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    [QUOTE=Lord of the Flies;44345359]Solution 330

    Straightforward, \text{lcm} (4,5,9,11)=1980 minutes.
    I put this question here in case any gcse student was lurking around [so they might be able to solve it] :yep:[\QUOTE]

    Problem 333**

    Let ABCD be a fixed convex quadrilateral with BC = DA and BC not parallel with DA. Let two variable points E and F lie of the sides BC and DA, respectively and satisfy BE = DF. The lines AC and BD meet at P, the lines BD and EF meet at Q, the lines EF and AC meet at R. Show that the circumcircles of the triangles PQR , as E and F vary, have a common point other than P.

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    (Original post by Smaug123)
    This was on one of our example sheets
    That is where I got it from
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    Solution 332

    Assume y <x.
    We consider t^{2}-tyz+y^{2}+1=0 with a given root x. By Vieta we obtain xx_{1}=y^{2}+1 and x+x_{1}=yz; whence x_{1} is a positive integer. If y >1, we get x_{1} < y. Hence we can replace (x,y) with (y,x_{1}). Thus we have y=1, giving x=2, since x>y; z=3
    If x=y, we get x^{2} |1, and so x=y=1; z=3.

    Solution 331

    Since the limit \displaystyle \lim_{x\to 0} \frac{f(x)}{x} exists and is non-zero, we have \displaystyle \sum_{k} f(\frac{1}{k}) converges iff \displaystyle \sum_{k} \frac{1}{k} converges.

    The proof is routine.

    Assume the limit is l. Abbreviate \displaystyle \frac{f(x)}{x}=g(x). Then for each \epsilon there exists \delta (both >0) such that l-\epsilon < g(x) < l+\epsilon whenever |x| < \delta. This gives us (l-\epsilon)x < f(x) < (l+\epsilon)x.
    Choosing \epsilon sufficiently small, we are done.

    Solution 333

    Let the circumcircles of APB and CDP meet at O. We know that O is the center of the spiral similarity that sends D to B and A to C (well-known lemma, proved via directed angles \pmod {\pi}). Our similarity sends also F to E and we see that the circumcircles of FDQ and BQE meet at O. Now applying Miquel's theorem to APD and DFQ, we see that the circumcircle of PRQ passes through O.
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    (Original post by The Sorcerer)
    Ah many commendations my mathematical friend. There was me thinking that you were just going through a mundane mathematical question via conventional methods... but then you just shock your dedicated audience tremendously by unveiling the beautiful esoteric technique that is absorption... ABSORPTION! I ask, WHO IN THE RIGHT MIND WOULD HAVE DARED TO HAVE UTILIZED SUCH AN IMMENSELY DIFFICULT TECHNIQUE?! and what an elegant sponge you chose to use for the job too! Wonderful. Keep up the good work. :pierre: I will be greatly anticipating and anxiously awaiting your next ground-breaking mathematical display.
    :lolwut:

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    EDIT: I'm satisfied that this reasoning is wrong; please ignore it.

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    x, y and z are positive integers satisfying  x^2+y^2+1 = xyz . Show that z=3.

    I did not see a specific way to eliminate to get z=3 so decided to try to deduce it using inequalities. Firstly, note that  x,y \geq 1 . Firstly, consider the following inequality:

     (x-y)^2 \geq 0

     x^2+y^2 \geq 2xy (1)

     x^2 +y^2 = xyz - 1

     xyz - 1 \geq 2xy

     xyz \geq 2xy+1 so z > 2.

    Next, add 2xy to both sides of (1) to get  x^2+y^2+2xy \geq 4xy

    Also, note that 2xy > 1, and so:

     x^2+y^2+2xy > x^2+y^2+1

     x^2+y^2+2xy > zxy

    Now, we have the following two inequalities:

     zxy < (x+y)^2 ,  4xy \leq (x+y)^2

    Now, if z is 4 or more, we can see that x and y can be chosen so that equality holds, which is a contradiction, so z < 4. This argument is the bit that concerns have been expressed over.

    This gives the inequality 2 < z < 4 so z = 3.

