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International Mathematical Olympiad '59, First Day

The goal of this set of notes is to improve our problem solving and proof writing skills. You are encouraged to submit a solution to any of these problems, and join in the discussion in #imo-discussion at Saturday 10/26/2015 at 10pm IST, 930am PDT. For more details, see IMO Problems Discussion Group.


These are the problems from the first ever International Mathematical Olympiad. They are pretty easy and I ask every one to try and solve them! Remember that the contestants were given Four and a Half Hours, and are required to prove these statements completely. So here Goes:

Q1. For every integer \(n\), prove that the fraction \(\dfrac{21n+4} {14n+3}\) cannot be reduced any further.(POL)

Q2. For which real numbers \(x\) do the following equations hold: \( (a) \sqrt{x+\sqrt{2x-1}}+ \sqrt{x-\sqrt{2x-1}}=√2, \\\ (b) \sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=1, \\\ (c) \sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}} =2.\)

(ROM)

Q3. Let \(x\) be an angle and let the real numbers \(a, b, c, \cos x\) satisfy the following equation: \[a\cos^2 {x}+ b \cos {x} + c =0.\] Write the analogous quadratic equation for \(a, b, c, \cos {2x} \). Compare the given and the obtained equality for \(a = 4,b = 2,c = −1\). (HUN)

Note by Sualeh Asif
1 year, 12 months ago

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Q1. Let g.c.d(\(a,b\)) be represented by (\(a,b\)).

(\(21n+4,14n+3\)) \(=\) (\(7n+1,14n+3\)) \(=\) (\(7n+1, 7n+2\)) \(=\) (\(7n+1,1\)) \(=\) \(1\)

Therefore, it can't it reduced for any integer \(n\). Surya Prakash · 1 year, 12 months ago

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@Surya Prakash This is absolutely correct! From a collary of the Euclidean Algorithm we know that \((a,b)=(b,a-b)\) and this makes the problem trivial! Sualeh Asif · 1 year, 12 months ago

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@Surya Prakash Same approach! Vinayak Verma · 1 year, 10 months ago

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@Surya Prakash Nice solution. For completeness, you should explain why \( (a,b) = (a-b, b) \), and state that's the reason why we have equality throughout.

At the IMO, this solution is worth 7-. Calvin Lin Staff · 1 year, 12 months ago

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@Surya Prakash Simple standard solution. Mehul Arora · 1 year, 12 months ago

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@Surya Prakash Same solution Anik Mandal · 1 year, 12 months ago

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I'll give my solution to Q3 (which is probably the long way around)

Note that \(\cos 2x = 2 \cos^2 x - 1\). Let \(\cos x = y\) and \(\cos 2x = z\).

We have

\[y = \dfrac {-b \pm \sqrt{b^2 - 4ac}}{2a}\]

\[y^2 = \dfrac{2b^2-4ac \pm 2b\sqrt{b^2 - 4ac}}{4a^2}\]

\[2y^2 = \dfrac{b^2-2ac \pm b\sqrt{b^2 - 4ac}}{a^2}\]

\[2y^2 - 1 = \dfrac{b^2-a^2-2ac \pm b\sqrt{b^2-4ac}}{a^2}\]

\[z = \dfrac{b^2-a^2-2ac \pm b\sqrt{b^2-4ac}}{a^2}\]

Now, we will reverse engineer completing the square (which is used to get the quadratic equation. I just reversed it)

\[\left (z + \dfrac {2ac+a^2-b^2}{a^2} \right )^2 = \dfrac {b^4 - 4ab^2 c}{a^4}\]

\[z^2 + \dfrac {4ac + 2a^2 - 2b^2}{a^2} z + \dfrac {4a^2c^2+a^4 + b^4+4a^3c-4ab^2c-2a^2b^2}{a^4} = \dfrac {b^4 - 4ab^2 c}{a^4}\]

\[z^2 + \dfrac {4ac + 2a^2 - 2b^2}{a^2} z + \dfrac {4a^2c^2+a^4+4a^3c-2a^2b^2}{a^4} = 0\]

\[a^2 z^2 + (2a^2-2b^2+4ac)z + (a+2c)^2-2b^2=0\]

\[a^2 \cos^2 2x + (2a^2-2b^2+4ac) \cos 2x + (a+2c)^2-2b^2=0\]

Therefore, the analogous equation is

\[a^2 \cos^2 2x + (2a^2-2b^2+4ac) \cos 2x + (a+2c)^2-2b^2=0\]

Comparing for \(a=4, b=2, c=-1\), we have

\[4 \cos^2 x + 2 \cos x - 1 = 0\]

\[4 \cos^2 2x + 2 \cos 2x - 1 = 0\]

Clearly, both equations have the same coefficients. The value of \(x\) in this case are certain multiples of \(\dfrac {\pi}{5}\).

