A4 BMO SHL 2024

by mihaig, May 15, 2025, 5:33 AM

Let $a\ge b\ge c\ge0$ be real numbers such that $ab+bc+ca=3.$
Prove
$$3+\left(2-\sqrt 3\right)\cdot\frac{\left(b-c\right)^2}{b+\left(\sqrt 3-1\right)c}\leq a+b+c.$$Prove that if $k<\sqrt 3-1$ is a positive constant, then
$$3+\left(2-\sqrt 3\right)\cdot\frac{\left(b-c\right)^2}{b+kc}\leq a+b+c$$is not always true
inequalities
L

Geometry

by AlexCenteno2007, May 15, 2025, 4:42 AM

Let ABC be an acute triangle. The altitudes from B and C intersect the sides AC and AB at E and F, respectively. The internal bisector of ∠A intersects BE and CF at T and S, respectively. The circles with diameters AT and AS intersect the circumcircle of ABC at X and Y, respectively. Prove that XY, EF, and BC meet at the exsimilicenter of BTX and CSY
geometry
L

Inspired by xytunghoanh

by sqing, May 15, 2025, 3:04 AM

Let $ a,b,c\ge 0, a^2 +b^2 +c^2 =3. $ Prove that
$$ \sqrt 3 \leq a+b+c+ ab^2 + bc^2+ ca^2\leq 6$$Let $ a,b,c\ge 0, a+b+c+a^2 +b^2 +c^2 =6. $ Prove that
$$ ab+bc+ca+ ab^2 + bc^2+ ca^2 \leq 6$$
inequalities proposed
algebra
L

inequality

by danilorj, May 14, 2025, 9:08 PM

Let $a, b, c$ be nonnegative real numbers such that $a + b + c = 3$. Prove that
\[ \frac{a}{4 - b} + \frac{b}{4 - c} + \frac{c}{4 - a} + \frac{1}{16}(1 - a)^2(1 - b)^2(1 - c)^2 \leq 1, \]and determine all such triples $(a, b, c)$ where the equality holds.
inequalities
L

Based on IMO 2024 P2

by Miquel-point, May 14, 2025, 6:15 PM

Prove that for any positive integers $a$, $b$, $c$ and $d$ there exists infinitely many positive integers $n$ for which $a^n+bc$ and $b^{n+d}-1$ are not relatively primes.

Proposed by Géza Kós
komal
number theory
L

Nice one

by imnotgoodatmathsorry, May 2, 2025, 2:10 PM

With $x,y,z >0$.Prove that: $\frac{xy}{4y+4z+x} + \frac{yz}{4z+4x+y} +\frac{zx}{4x+4y+z} \le \frac{x+y+z}{9}$
inequalities
inequalities proposed
inequalities unsolved
L

Dou Fang Geometry in Taiwan TST

by Li4, Apr 26, 2025, 5:03 AM

Let $\omega$ and $\Omega$ be the incircle and circumcircle of the acute triangle $ABC$, respectively. Draw a square $WXYZ$ so that all of its sides are tangent to $\omega$, and $X$, $Y$ are both on $BC$. Extend $AW$ and $AZ$, intersecting $\Omega$ at $P$ and $Q$, respectively. Prove that $PX$ and $QY$ intersects on $\Omega$.

Proposed by kyou46, Li4, Revolilol.
geometry
circumcircle
L

functional equation

by hanzo.ei, Apr 6, 2025, 6:08 PM

Find all functions \( f : \mathbb{R} \to \mathbb{R} \) satisfying the equation
\[ (f(x+y))^2= f(x^2) + f(2xf(y) + y^2), \quad \forall x, y \in \mathbb{R}. \]
functional equation
algebra
L

Equal segments in a cyclic quadrilateral

by a_507_bc, Jul 29, 2023, 12:15 PM

Consider a cyclic quadrilateral $ABCD$ in which $BC = CD$ and $AB < AD$. Let $E$ be a point on the side $AD$ and $F$ a point on the line $BC$ such that $AE = AB = AF$. Prove that $EF \parallel BD$.
geometry
cyclic quadrilateral
L

Iran geometry

by Dadgarnia, Apr 7, 2018, 3:26 PM

In triangle $ABC$ let $M$ be the midpoint of $BC$. Let $\omega$ be a circle inside of $ABC$ and is tangent to $AB,AC$ at $E,F$, respectively. The tangents from $M$ to $\omega$ meet $\omega$ at $P,Q$ such that $P$ and $B$ lie on the same side of $AM$. Let $X \equiv PM \cap BF $ and $Y \equiv QM \cap CE $. If $2PM=BC$ prove that $XY$ is tangent to $\omega$.