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    (Original post by DJMayes)
    Guys, could you check my logic to my solution of problem 332? I have done it a different way and obtained the solution but there is a concern that my solution is wrong. Could you see what you think?

    Solution 332 (Alternate)

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    x, y and z are positive integers satisfying  x^2+y^2+1 = xyz . Show that z=3.

    I did not see a specific way to eliminate to get z=3 so decided to try to deduce it using inequalities. Firstly, note that  x,y \geq 1 . Firstly, consider the following inequality:

     (x-y)^2 \geq 0

     x^2+y^2 \geq 2xy (1)

     x^2 +y^2 = xyz - 1

     xyz - 1 \geq 2xy

     xyz \geq 2xy+1 so z > 2.

    Next, add 2xy to both sides of (1) to get  x^2+y^2+2xy \geq 4xy

    Also, note that 2xy > 1, and so:

     x^2+y^2+2xy &gt; x^2+y^2+1

     x^2+y^2+2xy &gt; zxy

    Now, we have the following two inequalities:

     zxy &lt; (x+y)^2 ,  4xy \leq (x+y)^2

    Now, if z is 4 or more, we can see that x and y can be chosen so that equality holds, which is a contradiction, so z < 4. This argument is the bit that concerns have been expressed over.

    This gives the inequality 2 < z < 4 so z = 3.

    This is dodgy, I think. Think of it as:
    Spoiler:
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    z x y &lt; (x+y)^2, 4 x y \leq (x+y)^2 means
    z &lt; \dfrac{(x+y)^2}{x y}, 4 \leq \dfrac{(x+y)^2}{x y}
    so we are trying to make a<b out of a<c, b<=c.
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    (Original post by Mladenov)
    Solution 332

    Assume y &lt;x.
    We consider t^{2}-tyz+y^{2}+1=0 with a given root x. By Vieta we obtain xx_{1}=y^{2}+1 and x+x_{1}=yz; whence x_{1} is a positive integer. If y &gt;1, we get x_{1} &lt; y. Hence we can replace (x,y) with (y,x_{1}). Thus we have y=1, giving x=2, since x&gt;y; z=3
    If x=y, we get x^{2} |1, and so x=y=1; z=3.
    Forgive me but I don't see how this method gives a sufficient proof as there are infinitely many solutions without x or y equal to 1. For example, (5,2,3) and (13,5,3) satisfy the equation and using the second root of the quadratic as you did you can find infinitely many larger and larger solutions. I have been bashing my head against this for the better part of a day now and can't see how any of the solutions I've seen so far work as they rely on x or y equalling 1 which is demonstrably not necessary.
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    (Original post by DJMayes)
    Forgive me but I don't see how this method gives a sufficient proof as there are infinitely many solutions without x or y equal to 1. For example, (5,2,3) and (13,5,3) satisfy the equation and using the second root of the quadratic as you did you can find infinitely many larger and larger solutions. I have been bashing my head against this for the better part of a day now and can't see how any of the solutions I've seen so far work as they rely on x or y equalling 1 which is demonstrably not necessary.
    We replace (x,y) with (y,x_{1}) till we get (x_{s},x_{s+1}) with x_{s+1}=1. Throughout the process z remains constant, hence the conclusion.
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    (Original post by Mladenov)
    We replace (x,y) with (y,x_{1}) till we get (x_{s},x_{s+1}) with x_{s+1}=1. Throughout the process z remains constant, hence the conclusion.
    OK, that makes more sense. Thanks!
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    I am so out of this business now, but here is one Diophantine equation.

