BMO1 2009/10 Q5 functional equation: f(x)f(y)=f(x+y)+xyf(x)f(y) = f(x + y) + xy

Find all functions f, defined on the real numbers and taking real values, which satisfy the equation f(x)f(y)=f(x+y)+xy for all real numbers x and y. I worked out f(0)=1, and f(−1)f(1)=0, but then I hit a wall. Answer Setting y=0 gives f(x)f(0)=f(x). Since f cannot be identically zero, it follows that f(0)=1. Setting x=1,y=−1 then … Read more

Solving for f(2004)f(2004) in a given functional equation

Given that f(1)=2005 and f(1)+f(2)+…f(n)=n2f(n) for all n>1. Determine the value of f(2004). My progress: I first substituted n−1 into the equation to get f(1)+f(2)+…+f(n−1)=(n−1)2f(n−1) and then I subtracted it from the first equation to get f(n)=n2f(n)−(n−1)2f(n−1). I got a little stuck so I checked the answers for a hint and it said that: f(n)=(n−1)2n2−1f(n−1). … Read more

Suppose a continuous function f:R→Rf:\mathbb{R} \rightarrow \mathbb{R} is nowhere monotone. Show that there exists a local minimum for each interval.

Suppose a continuous function f:R→R is nowhere monotone. Show that there exists a local minimum for each interval. This question is from Moscow institute. First of all, I can’t even construct a nowhere monotone function. What I can think of, linear functions, clearly do not satisfy this, as we can always zoom in to get … Read more

Find sum with binomial coefficients and powers of 2

Find this sum for positive n and m: S(n, m) = \sum_{i=0}^n \frac{1}{2^{m+i+1}}\binom{m+i}{i} + \sum_{i=0}^m \frac{1}{2^{n+i+1}}\binom{n+i}{i}. Obviosly, S(n,m)=S(m,n). Therefore I’ve tried find T(n,m) = \sum_{i=0}^n \frac{1}{2^{m+i}}\binom{m+i}{i} by T(n, m+1), but in binomial we have \binom{m+i+1}{i} = \binom{m+i}{i} + \binom{m+i}{i-1}, and this “i-1” brings nothing good. Other combinations like T(n+1,m+1)+T(n,m) also doesn’t provide advance. Any ideas? … Read more

BMO1 2005/06 Question 4 Combinatorics Problem

The equilateral triangle ABC has sides of integer length N. The triangle is completely divided (by drawing lines parallel to the sides of the triangle) into equilateral triangular cells of side length 1. A continuous route is chosen, starting inside the cell with vertex A and always crossing from one cell to another through an … Read more

Ghosts closing and opening doors [duplicate]

This question already has answers here: Prison problem: locking or unlocking every nth door for n=1,2,3,… (2 answers) Closed 6 years ago. There are 1000 doors D1,D2,D3,…,D1000 and 1000 persons P1,P2,…,P1000. Initially all the doors were closed. Person P1 goes and opens all the doors. Then person P2 closes doors D2,D4,…,D1000 and leaves the odd-numbered … Read more

Prove that ∑cyc√a+bc≥2∑cyc√ca+b\sum\limits_{cyc}\sqrt{\frac{a+b}{c}}\ge2\sum\limits_{cyc}\sqrt{\frac{c}{a+b}}

Let a,b,c be positive numbers. Then we need to prove √a+bc+√b+ca+√c+ab≥2(√ca+b+√ab+c+√bc+a). I have an idea to set x=ab+c, y=bc+a,z=ca+b then 11+x+11+y+11+z=2 and we need to prove 1√x+1√y+1√z≥2(√x+√y+√z) But I could not go further. Answer It is a consequence of Chebychev’s inequality: ∑cyc√a+bc≥2∑cyc√ca+b⟺∑cyca+b−2c√c(a+b)≥0 Since the a+b−2c and 1√c(a+b) are ordered in the same way, we can … Read more

How many sides from diagonals?

A polygon has 100 diagonals, then it has at least: A-15, B-16, C-17, D-18 Sides? Using simple patterns, I noticed that all figures (even sides) have n2 sides for n diagonals; this makes me believe that there must be 50 sides, the answer 18 seems right then? But it cannot be <50 right? Answer Hint: … Read more

Find tanC\tan C in a triangle satisfying the constraint

Given a triangle with angles A,B,C and sides a,b,c opposite to their respective angles, how can I find tanC such that c2=a3+b3+c3a+b+c I used the law of Cosines on the LHS as well as on the cube terms but I don’t know exactly I’m looking for in manipulating the expression. Should I try to get … Read more