circle C_17 larger than other 16 circles?(2015 UQ/QAMT PS Competition 11-12 p3)

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0 replies

jlacosta

Apr 2, 2025

0 replies

Eazy equation clap

giangtruong13 0

4 hours ago

Find all $x,y,z$ satisfy that: $\frac{x}{y+z}=2x-1; \frac{y}{x+z}=3y-1;\frac{z}{x+y}=5x-1$

0 replies

giangtruong13

4 hours ago

0 replies

inequality

revol_ufiaw 3

N 5 hours ago by MS_asdfgzxcvb

Prove that that for any real $x \ge 0$ and natural number $n$ ,
$x^n (n+1)^{n+1} \le n^n (x+1)^{n+1}.$

3 replies

revol_ufiaw

6 hours ago

MS_asdfgzxcvb

5 hours ago

How to judge a number is prime or not?

mingzhehu 0

5 hours ago

A=（10X1+1）(10X+1），X1，X∈N+
B=（10 X1+3）（10X+7），X∈N，X1∈N
C=(10 X1+9)(10X+9), X∈N，X1∈N
D=(10 X1+1)(10X+3), X1∈N+，X∈N
E=（10 X1+7）(10X+9），X∈N，X1∈N
F=（10 X1+1）（10X+7），X1∈N+，X∈N
G=（10 X1+3）（10X+9），X∈N，X1∈N
H=（10 X1+1）10X+9），X1∈N+，X∈N
I=（10 X1+3）（10X+3），X1∈N，X∈N
J=( 10X1+7)(10X+7),X∈N，X1∈N

For any natural number P∈{P=10N+1，n∈N}，make P=A or B or C
If P can make the roots of function group(ABC) without any root group completely made up of integer, P will be a prime
For any natural number P∈{P=10N+3，n∈N}，make P=D or E
If P can make the roots of function group(DE) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+7，n∈N}，make P=F or G
If P can make the roots of function group(FG) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+9，n∈N}，make P=H or I or J
If P can make the roots of function group(GIJ) without any root group completely made up
of integer, P will be a prime

0 replies

mingzhehu

5 hours ago

0 replies

What is an isogonal conjugate and why is it useful?

EaZ_Shadow 6

N 5 hours ago by maxamc

What is an isogonal conjugate and why is it useful? People use them in Olympiad geometry proofs but I don’t understand why and what is the purpose, as it complicates me because of me not understanding it.

6 replies

EaZ_Shadow

Dec 28, 2024

maxamc

5 hours ago

No more topics!

circle C_17 larger than other 16 circles?(2015 UQ/QAMT PS Competition 11-12 p3)

parmenides51 1

N Jan 22, 2021 by sotpidot

Take a circle and inscribe $4$ non-overlapping equal sized circles $A, B, C, D$ such that $A$ is tangent to $B, B$ is tangent to $C, C$ is tangent to $D$ and $D$ is tangent to $A$ such that each is also tangent to the large circle. In each of these $4$ circles, inscribe $4$ smaller tangent circles in exactly the same way, to get $16$ small circles. Finally, inscribe a small circle $C_{17}$ in the space around the centre of the large circle and tangent to each of $A, B, C, D$ . Which is larger, $C_{17}$ or one of the $16$ circles inside $A, B, C, D$ ?

1 reply

parmenides51

Jan 22, 2021

sotpidot

Jan 22, 2021

circle C_17 larger than other 16 circles?(2015 UQ/QAMT PS Competition 11-12 p3)

G H J

Reveal topic tags

geometry

tangent

circles

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parmenides51

30629 posts

parmenides51

#1 h Jan 22, 2021, 5:52 AM • 1 Y

Y by Mango247

This post has been edited 1 time. Last edited by parmenides51, Jan 22, 2021, 5:59 AM

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sotpidot

290 posts

sotpidot

#2 h Jan 22, 2021, 6:53 AM • 1 Y

Y by Mango247

solution

Let the radius of the original circle be $r$ . We know that the centers of $A, B, C, D$ form a square, by symmetry. Let these radii be $r_1$ . Then we have $\sqrt{2}(r - r_1) = 2r_1$ . Solving, we get $r_1 = \frac{r}{\sqrt2 + 1}$ . Let $r_2$ be the radii of the $16$ smaller circles. Then we similarly have $r_2 = \frac{r_1}{\sqrt2 + 1}$ . Substituting for $r_1$ gives $r_2 = \frac{r}{3 + \sqrt2}$ .

Let $C_{17}$ have radius $r_3$ . We see that $r_3 = r - 2r_1 = \frac{r \cdot (\sqrt2 - 1)}{\sqrt2 + 1}$ .

We claim that $r_2 > r_3$ .

We have $\frac{r}{3 + \sqrt2} > \frac{r \cdot (\sqrt2 - 1)}{\sqrt2 + 1}$ . Dividing by $r$ and multiplying by denominators gives us $\sqrt2 + 1 > 2\sqrt2 - 1$ , which simplifies to $2 > \sqrt2$ . Thus, $r_2 > r_3$ and the $16$ circles inside of $A, B, C, D$ are each bigger than $C_{17}$

This post has been edited 3 times. Last edited by sotpidot, Jan 22, 2021, 6:54 AM
Reason: im bad at latex

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