    Problem 334**

    Find all pairs (x,y) of positive integers satisfying the equation x^{2}+xy-y^{2}=1.
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    Problem 335**/***

    Let x\sin(2x) be a solution of a homogeneous fourth order ODE with real constant coefficients. Determine the general solution and the ODE it solves.
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    (Original post by DJMayes)
    Solution 334: (I think)

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     x^2 +xy - y^2 - 1 = 0

    In order to look for solutions, try to solve for x in terms of y. By applying the quadratic formula, we get:

     2x = -y \pm \sqrt{5y^2+4}

    Now, if integer solutions exist, we must have the bit inside the square root as a perfect square - that is; any possible solutions for x and y must satisfy:

     5y^2 + 4 = z^2

     5y^2 = z^2 - 4

     5y^2 = (z-2)(z+2)

    Now, the RHS must be divisible by y. This opens up two possibilities:

    1 - Each factor of (z-2)(z+2) contains a factor of y. As these are four apart, this is only possible if y = 1, 2 or 4. By checking we find that y=1 is a solution but y=2 or 4 are not.

    2 - One of the factors of (z-2)(z+2) is divisible by y^2. Now, if one factor is divisible by y^2, and z-2 is greater than 5 ( z > 7) then z^2-4 > 5y^2 and there are no solutions. This means we are only interested in values of z up to and including 6.

    Testing these, we get z=2 and z=3 as solutions with corresponding y values of 0 and 1. As we are interested in integer solutions, the only possible solution is y=1. Now, we are left with solving the following equation:

     x^2+x-2 = 0

     (x+2)(x-1) = 0 and so our only positive integer solution is (1, 1)

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    There are other solutions, consider (5,8) and (13,21). In fact I think there are an infinite number of solutions. This kind of explains how to solve it along with the rest of that page, it's not very clear though. Any other links on the subject would be useful!
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    (Original post by DJMayes)
    Solution 334: (I think)

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     x^2 +xy - y^2 - 1 = 0

    In order to look for solutions, try to solve for x in terms of y. By applying the quadratic formula, we get:

     2x = -y \pm \sqrt{5y^2+4}

    Now, if integer solutions exist, we must have the bit inside the square root as a perfect square - that is; any possible solutions for x and y must satisfy:

     5y^2 + 4 = z^2

     5y^2 = z^2 - 4

     5y^2 = (z-2)(z+2)

    Now, the RHS must be divisible by y. This opens up two possibilities:

    1 - Each factor of (z-2)(z+2) contains a factor of y. As these are four apart, this is only possible if y = 1, 2 or 4. By checking we find that y=1 is a solution but y=2 or 4 are not.

    2 - One of the factors of (z-2)(z+2) is divisible by y^2. Now, if one factor is divisible by y^2, and z-2 is greater than 5 ( z > 7) then z^2-4 > 5y^2 and there are no solutions. This means we are only interested in values of z up to and including 6.

    Testing these, we get z=2 and z=3 as solutions with corresponding y values of 0 and 1. As we are interested in integer solutions, the only possible solution is y=1. Now, we are left with solving the following equation:

     x^2+x-2 = 0

     (x+2)(x-1) = 0 and so our only positive integer solution is (1, 1)

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    Your reasoning assumes y prime or 1, which is not necessary. For example, 5 \times 144^{2} = 322^{2}-4. Also, you have missed the case z=7, which corresponds to y=3. Not a hint, but
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    There are infinitely many solutions, and one should look for a family of solutions. Think of some recurrence relation and then infinite descent.
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    (Original post by Mladenov)
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    Your reasoning assumes y prime or 1, which is not necessary. For example, 5 \times 144^{2} = 322^{2}-4. Also, you have missed the case z=7, which corresponds to y=3. Not a hint, but
    Spoiler:
    Show
    There are infinitely many solutions, and one should look for a family of solutions. Think of some recurrence relation and then infinite descent.
    I had added in the case z=7 before then but as the reasoning is wrong I've gotten rid of it. Thank you for pointing out the flaw.

    OK, now properly done the question.

    Solution 334:

    Spoiler:
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    Some solutions are (1,1), (2,3), (5,8) and so on. This looks very similar to the Fibonacci sequence. As such, we will guess that, if (x,y) is a solution, (x+y, x+2y) is a solution. This is for obtaining larger solutions. Reversing this, we want to show that if (x,y) is a solution, then we can generate a smaller solution (2x-y, y-x):

     (2x-y)^2 +(2x-y)(y-x) - (y-x)^2 = 1

     4x^2 -4xy +y^2 +3xy -y^2 -2x^2 -y^2 - x^2 +2xy = 1

     x^2 +xy - y^2 = 1

    Hence (2x-y, y-x) is also a solution. We should now prove the validity of this solution.

