2010 JBMO Shortlist G2

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies

jlacosta

Apr 2, 2025

0 replies

inquequality

ngocthi0101 8

N 15 minutes ago by sqing

let $a,b,c > 0$ prove that
$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$

8 replies

2 viewing

ngocthi0101

Sep 26, 2014

sqing

15 minutes ago

Tangent.

steven_zhang123 2

N 18 minutes ago by AshAuktober

Source: China TST 2001 Quiz 6 P1

In $\triangle ABC$ with $AB > BC$ , a tangent to the circumcircle of $\triangle ABC$ at point $B$ intersects the extension of $AC$ at point $D$ . $E$ is the midpoint of $BD$ , and $AE$ intersects the circumcircle of $\triangle ABC$ at $F$ . Prove that $\angle CBF = \angle BDF$ .

2 replies

steven_zhang123

Mar 23, 2025

AshAuktober

18 minutes ago

IMO ShortList 1998, algebra problem 1

orl 37

N 33 minutes ago by Marcus_Zhang

Source: IMO ShortList 1998, algebra problem 1

Let $a_{1},a_{2},\ldots ,a_{n}$ be positive real numbers such that $a_{1}+a_{2}+\cdots +a_{n}<1$ . Prove that

$\frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}.$

37 replies

orl

Oct 22, 2004

Marcus_Zhang

33 minutes ago

Integer Coefficient Polynomial with order

MNJ2357 9

N 42 minutes ago by v_Enhance

Source: 2019 Korea Winter Program Practice Test 1 Problem 3

Find all polynomials $P(x)$ with integer coefficients such that for all positive number $n$ and prime $p$ satisfying $p\nmid nP(n)$ , we have $ord_p(n)\ge ord_p(P(n))$ .

9 replies

MNJ2357

Jan 12, 2019

v_Enhance

42 minutes ago

No more topics!

2010 JBMO Shortlist G2

parmenides51 3

N Feb 21, 2020 by Math-wiz

Source: 2010 JBMO Shortlist G2

Consider a triangle ${ABC}$ and let ${M}$ be the midpoint of the side ${BC.}$ Suppose ${\angle MAC=\angle ABC}$ and ${\angle BAM=105^{\circ}.}$ Find the measure of ${\angle ABC}$ .

3 replies

parmenides51

Oct 8, 2017

Math-wiz

Feb 21, 2020

2010 JBMO Shortlist G2

G H J

Reveal topic tags

geometry

JBMO

G H BBookmark kLocked kLocked NReply

Source: 2010 JBMO Shortlist G2

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parmenides51

30629 posts

parmenides51

#1 h Oct 8, 2017, 12:54 PM • 2 Y

Y by Adventure10, Mango247

Consider a triangle ${ABC}$ and let ${M}$ be the midpoint of the side ${BC.}$ Suppose ${\angle MAC=\angle ABC}$ and ${\angle BAM=105^{\circ}.}$ Find the measure of ${\angle ABC}$ .

Z K Y

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mUnLoI

2 posts

mUnLoI

#2 h Feb 12, 2020, 5:59 PM • 2 Y

Y by Adventure10, Wild

Can someone solve this problem? Pls

Z K Y

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augustin_p

170 posts

augustin_p

#3 h Feb 21, 2020, 9:52 AM • 2 Y

Y by microsoft_office_word, Adventure10

Denote $\angle ABC = x$ , then $\angle AMC = 105+x$ . We have $\Delta ABC \sim \Delta AMC \implies\frac{AM}{AB}=\frac{CM}{CA}=\frac{CA}{BC} \implies \sqrt2CA=BC.$ Using the law of sines in $\Delta AMC$ we have $\frac{\sin{x}}{CM}=\frac{\sin(105+x)}{AC} \iff \frac{2\sin{x}}{BC}=\frac{\sin(75-x)}{AC} \iff \sqrt2\sin{x}=\sin(75-2x)\iff\tan{x}=\frac{1}{\sqrt3} \iff x=30.$