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Contests & Programs AMC and other contests, summer programs, etc.

AMC and other contests, summer programs, etc.

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Contests & Programs AMC and other contests, summer programs, etc.

AMC and other contests, summer programs, etc.

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2-var inequality

sqing 1

N 42 minutes ago by sqing

Source: Own

Let $a,b> 0 , ab(a+b+1) =3.$ Prove that $\frac{1}{a^2}+\frac{1}{b^2}+\frac{24}{(a+b)^2} \geq 8$ $\frac{a}{b^2}+\frac{b}{a^2}+\frac{49}{(a+ b)^2} \geq \frac{57}{4}$ Let $a,b> 0 , (a+b)(ab+1) =4.$ Prove that $\frac{1}{a^2}+\frac{1}{b^2}+\frac{40}{(a+b)^2} \geq 12$ $\frac{a}{b^2}+\frac{b}{a^2}+\frac{76}{(a+ b)^2} \geq 21$

1 reply

sqing

an hour ago

sqing

42 minutes ago

Balkan Mathematical Olympiad

ABCD1728 1

N an hour ago by ABCD1728

Can anyone provide the PDF version of the book "Balkan Mathematical Olympiads" by Mircea Becheanu and Bogdan Enescu (published by XYZ press in 2014), thanks!

1 reply

ABCD1728

Yesterday at 11:27 PM

ABCD1728

an hour ago

Functional equation

shactal 0

an hour ago

Let $f:\mathbb R\to \mathbb R$ a function satifying $f(x+2xy) = f(x) + 2f(xy)$ for all $x,y\in \mathbb R$ .
If $f(1991)=a$ , then what is $f(1992)$ , the answer is in terms of $a$ .

0 replies

shactal

an hour ago

0 replies

area of quadrilateral

AlanLG 1

N an hour ago by Altronrren

Source: 3rd National Women´s Contest of Mexican Mathematics Olympiad 2024 , level 1+2 p5

Consider the acute-angled triangle $ABC$ . The segment $BC$ measures 40 units. Let $H$ be the orthocenter of triangle $ABC$ and $O$ its circumcenter. Let $D$ be the foot of the altitude from $A$ and $E$ the foot of the altitude from $B$ . Additionally, point $D$ divides the segment $BC$ such that $\frac{BD}{DC} = \frac{3}{5}$ . If the perpendicular bisector of segment $AC$ passes through point $D$ , calculate the area of quadrilateral $DHEO$ .

1 reply

AlanLG

Jun 14, 2024

Altronrren

an hour ago

Inspired by 2025 SXTB

sqing 1

N an hour ago by sqing

Source: Own

Let $a,b$ be real number such that $a^2+b^2=\frac12.$ Prove that
$-\frac{\sqrt{9+6\sqrt 3}}{2}\leq(a+1)^2- (b-1)^2\leq\frac{\sqrt{9+6\sqrt 3}}{2}$ Let $x$ be real number . Prove that
$-\frac{2\sqrt 2}{3}\leq \frac{x}{x^2+1}+ \frac{ 2x}{x^2+4} \leq\frac{2\sqrt 2}{3}$

1 reply

sqing

2 hours ago

sqing

an hour ago

IMO Shortlist 2014 G2

hajimbrak 14

N 2 hours ago by ezpotd

Let $ABC$ be a triangle. The points $K, L,$ and $M$ lie on the segments $BC, CA,$ and $AB,$ respectively, such that the lines $AK, BL,$ and $CM$ intersect in a common point. Prove that it is possible to choose two of the triangles $ALM, BMK,$ and $CKL$ whose inradii sum up to at least the inradius of the triangle $ABC$ .

Proposed by Estonia

14 replies

hajimbrak

Jul 11, 2015

ezpotd

2 hours ago

Divisiblity...

TUAN2k8 0

2 hours ago

Source: Own

Let $m$ and $n$ be two positive integer numbers such that $m \le n$ .Prove that $\binom{n}{m}$ divides $lcm(1,2,...,n)$

0 replies

TUAN2k8

2 hours ago

0 replies

interesting diophantiic fe in natural numbers

skellyrah 4

N 2 hours ago by aidan0626

Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that for all $m, n \in \mathbb{N}$ ,
$mn + f(n!) = f(f(n))! + n \cdot \gcd(f(m), m!).$

4 replies

skellyrah

Yesterday at 8:01 AM

aidan0626

2 hours ago

IMO 2010 Problem 4

mavropnevma 128

N 2 hours ago by ezpotd

Let $P$ be a point interior to triangle $ABC$ (with $CA \neq CB$ ). The lines $AP$ , $BP$ and $CP$ meet again its circumcircle $\Gamma$ at $K$ , $L$ , respectively $M$ . The tangent line at $C$ to $\Gamma$ meets the line $AB$ at $S$ . Show that from $SC = SP$ follows $MK = ML$ .

Proposed by Marcin E. Kuczma, Poland

128 replies

mavropnevma

Jul 8, 2010

ezpotd

2 hours ago

Simple Geometry

AbdulWaheed 5

N 2 hours ago by Adywastaken

Source: EGMO

Try to avoid Directed angles
Let ABC be an acute triangle inscribed in circle $\Omega$ . Let $X$ be the midpoint of the arc $\overarc{BC}$ not containing $A$ and define $Y, Z$ similarly. Show that the orthocenter of $XYZ$ is the incenter $I$ of $ABC$ .

5 replies

AbdulWaheed

May 23, 2025

Adywastaken

2 hours ago

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