IMO 2017 Day 2

Discussions of these problems have already started on AoPS. Let's start a few over here. :)

Problem 4

Let $R$ and $S$ be different points on a circle $\Omega$ such that $RS$ is not a diameter. Let $\ell$ be the tangent line to $\Omega$ at $R$ . Point $T$ is such that $S$ is the midpoint of the line segment $RT$ . Point $J$ is chosen on the shorter arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $JST$ intersects $\ell$ at two distinct points. Let $A$ be the common point of $\Gamma$ and $\ell$ that is closer to $R$ . Line $AJ$ meets $\Omega$ again at $K$ . Prove that the line $KT$ is tangent to $\Gamma$ .

Problem 5

An integer $N > 2$ is given. A collection of $N(N + 1)$ soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove $N(N - 1)$ players from this row leaving a new row of $2N$ players in which the following $N$ conditions hold:

( $1$ ) no one stands between the two tallest players,

( $2$ ) no one stands between the third and fourth tallest players,

( $N$ ) no one stands between the two shortest players.

Show that this is always possible.

Problem 6

An ordered pair $(x, y)$ of integers is a primitive point if the greatest common divisor of $x$ and $y$ is $1$ . Given a finite set $S$ of primitive points, prove that there exist a positive integer $n$ and integers $a_0, a_1, \ldots , a_n$ such that, for each $(x, y)$ in $S$ , we have:

$a_0x^n + a_1x^{n-1} y + a_2x^{n-2}y^2 + \cdots + a_{n-1}xy^{n-1} + a_ny^n = 1.$

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Easy Math Editor

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

Math	Appears 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}`	$\boxed{123}$

Comments

How I solved P4:

Firstly, wow, this was an easy geometry problem compared to normal geo questions.

Firstly, we prove $AT \parallel RK$ . Since $\angle SRK = \angle SJK = \angle STA$ , this follows by alternate angles.

Now, the midpoint condition felt weird, so I thought about how I could exploit it. This was through the construction of a parallelogram.

Let $RK$ intersect the circumcircle of $STK$ at $B$ . We will show $BTAR$ is a parallelogram. SInce $\angle ARS = \angle BKR$ by alternate segment theorem, and $\angle STB = \angle BKR$ , we have $\angle ARS = \angle STB$ , so $AR \parallel BT$ , so $BTAR$ is a parallelogram.

Since $S$ is the midpoint of $RT$ , we have $A, S, B$ collinear. Thus, $\angle TAB = \angle SBK = \angle STK$ , it follows $KT$ is tangent to $\Gamma$ by alternate segment theorem.

Sharky Kesa - 3 years, 11 months ago

Mine is a bit different than Sharky and here is my solution to problem $4$ .

Let $KS \cap TA = L.$ Join $RN$ and $AS.$ Now, $\angle SRK = \angle SJK = \angle ATS.$ Therefore , $AT || RK$ by alt. int. $\angle$ . Now, since $RT || AK$ and $RS = ST$ $=> KTLR$ is a $||gm$ . $=>$ $\angle LRT = \angle KTR$ $--1$ and $\angle TNK = \angle RKL$ also, $\angle ARS = \angle RKL$ - By Alt. Segment Theorem. $=>$ $\angle ARS = \angle TNK => ANSR$ is cyclic $=> \angle LAS = \angle LRS --2.$ Therefore, by $1$ and $2$ we have $\angle TAS = \angle LRS = \angle RTK.$ So done by the Alt. Segment Theorem.

Comments -- As is obvious from the problem, it is a $5$ minutes angle chasing. I highly doubt that this question is shortlisted from $1959$ Romania.[Joke] It's very disappointing especially when there was no geometry in first day.

Vishwash Kumar ΓΞΩ - 3 years, 11 months ago

I remembered seeing a comment saying that "Questions 1 and 4 have become easier in order for people to get the HM". Maybe this will explain.

Steven Jim - 3 years, 10 months ago

Math	Appears 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}`	$\boxed{123}$