International Mathematical Olympiad '60, 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 on Saturday at 9:00 pm IST ,8 30 PDT. For more details, see IMO Problems Discussion Group.


Here are the problems from the first day of the 1960 International mathematical Olympiad. They range from easy to harder Number Theory, Algebra and Geometry. Try your hand at them. Don't be discouraged if you can completely solve them. Do post your inspirations and ideas towards the problems. The discussion for these questions will be held soon. Happy Problem Solving!

Q1. (BUL) Find all the three-digit numbers for which one obtains, when dividing the number by 11, the sum of the squares of the digits of the initial number.

Q2. (HUN) For which real numbers xx does the following inequality hold: 4x2(11+2x)2<2x+9\dfrac{4x^2}{(1-\sqrt{1+2x})^2} < 2x +9?

Q3. (ROM) A right-angled triangle ABCABC is given for which the hypotenuse BCBC has length aa and is divided into nn equal segments, where nn is odd. Let α\alpha be the angle with which the point AA sees the segment containing the middle of the hypotenuse. Prove that tanα=4nh(n21)a,\tan{\alpha} = \dfrac{4nh}{ (n^2 -1)a}, where hh is the height of the triangle.

This is part of the set International Mathematical Olympiads

Note by Sualeh Asif
4 years, 1 month ago

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In third question I am getting an answer (4nh/n^2-1)*b/a^2. Where b is length of other leg. Also if we put n=3 we do not get alpha as 3h/2a. Please post a solution as early as possible. I have checked my solution a lot of times.

Aakash Khandelwal - 4 years, 1 month ago

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Can you post your solution?

Calvin Lin Staff - 4 years, 1 month ago

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Q#2 For the L.H.S to be defined, (11+2x)20    x0(1-\sqrt{1+2x})^2\neq 0 \implies x\neq 0 Also,for the inequality to be defined, 1+2x01+2x0x12\sqrt{1+2x}\geq 0 \rightarrow 1+2x\geq 0\rightarrow x\geq \frac{-1}{2}. Moreover, either 1+2x>1\sqrt{1+2x}>1 or 1+2x<1\sqrt{1+2x}<1. Case #1 1+2x>11+2x>12x>0x>0\sqrt{1+2x}>1\\ 1+2x>1 \\ 2x >0\rightarrow x>0 Case #2 1+2x<11+2x<12x<0x<0\sqrt{1+2x}<1\\ 1+2x<1\\ 2x<0 \rightarrow x <0 Therefore, either 12x<0\frac{-1}{2}\leq x <0 or x>0x>0 Now: 4x2(11+2x)2<2x+94x2×(1+1+2x)2(11+2x)2×(1+1+2x)2<2x+94x2×(1+1+2x)2(112x)2<2x+9(1+1+2x)2<2x+9\frac{4x^2}{(1-\sqrt{1+2x})^2}<2x+9\\ \frac{4x^2 \times (1+\sqrt{1+2x})^2}{(1-\sqrt{1+2x})^2\times (1+\sqrt{1+2x})^2}<2x+9\\ \frac{4x^2 \times (1+\sqrt{1+2x})^2}{(1-1-2x)^2} < 2x+9\\ (1+\sqrt{1+2x})^2<2x+9 Simplifying further,we get: 1+1+2x+21+2x<2x+921+2x<71+2x<721+2x<4942x<454x<4581+1+2x+2\sqrt{1+2x}<2x+9\\ 2\sqrt{1+2x}<7\\ \sqrt{1+2x}<\frac{7}{2}\\ 1+2x<\frac{49}{4} \\ 2x<\frac{45}{4} \\ x<\frac{45}{8} If x>0x>0 we get 0<x<4580 <x<\frac{45}{8} If 12x<0\frac{-1}{2}\leq x<0 ,then 12x<458,x0\frac{-1}{2}\leq x<\frac{45}{8},x\neq 0. Hence,all xx where 12x<458,x0\frac{-1}{2}\leq x<\frac{45}{8},x\neq 0 satisfy the inequality. .............................................................................................................................................................................................

Abdur Rehman Zahid - 4 years, 1 month ago

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Really good solution other than a few typos in the calculation and rationalisation!

Sualeh Asif - 4 years, 1 month ago

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  1. Let the 3 digit number be abc \overline{abc} . The remainder when it is divided by 11 is ab+c a - b + c . We are told that this is equal to a2+b2+c2 a^2 + b^2 + c ^2 , thus ab+c=a2+b2+c2 a - b + c = a^2 + b^2 + c^2 . Multiplying by 4 and completing the square, we obtain:
(4a24a+1)+(4b2+4b+1)+(4c24c+1)=3 (4a^2 - 4a + 1 ) + ( 4b^2 + 4b + 1 ) + ( 4 c^2 - 4c + 1) = 3

Hence, we must have 2a1=2b+1=2c1=1 |2a-1 | = |2b+1 | = |2c - 1 | = 1 , or that a=1,b=0,c=1 a = 1, b = 0 , c = 1 .

Conversely, we can verify that 101 101 satisfies the conditions of the question, since 1012(mod11) 101 \equiv 2 \pmod{11} and 12+02+12=2 1^2 + 0^2 + 1 ^2 = 2 .


Is the above solution correct?

Calvin Lin Staff - 4 years, 1 month ago

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I think not the remainder , but the quotient since 11  abc11 \ | \ \overline{abc} (The problem statement is a bit ambiguous)

Nihar Mahajan - 4 years, 1 month ago

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I thought so too! And the problem statement seems to imply this too

Sualeh Asif - 4 years, 1 month ago

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In addition to what Nihar said, while it is true that ab+cabc(mod11)a-b+c \equiv \overline{abc} \pmod{11}, that does not necessarily mean that the remainder is ab+ca-b+c. For example, what is the remainder when 909909 is divided by 1111?

Mursalin Habib - 4 years, 1 month ago

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Great! That's the point I was trying to make. I slid the "seemingly obvious fact" without substantiating why it is true. As it turns out, that is not the remainder and we often have to add / subtract 11 from it.

Didn't realize that I misinterpreted the question lol.

Calvin Lin Staff - 4 years, 1 month ago

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Third one can be done by applying coordinates

Aakash Khandelwal - 4 years, 1 month ago

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@Sualeh Asif Can you post what we discussed for Q1? Thanks!

Calvin Lin Staff - 4 years, 1 month ago

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I was really busy the last whole week! I will In Sha Allah post it in a few days!

Sualeh Asif - 4 years ago

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