angle chasing, circle with diameter, tangents related

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0 replies

jlacosta

Apr 2, 2025

0 replies

Interesting inequalities

sqing 1

N 4 minutes ago by lbh_qys

Source: Own

Let $a,b$ be real numbers . Prove that
$-\frac{1}{12}\leq \frac{ab+a+b+1}{(a^2+3)(b^2+3)}\leq \frac{1}{4}$ $-\frac{5}{96}\leq \frac{ab+a+b+2}{(a^2+4)(b^2+4)}\leq \frac{1}{5}$ $-\frac{1}{16}\leq \frac{ab+a+b+1}{(a^2+4)(b^2+4)}\leq \frac{3+\sqrt 5}{32}$ $-\frac{1}{18}\leq \frac{ab+a+b+3}{(a^2+3)(b^2+3)}\leq \frac{2+\sqrt[3] 2+\sqrt[3] {4}}{12}$

1 reply

+1 w

sqing

39 minutes ago

lbh_qys

4 minutes ago

FE solution too simple?

Yiyj1 3

N 40 minutes ago by AshAuktober

Source: 101 Algebra Problems from the AMSP

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that the equality $f(f(x)+y) = f(x^2-y)+4f(x)y$ holds for all pairs of real numbers $(x,y)$ .

My solution

Clearly, $f(x)=0$ is an obvious solution. Now, let $y=0$ . Then, we have $f(f(x)) = f(x^2),$ or $f(x)=x^2$ . Therefore, the solutions are $f(x)=0, x^2$ .

I feel like my solution is too simple. Is there something I did wrong or something I missed?

3 replies

Yiyj1

2 hours ago

AshAuktober

40 minutes ago

Inspired by Ruji2018252

sqing 3

N an hour ago by sqing

Source: Own

Let $a,b,c>1$ and $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=a+b+c-8$ . Prove that
$ab+bc+ca+a+b+c \leq 36$ $ab+bc+ ca\leq 27$

3 replies

sqing

3 hours ago

sqing

an hour ago

Interesting inequalities

sqing 1

N 2 hours ago by sqing

Source: Own

Let $a,b$ be real numbers . Prove that
$-\frac{11+8\sqrt 2}{7}\leq \frac{ab+a+b-1}{(a^2+a+1)(b^2+b+1)}\leq \frac{2}{9}$ $-\frac{37+13\sqrt{13}}{414}\leq \frac{ab+a+b-2}{(a^2+a+4)(b^2+b+4)}\leq \frac{3}{50}$ $-\frac{5\sqrt 5+9}{22}\leq \frac{ab+a+b-2}{(a^2+a+2)(b^2+b+2)}\leq \frac{5\sqrt 5-9}{22}$

1 reply

sqing

2 hours ago

sqing

2 hours ago

No more topics!

angle chasing, circle with diameter, tangents related

parmenides51 1

N Feb 15, 2021 by badumtsss

Source: 2018 SPbU finals, grades 10-11 p3 v5 - Saint Petersburg State University School Olympiad

You are given an acute-angled triangle $ABC$ . The circle with diameter $BC$ meets sides $AB$ and $AC$ at points D and $E$ , respectively. The tangents drawn to the circle at points $D$ and $E$ meet at point $K$ . Find the angle between lines $AK$ and $BC$ .

1 reply

parmenides51

Feb 15, 2021

badumtsss

Feb 15, 2021

angle chasing, circle with diameter, tangents related

G H J

Reveal topic tags

geometry

angles

Tangents

G H BBookmark kLocked kLocked NReply

Source: 2018 SPbU finals, grades 10-11 p3 v5 - Saint Petersburg State University School Olympiad

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parmenides51

30630 posts

parmenides51

#1 h Feb 15, 2021, 5:42 PM • 1 Y

Y by son7

Z K Y

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badumtsss

61 posts

badumtsss

#2 h Feb 15, 2021, 10:44 PM • 3 Y

Y by Mango247, Mango247, Mango247

We see that $D$ and $E$ are the feet of altitudes from $C$ and $B$ , respectively. so $\angle KED = \angle KDE = \angle DCE =90-\angle A$ so $K$ is the circumcenter of $ADE$ . We know that $DE$ is the antiparallel of $BC$ , so line $AK$ is actually the altitude in $\triangle ABC$ so the angle between $AK$ and $BC$ is $90^{\circ}$

Z K Y

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