I hope that's what I had to do when I had to compare.


Edit: This solution is only valid as long as \(a \neq 0\) because otherwise, we have division by 0. If \(a\) does equal 0, we have

\[by+c=0\]

\[y= \dfrac{-c}{b}\]

\[y^2=\dfrac{c^2}{b^2}\]

\[2y^2=\dfrac{2c^2}{b^2}\]

\[2y^2-1=\dfrac{2c^2-b^2}{b^2}\]

\[z=\dfrac{2c^2-b^2}{b^2}\]

\[b^2 z = 2c^2-b^2\]

\[b^2 z + b^2 - 2c^2 = 0\]

\[b^2 \cos 2x + b^2 - 2c^2 = 0\] Sharky Kesa · 1 year, 12 months ago

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@Sharky Kesa Good approach, though you should justify some of the steps. For example, what happens in the case that \( a = 0 \)? Then your solution is no longer valid.

At the IMO, your solution would be worth a 7-. Calvin Lin Staff · 1 year, 12 months ago

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@Sharky Kesa Another thing to note is that the coefficients of both the equations are proportional. That is that the coefficients of the L.H.S of the \(\cos{2x}\) quadratic are \(4\times\) that of the original equation.

This ends our questions for the First (ever) day of the IMO! Great work @Sharky Kesa ,@Surya Prakash @Chew-Seong Cheong and all others! Sualeh Asif · 1 year, 12 months ago

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Solution for Q1:

Lemma - (1) : Let (a,b) denote the GCD of a and b. Then, (a,b) = (a-b,b).

Proof : Let (a,b) = d and (a-b,b) = e. Then, from the definition of GCD, we have (a,b) = d \( \Rightarrow \) d|a and d|b \( \Rightarrow \) d|(a-b) and d|b \( \Rightarrow \) d|(a-b,b), i.e. d|e. Similarly, (a-b,b) = e \( \Rightarrow \) e|(a-b) and e|b \( \Rightarrow \) e|a and e|b \( \Rightarrow \) e|(a,b), i.e. e|d. Hence, e|d and d|e implies |e| = |d|, thereby proving that |(a,b)| = |(a-b,b)|. But the greatest common divisor of any two integers is the largest positive integer that divides both of them, i.e. GCD range is the set of positive integers. Hence, we can say |(a,b)| = |(a-b,b)| \( \Rightarrow \boxed{(a,b) = (a-b,b)} \).

Now, from repeated use of Lemma-(1), we can write (21n+4, 14n+3) = (7n+1, 14n+3) = (7n+1, 7n+2) = (7n+1,1) = 1. Therefore, the fraction is irreducible for any integer n. Karthik Venkata · 1 year, 12 months ago

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Answer to Question 3:

\(\color{blue}{a\cos^2{x} + b\cos{x} + c = 0 \quad \quad ...(1) }\) \(\begin{equation} \begin{split} \Rightarrow a\cos^2{x} & = - b\cos{x} - c \\ \frac{a}{2}(\cos{2x}+1) & = - b\cos{x} - c \\ a \cos{2x} & = -a - 2b\cos{x} - 2c \\ a^2 \cos^2{2x} & = a^2 + 4b^2\cos^2{x} + 4c^2 + 2(2ab \cos{x} + 4bc \cos{x} + 2ca) \\ & = \frac{4b^2}{2}(\cos{2x}+1) + 4(a+2c)b\cos{x} + a^2 + 4c^2+4ca \\ & = 2b^2\cos{2x} - 2(a+2c)(-a-2b\cos{x} -2c) - 2a^2 - 8c^2 - 8ca + a^2 + 2b^2 + 4c^2+4ca \\ & = 2b^2\cos{2x} - 2a(a+2c)\cos{2x} - a^2 + 2b^2 - 4c^2 - 4ca \end{split} \end{equation} \)

The required equation is:

\[\color{blue}{a^2 \cos^2{2x}+ (2a^2-2b^2+4ca)\cos{2x} + a^2 - 2b^2 + 4c^2 + 4ca = 0 \quad \quad ...(2)} \]

