IMOmathForum Math Questions
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:
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
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
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?
Thank you in advance.
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 wouldve 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.
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
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