    Firstly, if  y \leq x then  x^2+xy-y^2 \geq xy and the only solution with x and y equal is (1,1) so for any other solution we have y>x, making y-x positive and valid. If  y \geq 2x then  x^2 +xy-y^2 \leq -x^2 and no solutions exist, and so the x solution is valid.

    Hence we have shown that the existence of a solution (x,y) implies the existence of a smaller solution (2x-y, y-x). We can repeat this process indefinitely until either the new x or y value equals 1, giving the solution (1,1) and so all solutions are generated from these. Working backwards, our family of solutions is  (F_{2n}, F_{2n+1}) Where F_n represents the nth Fibonacci number with F_0 = F_1 = 1.

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    (Original post by Mladenov)
    Solution 332

    Assume y &lt;x.
    We consider t^{2}-tyz+y^{2}+1=0 with a given root x. By Vieta we obtain xx_{1}=y^{2}+1 and x+x_{1}=yz; whence x_{1} is a positive integer. If y &gt;1, we get x_{1} &lt; y. Hence we can replace (x,y) with (y,x_{1}). Thus we have y=1, giving x=2, since x&gt;y; z=3
    If x=y, we get x^{2} |1, and so x=y=1; z=3.

    Solution 331

    Since the limit \displaystyle \lim_{x\to 0} \frac{f(x)}{x} exists and is non-zero, we have \displaystyle \sum_{k} f(\frac{1}{k}) converges iff \displaystyle \sum_{k} \frac{1}{k} converges.

    The proof is routine.

    Assume the limit is l. Abbreviate \displaystyle \frac{f(x)}{x}=g(x). Then for each \epsilon there exists \delta (both &gt;0) such that l-\epsilon &lt; g(x) &lt; l+\epsilon whenever |x| &lt; \delta. This gives us (l-\epsilon)x &lt; f(x) &lt; (l+\epsilon)x.
    Choosing \epsilon sufficiently small, we are done.

    Solution 333

    Let the circumcircles of APB and CDP meet at O. We know that O is the center of the spiral similarity that sends D to B and A to C (well-known lemma, proved via directed angles \pmod {\pi}). Our similarity sends also F to E and we see that the circumcircles of FDQ and BQE meet at O. Now applying Miquel's theorem to APD and DFQ, we see that the circumcircle of PRQ passes through O.
    You should probably also justify that O \neq P explicitly, unless I'm missing something.
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    (Original post by BlueSam3)
    You should probably also justify that O \neq P explicitly, unless I'm missing something.
    But this follows directly from the condition that BC \cap AD \not= \O.
    To prove it, suppose that the circumscribed circles of APB andCPD are tangent at P. Then, by angle chasing, more specifically, we let T be the tangent through P, \alpha - the angle between CP and T, and \beta - the angle between AP and T, thus we see \angle PBC = \alpha = \beta = \angle PDA, a contradiction.
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    (Original post by Mladenov)
    But this follows directly from the condition that BC \cap AD \not= \O.
    To prove it, suppose that the circumscribed circles of APB andCPD are tangent at P. Then, by angle chasing, more specifically, we let T be the tangent through P, \alpha - the angle between CP and T, and \beta - the angle between AP and T, thus we see \angle PBC = \alpha = \beta = \angle PDA, a contradiction.
    Fair enough, now I'm happy with the solution.
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    Problem 336 (**)

    Prove that for all positive a,b,c:

     \displaystyle \frac{a^3}{b} + \frac{b^3}{c} + \frac{c^3}{a} \geq ab + bc + ac
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    Solution 336

    AM-GM: \displaystyle \sum_{a,b,c} \left(\frac{a^3}{b}+ab\right) \geq \sum_{a,b,c} 2a^2 which along with \displaystyle\sum_{a,b,c} a^2\geq \sum_{a,b,c} ab yields the inequality.
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    LOTF Solution 336

    By maths we have the inequality.
 
 
 
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