variation of billiard, circular hockey rink PUMaC 2017 Live

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0 replies

jlacosta

Apr 2, 2025

0 replies

JEE Related ig?

mikkymini2 10

N 2 hours ago by Idiot_of_the64squares

Hey everyone,

Just wanted to see if there are any other JEE aspirants on this forum currently prepping for it[mention year if you can]

I am actually entering 10th this year and have decided to try for it...So this year is just going to go in me strengthening my math (IOQM level (heard its enough till Mains part, so will start from there) for the problem solving part, and learn some topics from 11th and 12th as well)

It would be great to connect with others who are going through the same thing - share study strategies, tips, resources, discuss, and maybe even form study groups(not sure how to tho :maybe: ) and motivate each other ig?. :D
So yea, cya later

10 replies

mikkymini2

Thursday at 2:54 PM

Idiot_of_the64squares

2 hours ago

Inequalities

sqing 0

2 hours ago

Let $a,b,c\in [0,1]$ . Prove that
$(a+b+c)\left(\frac{1}{a^2+3}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{19}{4}$ $(a+b+c)\left(\frac{1}{a^2+ 4}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{23}{5}$ $(a+b+c)\left(\frac{1}{a^2+ \frac{5}{2}}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{34}{7}$ $(a+b+c)\left(\frac{1}{a^2+ \frac{7}{2}}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{14}{3}$

0 replies

sqing

2 hours ago

0 replies

lcm(1,2,3,...,n)

lgx57 5

N 3 hours ago by Kempu33334

Let $M=\operatorname{lcm}(1,2,3,\cdots,n)$ .Estimate the range of $M$ .

5 replies

lgx57

Apr 9, 2025

Kempu33334

3 hours ago

Inequality

math2000 7

N 3 hours ago by imnotgoodatmathsorry

Let $a,b,c>0$ .Prove that $\dfrac{1}{(a+b)\sqrt{(a+2c)(b+2c)}}>\dfrac{3}{2(a+b+c)^2}$

7 replies

math2000

Jan 22, 2021

imnotgoodatmathsorry

3 hours ago

No more topics!

variation of billiard, circular hockey rink PUMaC 2017 Live_Round 5.2

parmenides51 2

N Jan 1, 2022 by parmenides51

Kapil the Kingly, Casimir the Conjurer, and Zack the Zombie all stand in clockwise order on the perimeter of a circular hockey rink with radius $10$ meters. The central angle between Kapil and Casimir is $30^o$ and the central angle between Casimir and Zack is $130^o$ . Kapil smacks a hockey puck in a certain direction, and the puck bounces against the edge of the rink without losing velocity. Kapil aims his shot so that it takes the minimal number of bounces to reach both Casimir and Zack. If the total number of meters traveled by the puck when it has reached the final person is a $\sin (b^o)$ , where a and b are positive integers and $0 \le b < 90^o$ , find $a + b$ .

Note: the puck bounces off of the rink according to the rule “angle of incidence equals angle of reflection” with respect to the tangent line.

2 replies

parmenides51

Dec 31, 2021

parmenides51

Jan 1, 2022

variation of billiard, circular hockey rink PUMaC 2017 Live_Round 5.2

G H J

Reveal topic tags

geometry

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parmenides51

30630 posts

parmenides51

#1 h Dec 31, 2021, 8:26 AM

Y by

This post has been edited 1 time. Last edited by parmenides51, Dec 31, 2021, 11:40 AM

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HappyMathEducation

181 posts

HappyMathEducation

#2 h Dec 31, 2021, 5:03 PM

Y by

solution?

Kapil should shoot at the point that form $130^\circ$ central angle with Kapil on the clockwise direction. That is 3 bounces in total.

$4 \times 20 \times \sin(65^\circ) \Rightarrow a+b = \boxed {145}$

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parmenides51

30630 posts

parmenides51

#3 h Jan 1, 2022, 9:11 PM

Y by

Besides this above, here was another solution accepted, due to a different interpretation of the problem.

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