# IMOmath

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@Neelabh Deka

We first notice that $$f(x)$$ cannot be $$0$$ for every $$x$$.

Placing $$y=0$$ in the given equation yields $$f(x)=f(x)f(0)$$. If we place a value $$x$$ for which $$f(x)\neq 0$$ in the last equation we obtain that $$f(0)=1$$.

We now set $$y=-x$$ in the functional equation to obtain $$1-x^2=f(0)-x^2=f(x)f(-x)$$. This specifically implies that $$f(1)f(-1)=0$$.

Consider now the two cases:

• Case 1. $$f(1)=0$$. Set $$x=1$$ in the functional equation. Then we get $$f(y+1)+y=0$$ which means that $$f(y+1)=-y$$. Substituting $$x=y+1$$ yields $$f(x)=1-x$$.

• Case 2. $$f(-1)=0$$. We now set $$x=-1$$ in the original equation and obtain $$f(y-1)-y=0$$ which means that $$f(y-1)=y$$. Substituting $$x=y-1$$ yields $$f(x)=x+1$$.

It is easy to verify that each of the functions $$f(x)=1-x$$ and $$f(x)=x+1$$ is a solution to the given functional equation.

Posted on: 11/25/2014 at 15:11:55     Posted by maticivan

O is the circumcentre of ∆ ABC and K is the circumcentre of ∆ AOC. The lines AB, BC meet the circle at M and N respectively. L is the reflection of K in the line MN. Find the angle between BL and AC.

Posted on: 02/22/2015 at 06:02:21     Posted by SRIDEV

Sir
I am studying Problem 5 under inversion. The solution says " that $$M\prime$$ is on the polar of point $$B$$ with respect to circle $$A\prime C\prime N\prime K\prime$$ ". Why is it so?

Posted on: 04/24/2015 at 14:04:39     Posted by wyjhyd

Hi everybody
I have a seemingly simple inequality to prove:

If a,b are Natural numbers greater than 1 show that: ab>a+b

Can anyone help solve it?

Posted on: 07/25/2015 at 20:07:25     Posted by Ebrahimi

Prove that 1+1/2^3+1/3^3+...+1/n^3 < 5/4. This was question 35 in the 1969 Short List for the IMO. I solved it as follows using the integral test:

int_2^infty 1/x^3 = 1/8 > sum_3^infty 1/n^3. Thus, 1+1/8+1/8 = 5/4 > sum_3^infty 1/n^3.

Is this the way you would‘ve also solved it?

Posted on: 11/01/2015 at 10:11:23     Posted by jackcornish

Consider a gathering of more than three people. Assume that knowing is
a symmetric relation i.e., if person A knows person B then B knows A. Given any two
persons, number of people they both know is exactly one. Prove that if two persons do
not know each other then they know the same number of people.

Posted on: 12/04/2015 at 20:12:42     Posted by Mathzoo

In the following theorem from multivariable calculus for finding a tangent plane passing through a point the following theorem is stated \overrightarrow n(x_0,y_0)=\langle -f_x(x_0,y_0), -f_y(x_0,y_0),1\rangle. While working with the respective examples how does one find the f_y, while working through this I took the derivative of the respective function and got a different answer from the solution stated in the example page.

http://imomath.com/index.php?options=296&lmm=0

Posted on: 07/27/2016 at 15:07:41     Posted by Zophike1

Find the value of ‘iota‘ raised to the power of ‘iota‘, the power in turn raised to the power of ‘iota‘ and so on, till infinity.

Posted on: 09/11/2016 at 05:09:04     Posted by mohitmac26

How do I solve the following system of equations?
$$x^2+y^2+xy=19$$, $$y^2+z^2+yz=49$$ and $$z^2+z^2+zx=39$$.

Posted on: 02/23/2017 at 03:02:45     Posted by luimichael

Why we don‘t consider the LCM and HCF of negative integers. If we consider then we will see that the factors are larger than the number and LCM is less than the number

Posted on: 11/14/2017 at 09:11:01     Posted by Swapnil

Hi!I need a help with one math problem:
All zeros of polynomial P(x)=x^n+an-1*x^(n-1)+...+a1*x1€R[x],n>1,with non-negativ coefficients,are real and different.
Prove inequalitie:
P(1)*P(2)*...*P(2016)>2017!^n