# 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

Posted on: 11/16/2017 at 05:11:27     Posted by bbbmiljana

See the graph here https://i.stack.imgur.com/sjSHq.jpg

1) Determine the actual bandwiths of all five communication channels? 2) Determine the total throughput between P and S cities in the case when all five channels are working properly?

Our mission is to determine the actual bandwith of data channels between the magior cities. we know the network topology between cities C , S , P , Y as shown in the picture and it consists of five channels. We got the information from three different intelligence source. At first we got the partial results of one network exercise when the actual throughput between P city and S city was measured in five cases when one channel (from total five channels) was switched off. But we only got the finve numbers - 258.9, 305.3, 141.5, 163.8 and 241.4 Mbits/s - and we don‘t know, which actual channels were actually switched off (in order) in these cases. From the second intelligence souce we got the information that the bandwith of Y - S cities channel is about 59% from the bandwith of C-S cities channel. Third intelligence source told us that if the direct C-Y cities channel will be out of order, then the total troughput between C-Y cities will decrease 1.78 times.

Posted on: 01/31/2019 at 16:01:17     Posted by niya diba

Find all $$f: \mathbb{R} \to \mathbb{R}$$ such that,
$f(xf(x)+yf(y))=x^2+y^2$

Posted on: 07/16/2019 at 19:07:42     Posted by Tuhin Bose

Transformation of rectangular coordinates to elliptic coordinates with focal point or major axis along y-direction.

The traditional coordinate transform, which gives the elliptic coordinates, sets the focal points on the x-axis (a,0) and (-a,0). And we have the following transformations

x = acoshηcosψ
y = asinhηsinψ
I need to know, how we need to change the coordinates such that the focal points are on the y-axis i.e, (0,a) and (0,-a) . In that case what will be these equations?

Posted on: 09/16/2019 at 11:09:43     Posted by Ayesha

Hello，guys

A triangle ABC satisfies BC = AC+ 1
2AB. Point P on side AB is taken so that
AP = 3PB. Prove that \PAC = 2\CPA.

Posted on: 04/01/2020 at 03:04:10     Posted by Dennis

Hi,everybody

please teach me how to solve the problem below：

A triangle ABC satisfies BC = AC+ 1/2AB. Point P on side AB is taken so that
AP = 3PB. Prove that angle(PAC) = 2*angle(CPA).

Thanks a lot
Dennis

Posted on: 04/01/2020 at 03:04:18     Posted by Dennis

Hi guys!I need a help with one math problem:
Let S = {P1,P2, . . . ,P2000} be a set of 2000 points in the interior of a circle of
radius 1, one of which at its center. For i=1,2, . . . ,2000 denote by xi the distance
from Pi to the closest point Pj ≠ Pi. Prove that x_1^2+x_2^2+⋯+x_2000^2.
The source is Swiss IMO Team Selection Tests 2000.
I will really apreciate!
Codru

Posted on: 11/24/2020 at 17:11:08     Posted by codrutac

Hello all!

I have created my own math problem, and would request all of you to try it!

Problem: Let $$\Delta ABC$$ be a triangle and $$D$$, $$E$$, $$F$$ are points on $$BC$$, $$CA$$, $$AB$$ such that $$AD\cap BE\cap CF=X$$. If $$\frac{BD}{CD}$$, $$\frac{CE}{AE}$$, $$\frac{AF}{BF}$$ are reals in the interval $$[\frac{1}{5},5]$$, such that $\frac{BD}{CD}+\frac{CE}{AE}+\frac{AF}{BF}=\frac{31}{5}$ then, $$\frac{AX}{XD}+\frac{BX}{XE}+\frac{CX}{XF}$$ can be expressed as $$\frac{m}{n}$$ for relatively prime positive integers $$m$$ and $$n$$. Find $$m+n$$.

Posted on: 11/28/2020 at 08:11:21     Posted by Imran123

Solve a(a+1)=b!+(b+1)!

Posted on: 05/07/2021 at 19:05:19     Posted by Mohammadreza3

Hello everybody!

Determine all real-valued functions f on the set of real numbers satisfying

f(nx) = nf(x)

for all real numbers x and all integers n.

Posted on: 05/17/2021 at 14:05:13     Posted by k_chel

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