For \(a=4\), \(b=2\) and \(c=-1\),

\(\begin{cases} (1): & 4\cos^2{x} + 2\cos{x} - 1 = 0 \\ \\ (2): & 16\cos^2{2x} + 8\cos{2x} - 4 = 0 & \Rightarrow 4\cos^2{2x} + 2\cos{2x} - 1 = 0 \end{cases} \)

\(\begin{equation} \begin{split} \text{From }(1): \quad \cos{x} & = \frac{-2\pm\sqrt{20}}{8} = \frac{-1\pm\sqrt{5}}{4} \\ \Rightarrow \cos{2x} & = 2\cos^2{x} - 1 \\ & = 2 \left( \frac{-1\pm\sqrt{5}}{4} \right)^2 - 1 \\ & = 2 \left( \frac{6 \color{red}{\mp} 2 \sqrt{5}}{16} \right) - 1 \\ & = \frac{3 \color{red}{\mp} \sqrt{5} - 4}{4} = \frac{-1 \color{red}{\mp} \sqrt{5}}{4} \end{split} \end{equation} \)

Therefore, both \(\cos{x}\) and \(\cos{2x}\) satisfy the equation \(4\cos^2{x} + 2\cos{x} - 1 = 0 \).

Actually, \(\cos{72^\circ} = \frac{-1 + \sqrt{5}}{4}\) and \(\cos{144^\circ} = \frac{-1 - \sqrt{5}}{4}\). Chew-Seong Cheong · 1 year, 12 months ago

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@Chew-Seong Cheong Another typo in the eleventh line. I think the equation in (2) is in \(\cos {2x}\) Sualeh Asif · 1 year, 12 months ago

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@Sualeh Asif Thanks. I have edited it. Chew-Seong Cheong · 1 year, 12 months ago

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@Chew-Seong Cheong Another great solution! Do note that the coefficients of \((1),(2)\) are proportional. Sualeh Asif · 1 year, 12 months ago

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@Chew-Seong Cheong In your third line, I think you mean \( \frac{a}{2} ( \cos 2x + 1 ) \)?

Good solution. At the IMO, this would be worth 7-. Calvin Lin Staff · 1 year, 12 months ago

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@Calvin Lin Yes, you are right. Edited it. Chew-Seong Cheong · 1 year, 12 months ago

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Q2. I will just write the solution for the first one. Try the remaining on your own.

Clearly, the root is satisfied iff \(x \geq \dfrac{1}{2}\). Squaring on both sides gives \(2x +2|x-1| = 2\).

So, \(x+ |x-1| = 1\).

If \(x > 1\), then \(2x - 1 = 1\) which gives \(x=1\) which is false since we assumed that \(x>1\).

If \(x \leq 1\), then we get \(1=1\) which is true. So, all reals less than or equal to \(1\) satisfied the above equation. But \(x \geq \dfrac{1}{2}\). So, all reals \(x \in \Big[ \dfrac{1}{2}, 1 \Big] \) satisfy this equation. Surya Prakash · 1 year, 12 months ago

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@Surya Prakash It might seem like a silly question to some, but how do you jump from the original equation to \(2x + 2|x - 1| = 2\) N Solomon · 1 year, 12 months ago

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@Surya Prakash At the IMO, avoid words like "Clearly" and "Obviously". You have to explain what you mean by "Clearly, the root is satisfied iff \( x \geq \frac{1}{2} \).

Furthermore, when you square an equation, you introduce extraneous roots. So, any solution that you obtain still has to be verified that it is a solution to the original problem. So far, we can only conclude that "the solution is a subset of ....".

At the IMO, this solution would be worth 0+. Calvin Lin Staff · 1 year, 12 months ago

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@Calvin Lin Got it Sir!! Thanks for suggestion. Surya Prakash · 1 year, 12 months ago

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@Surya Prakash This is a perfect solution. All reals \(\frac{1}{2} \leq x \leq 1\) satisfy the equation for the part one. It can be proved using the same method that no solutions exist for the second part. The main thing is to note that \(x\geq \frac{1}{2}\) and then square to obtain expressions. The third one is just a follow up from this. Sualeh Asif · 1 year, 12 months ago

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@Sualeh Asif Yes. U r right. Surya Prakash · 1 year, 12 months ago

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Let \(\dfrac{21n + 4}{ 14 n +3 }= k\) D H · 1 year ago

