r /R <= cos (180^o/n), 2 circles and a convex n-gon

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0 replies

jlacosta

May 1, 2025

0 replies

Interesting inequalities

sqing 1

N 8 minutes ago by sqing

Source: Own

Let $a,b>0$ and $a+b\leq 1$ . Prove that
$\left(\frac{4}{a^2}-1\right)\left(\frac{4}{b^2}-1\right)-k\left(\frac{a}{b}+\frac{b}{a}\right) \geq 225-2k$ Where $104\geq k\in N^+.$
$\left(\frac{4}{a^2}-1\right)\left(\frac{4}{b^2}-1\right) \geq 225$ $\left(\frac{4}{a^2}-1\right)\left(\frac{4}{b^2}-1\right)-106\left(\frac{a}{b}+\frac{b}{a}\right) \geq \frac{155}{12}$ $\left(\frac{4}{a^2}-1\right)\left(\frac{4}{b^2}-1\right)-108\left(\frac{a}{b}+\frac{b}{a}\right) \geq \frac{26}{3}$ $\left(\frac{4}{a^2}-1\right)\left(\frac{4}{b^2}-1\right)-110\left(\frac{a}{b}+\frac{b}{a}\right) \geq \frac{17}{4}$

1 reply

1 viewing

sqing

18 minutes ago

sqing

8 minutes ago

R to R, with x+f(xy)=f(1+f(y))x

NicoN9 2

N 20 minutes ago by dangerousliri

Source: Own.

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $x+f(xy)=f(1+f(y))x$ for all $x, y\in \mathbb{R}$ .

2 replies

NicoN9

39 minutes ago

dangerousliri

20 minutes ago

Combi Geo

Adywastaken 2

N an hour ago by lakshya2009

Source: NMTC 2024/8

A regular polygon with $100$ vertices is given. To each vertex, a natural number from the set $\{1,2,\dots,49\}$ is assigned. Prove that there are $4$ vertices $A, B, C, D$ such that if the numbers $a, b, c, d$ are assigned to them respectively, then $a+b=c+d$ and $ABCD$ is a parallelogram.

2 replies

1 viewing

Adywastaken

Yesterday at 3:58 PM

lakshya2009

an hour ago

Brilliant guessing game on triples

Assassino9931 1

N an hour ago by Sardor_lil

Source: Al-Khwarizmi Junior International Olympiad 2025 P8

There are $100$ cards on a table, flipped face down. Madina knows that on each card a single number is written and that the numbers are different integers from $1$ to $100$ . In a move, Madina is allowed to choose any $3$ cards, and she is told a number that is written on one of the chosen cards, but not which specific card it is on. After several moves, Madina must determine the written numbers on as many cards as possible. What is the maximum number of cards Madina can ensure to determine?

Shubin Yakov, Russia

1 reply

Assassino9931

Yesterday at 9:46 AM

Sardor_lil

an hour ago

No more topics!

r /R <= cos (180^o/n), 2 circles and a convex n-gon

parmenides51 1

N Nov 19, 2021 by rafaello

Source: 2010 Baltic Way Shortlist SWE4 BW https://artofproblemsolving.com/community/c2641329_baltic_way_shortlist__geometry

A convex $n$ -gon encloses a circle of radius $r$ , and is itself enclosed within a circle of radius $R$ . Prove that
$\frac{r}{R} \le \cos \frac{180^o}{n}.$

1 reply

parmenides51

Nov 18, 2021

rafaello

Nov 19, 2021

r /R <= cos (180^o/n), 2 circles and a convex n-gon

G H J

Reveal topic tags

convex polygon

geometric inequality

geometry

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Source: 2010 Baltic Way Shortlist SWE4 BW https://artofproblemsolving.com/community/c2641329_baltic_way_shortlist__geometry

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parmenides51

30652 posts

parmenides51

#1 h Nov 18, 2021, 4:10 PM

Y by

A convex $n$ -gon encloses a circle of radius $r$ , and is itself enclosed within a circle of radius $R$ . Prove that
$\frac{r}{R} \le \cos \frac{180^o}{n}.$

Z K Y

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rafaello

1079 posts

rafaello

#2 h Nov 19, 2021, 10:32 PM

Y by

Let $A_1A_2\ldots A_n$ be the $n$ -gon with $O$ as the center of circle $\omega$ , which encloses the $n$ -gon. Let $C_i=\overline{OA_i}\cap \omega$ . Let $B_1,B_2,\ldots, B_n$ be the feet from $O$ to $\overline{C_1C_2},\overline{C_2C_3},\ldots,\overline{C_nC_1}$ , respectively.
$\begin{align*} R\cos{\frac{180^\circ}{n}}&\geq R\cdot \frac{\cos{\angle C_1OB_1}+\cos{\angle B_1OC_2}+\ldots +\cos{\angle B_nOC_1}}{2n}\\&=\frac{OB_1+OB_2+\ldots+OB_n}{n}\\&\geq \frac{2S_{C_1C_2\ldots C_n}}{C_1C_2+\ldots + C_nC_1}\\&\geq r, \end{align*}$ where the the first inequality comes from Jensen's inequality and the second from Chebyshev's inequality.

This post has been edited 2 times. Last edited by rafaello, Dec 28, 2021, 12:36 AM
Reason: fixed

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