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Contests & Programs 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.

3 M G

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geometric transformation

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Prove the inequality

Butterfly 0

2 hours ago

Let $a,b,c$ be real numbers such that $a+b+c=3$ . Prove $a^3b+b^3c+c^3a\le \frac{9}{32}(63+5\sqrt{105}).$

0 replies

Butterfly

2 hours ago

0 replies

Functional equation

shactal 1

N 2 hours ago by ariopro1387

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 $a$ .

1 reply

shactal

4 hours ago

ariopro1387

2 hours ago

interesting diophantiic fe in natural numbers

skellyrah 5

N 2 hours ago by skellyrah

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

5 replies

skellyrah

Yesterday at 8:01 AM

skellyrah

2 hours ago

Non-linear Recursive Sequence

amogususususus 3

N 2 hours ago by SunnyEvan

Given $a_1=1$ and the recursive relation
$a_{i+1}=a_i+\frac{1}{a_i}$ for all natural number $i$ . Find the general form of $a_n$ .

Is there any way to solve this problem and similar ones?

3 replies

amogususususus

Jan 24, 2025

SunnyEvan

2 hours ago

Inspired by 2025 Beijing

sqing 6

N 3 hours ago by sqing

Source: Own

Let $a,b,c,d >0$ and $(a^2+b^2+c^2)(b^2+c^2+d^2)=36.$ Prove that
$ab^2c^2d \leq 8$ $a^2bcd^2 \leq 16$ $ab^3c^3d \leq \frac{2187}{128}$ $a^3bcd^3 \leq \frac{2187}{32}$

6 replies

sqing

Yesterday at 4:56 PM

sqing

3 hours ago

Serbian selection contest for the IMO 2025 - P4

OgnjenTesic 2

N 3 hours ago by sqing-inequality-BUST

Source: Serbian selection contest for the IMO 2025

For a permutation $\pi$ of the set $A = \{1, 2, \ldots, 2025\}$ , define its colorfulness as the greatest natural number $k$ such that:
- For all $1 \le i, j \le 2025$ , $i \ne j$ , if $|i - j| < k$ , then $|\pi(i) - \pi(j)| \ge k$ .
What is the maximum possible colorfulness of a permutation of the set $A$ ? Determine how many such permutations have maximal colorfulness.

Proposed by Pavle Martinović

2 replies

OgnjenTesic

May 22, 2025

sqing-inequality-BUST

3 hours ago

Nice "if and only if" function problem

ICE_CNME_4 14

N 3 hours ago by wh0nix

Let $f : [0, \infty) \to [0, \infty)$ , $f(x) = \dfrac{ax + b}{cx + d}$ , with $a, d \in (0, \infty)$ , $b, c \in [0, \infty)$ . Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
$f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.$ (For $n \in \mathbb{N}^*$ and $x \geq 0$ , the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$ . )

Please do it at 9th grade level. Thank you!

14 replies

ICE_CNME_4

Friday at 7:23 PM

wh0nix

3 hours ago

2-var inequality

sqing 1

N 3 hours 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

4 hours ago

sqing

3 hours ago

Balkan Mathematical Olympiad

ABCD1728 1

N 3 hours 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

3 hours ago

area of quadrilateral

AlanLG 1

N 4 hours 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

4 hours ago

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