b^3+c^3=1$. Prove that $\displaystyle a+b+c+a^2+b^2+c^2 \leq 4$. When does the equality hold ? Problem 4. The sequence $\displaystyle a_n$ is defined by $\displaystyle a_0=3$ and $\displaystyle a_{n+1}-a_n=n(a_n-1), n\geq0$ Find all positive integers m for which $\displaystyle \gcd(m,a_n)=1$ for all $\displaystyle n\geq0$. Problem 5 . Let us consider arrangements of the numbers 1 through 64 on the squares of an 8 × 8 chessboard, each square containing exactly one number and each number appearing exactly once. A number in such an arrangement is called interesting if it is both the largest number in its row and the smallest number in its column. Prove or disprove each of the following statements : (a) Each such arrangement contains at least one interesting number. (b) Each such arrangement contains at most one interesting number. Problem 6. Let ABC be a triangle. Its incircle meets the sides BC, CA and AB in the points D, E and F, respectively. Let P denote the intersection point of ED and the line perpendicular to EF and passing through F, and similarly let Q denote the intersection point of EF and the line perpendicular to ED and passing through D. Prove that B is the mid-point of the segment PQ.

 Tags algebra, geometry, number theory

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post bobsmith76 Number Theory 37 August 13th, 2014 06:47 AM hikjaf Linear Algebra 2 October 18th, 2013 08:49 AM flipsy618 Advanced Statistics 2 January 14th, 2013 04:53 PM CEL Calculus 3 September 19th, 2012 02:16 AM sivela Algebra 2 March 8th, 2010 06:14 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top