# 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 $x$ does the following inequality hold: $\dfrac{4x^2}{(1-\sqrt{1+2x})^2} < 2x +9$?

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

###### This is part of the set International Mathematical Olympiads

Note by Sualeh Asif
4 years, 9 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

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

Staff - 4 years, 9 months ago

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

- 4 years, 8 months ago

Q#2 For the L.H.S to be defined, $(1-\sqrt{1+2x})^2\neq 0 \implies x\neq 0$ Also,for the inequality to be defined, $\sqrt{1+2x}\geq 0 \rightarrow 1+2x\geq 0\rightarrow x\geq \frac{-1}{2}$. Moreover, either $\sqrt{1+2x}>1$ or $\sqrt{1+2x}<1$. Case #1 $\sqrt{1+2x}>1\\ 1+2x>1 \\ 2x >0\rightarrow x>0$ Case #2 $\sqrt{1+2x}<1\\ 1+2x<1\\ 2x<0 \rightarrow x <0$ Therefore, either $\frac{-1}{2}\leq x <0$ or $x>0$ Now: $\frac{4x^2}{(1-\sqrt{1+2x})^2}<2x+9\\ \frac{4x^2 \times (1+\sqrt{1+2x})^2}{(1-\sqrt{1+2x})^2\times (1+\sqrt{1+2x})^2}<2x+9\\ \frac{4x^2 \times (1+\sqrt{1+2x})^2}{(1-1-2x)^2} < 2x+9\\ (1+\sqrt{1+2x})^2<2x+9$ Simplifying further,we get: $1+1+2x+2\sqrt{1+2x}<2x+9\\ 2\sqrt{1+2x}<7\\ \sqrt{1+2x}<\frac{7}{2}\\ 1+2x<\frac{49}{4} \\ 2x<\frac{45}{4} \\ x<\frac{45}{8}$ If $x>0$ we get $0 If $\frac{-1}{2}\leq x<0$ ,then $\frac{-1}{2}\leq x<\frac{45}{8},x\neq 0$. Hence,all $x$ where $\frac{-1}{2}\leq x<\frac{45}{8},x\neq 0$ satisfy the inequality. .............................................................................................................................................................................................

- 4 years, 9 months ago

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

- 4 years, 9 months ago

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.

- 4 years, 9 months ago

Staff - 4 years, 9 months ago

Third one can be done by applying coordinates

- 4 years, 9 months ago

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

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

Conversely, we can verify that $101$ satisfies the conditions of the question, since $101 \equiv 2 \pmod{11}$ and $1^2 + 0^2 + 1 ^2 = 2$.

Is the above solution correct?

Staff - 4 years, 9 months ago

In addition to what Nihar said, while it is true that $a-b+c \equiv \overline{abc} \pmod{11}$, that does not necessarily mean that the remainder is $a-b+c$. For example, what is the remainder when $909$ is divided by $11$?

- 4 years, 9 months ago

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.

Staff - 4 years, 9 months ago

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

- 4 years, 9 months ago

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

- 4 years, 9 months ago