Proposed by Iman Maghsoudi
This post has been edited 2 times. Last edited by Dadgarnia, Apr 8, 2018, 10:46 AM
geometry
Pole and Polar
Pappus
L

Convex and concave functions in Real numbers -- Basic 1

by adityaguharoy, Mar 1, 2018, 1:44 PM

Convex functions
Let $f : \mathbb{R} \to \mathbb{R}$ be a function, and let $a,b$ be two real numbers with $a<b$. Then we say that $f$ is a convex function on the interval $[a,b]$ if and only if the following is true :
Given any $t \in [0,1]$ , and , any $x_1 , x_2 \in [a,b]$ then,
$$\boxed{f(tx_1 + (1-t)x_2) \le t \cdot f(x_1) + (1-t) \cdot f(x_2)}$$And we say that $f$ is strictly convex on $[a,b]$ if the above inequality is strict whenever $x_1 \ne x_2$ and $t \in (0,1)$.

Concave functions
Let $f : \mathbb{R} \to \mathbb{R}$ be a function, and let $a,b$ be two real numbers with $a<b$. Then we say that $f$ is a concave function on the interval $[a,b]$ if and only if the following is true :
Given any $t \in [0,1]$ , and , any $x_1 , x_2 \in [a,b]$ then,
$$\boxed{f(tx_1 + (1-t)x_2) \ge t \cdot f(x_1) + (1-t) \cdot f(x_2)}$$And we say that $f$ is strictly concave on $[a,b]$ if the above inequality is strict whenever $x_1 \ne x_2$ and $t \in (0,1)$.

Quick exercises



Let us celebrate
This post has been edited 2 times. Last edited by adityaguharoy, Mar 4, 2018, 7:25 AM
Convex Functions
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definitions

Relaxing the to-be satisfied conditions for Squeezing Monotone sequences

by adityaguharoy, Feb 28, 2018, 4:09 PM

The Sandwich theorem (or Squeeze Principle ) for sequences of real numbers says that :
If $\{x_n\}_{n=1}^{\infty}$, $\{y_n\}_{n=1}^{\infty}$ and $\{z_n\}_{n=1}^{\infty}$ be three sequences of real numbers which obeys all the following :
$\bullet$ $x_n \le y_n \le z_n$ $, \forall n \in \mathbb{Z}^+$
$\bullet$ Both the sequences $\{x_n\}_{n=1}^{\infty}$ and $\{z_n\}_{n=1}^{\infty}$ converges
$\bullet$ $\lim_{n \to \infty} ( x_n ) = \lim_{n \to \infty} ( z_n )$
Then, $\{y_n\}_{n=1}^{\infty}$ must converge, and also obey
$$\boxed{\lim_{n \to \infty} (x_n) = \lim_{n \to \infty} (y_n) = \lim_{n \to \infty} (z_n)}$$Forgetting the third condition (as given above) can lead us to errorenous results.
For example : If we define $\{x_n\}_{n=1}^{\infty}$ as $x_n = 1 + \frac{1}{n}$ $, \forall n \in \mathbb{Z}^+$ and, we define $\{z_n\}_{n=1}^{\infty}$ as $z_n = 5 + \frac{1}{n}$ $, \forall n \in \mathbb{Z}^+$, and we define the sequence $\{y_n\}_{n=1}^{\infty}$ as $y_n = 2+ (-1)^n$ $, \forall n \in \mathbb{Z}^+$, then $x_n \le y_n \le z_n$ $,\forall n \in \mathbb{Z}^+$ and both the sequences $\{x_n\}_{n=1}^{\infty}$ and $\{z_n\}_{n=1}^{\infty}$ converges, but however, $\{y_n\}_{n=1}^{\infty}$ diverges.


But, it seems we can relax the conditions for some special type of sequences. Let us see how, we can relax it for monotone sequences.
Note, that the Monotone Convergence Theorem (of Bolzano Weirestrass) says that :
Every monotone bounded sequence of real numbers must converge.

Thus, if $\{x_n\}_{n=1}^{\infty}$ , $\{y_n\}_{n=1}^{\infty}$ , and , $\{z_n\}_{n=1}^{\infty}$ be three monotone sequences such that
$\bullet$ $x_n \le y_n \le z_n$ $\forall n \in \mathbb{Z}^+$
$\bullet$ Both the sequences $\{x_n\}_{n=1}^{\infty}$ , and , $\{z_n\}_{n=1}^{\infty}$ converges
Then,
The sequence $\{y_n\}_{n=1}^{\infty}$ must converge , and ,
$$\boxed{\lim_{n \to \infty} (x_n) \le \lim_{n \to \infty} (y_n) \le \lim_{n \to \infty} (z_n)}$$
This post has been edited 1 time. Last edited by adityaguharoy, Mar 1, 2018, 1:41 PM
real analysis
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