RMO 2014 Delhi Region Q.5

Let $ABC$ be an acute-angeled triangle and let $H$ be its orthocenter. For any point $P$ on the circumcircle of triangle $ABC$ , let $Q$ be the point of the intersection of the line $BH$ with the line $AP$ . Show that there is a unique point $X$ on the circumcircle of $ABC$ such that for every point $P\not=A,B$ the circumcircle of $HQP$ pass through $X$ .

You can find rest of the problems here
You can find the solutions here

#Geometry #RMO #Proofs #Triangles #Circles

Easy Math Editor

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.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

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

I got this! I hope it is right.

Ranjana Kasangeri - 6 years, 6 months ago

How?

Aneesh Kundu - 6 years, 6 months ago

Take any other point say, $Z$ and let $AZ$ intersect $BH$ at $Y$ . And consider circle $HYZ$ meeting circle $ABC$ at $X.$ Now,prove $HQPX$ is a cyclic quad!

Ranjana Kasangeri - 6 years, 6 months ago

How many are you getting right? (Outta 6, right?)

Satvik Golechha - 6 years, 6 months ago

I solved 4

Aneesh Kundu - 6 years, 6 months ago

Dunno whether they are right or wrong

Aneesh Kundu - 6 years, 6 months ago

can someone give me the solution to this problem? plzz

Nihar Mahajan - 6 years, 6 months ago

Quite similar to one which came in Rajasthan paper, though more difficult.

Satvik Golechha - 6 years, 6 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}`	$\boxed{123}$