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Q#2 \[(a) \sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=1\] For the L.H.S to be real,\(\sqrt{2x-1}\) must be greater then or equal to zero.So \[\sqrt{2x-1}\geq 0\\2x-1\geq 0\\x\geq \frac {1}{2}\] ............................................................................................................................................................................................. \[\begin{align}\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=1\\ \text{Squaring both sides} 2x+2(\sqrt{x+\sqrt{2x-1}})(\sqrt{x-\sqrt{2x-1}})=1\\2x+2\sqrt{(x-1)^2}=1\\2x+2|x-1|=1\\x+|x-1|=\frac{1}{2}\end{align}\] \[\text{CASE 1:} x>1\\ x+|x-1|=\frac{1}{2}\rightarrow x+x-1=\frac{1}{2}\\2x=\frac{1+2}{2}\\x=\frac{3}{4}\] Here \(\frac{3}{4}<1\),but we assumed that (x>1/),a contradiction.Hence no solutions exist in this case. \[\text{CASE 2:} x<1 \\x+|x-1|=\frac{1}{2}\rightarrow x-x+1=\frac{1}{2}\\1=\frac{1}{2} \text{A contradiction}\] Therefore no solutions can exust in this case. As no solutions can exist in either case,so the equation has no solutions. Abdur Rehman Zahid · 1 year, 11 months ago

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@Abdur Rehman Zahid Great work, Though you made a mistake. In case 1 you assumed \(x>1\) while the answer you got was less than 1. Contradiction. No solutions for the first or second case. Put in a complete solutoon doing the third part too. Sualeh Asif · 1 year, 11 months ago

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@Sualeh Asif I just realized it now.sorry!!!! (Ican't edit my solution,can I?) Abdur Rehman Zahid · 1 year, 11 months ago

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@Abdur Rehman Zahid Sure you can! :D Just click the "EDIT" button at the bottom of your solution :) Mehul Arora · 1 year, 11 months ago

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@Mehul Arora Thanks!!!! Abdur Rehman Zahid · 1 year, 11 months ago

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@Sualeh Asif @Sualeh Asif please open your slack :) Mehul Arora · 1 year, 11 months ago

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1) gcd{21n + 4, 14n + 3} = 1 so they cant be reduced Dev Sharma · 1 year, 12 months ago

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@Dev Sharma Sorry that's not what I meant. Just went out of track! Hrithik Nambiar · 1 year, 12 months ago

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@Dev Sharma At the IMO, this solution is a 0+, because you have not proven the statement. Calvin Lin Staff · 1 year, 12 months ago

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@Calvin Lin Yes sir! Thanks' a lot. Hrithik Nambiar · 1 year, 12 months ago

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@Dev Sharma Ya, this can be proved by Euclidean Algorithm. Swapnil Das · 1 year, 12 months ago

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@Dev Sharma How did you declare gcd as 1 directly? Aditya Chauhan · 1 year, 12 months ago

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@Aditya Chauhan NO ! I USED THE METHOD AS THAT OF @Dev Sharma ! BUT THEN I HAD A SENSE OF DOING IT IN ANOTHER WAY! ANYWAYS MY BAD ! Hrithik Nambiar · 1 year, 12 months ago

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@Hrithik Nambiar Avoid using upper case of letters. It is considered as rude on brilliant. Nihar Mahajan · 1 year, 12 months ago

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@Hrithik Nambiar Please refrain from typing in all caps, as that is considered very rude on the internet. Thanks for your assistance! Calvin Lin Staff · 1 year, 12 months ago

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@Calvin Lin yes sir . Hrithik Nambiar · 1 year, 12 months ago

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@Dev Sharma Did it the same way .But , Let me ask you guys something - won't the numerator be even always if "n" is not equal to zero and the denominator would always be odd. So , obviously, it can not be further reduced. Hrithik Nambiar · 1 year, 12 months ago

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@Hrithik Nambiar No what guarantee do you have that say the numerator and denominator are divisible by 7.

I.e. The numerator is 14(yet even) and denominator 7(yet odd). The fraction can be reduced.

And This is to be proved for all primes! Sualeh Asif · 1 year, 12 months ago

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@Sualeh Asif by using the fact ax + by = 1 Dev Sharma · 1 year, 12 months ago

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@Sualeh Asif exactly! sorry Hrithik Nambiar · 1 year, 12 months ago

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