Inspired by hlminh

by sqing, Apr 22, 2025, 4:43 AM

Let $a,b,c$ be real numbers such that $a^2+b^2+c^2=1.$ Prove that $|a-kb|+|b-kc|+|c-ka|\leq \sqrt{3k^2+2k+3}$ Where $k\geq 0 .$

inequalities proposed

algebra

P(x) | P(x^2-2)

by GreenTea2593, Apr 22, 2025, 3:27 AM

Let $P(x)$ be a monic polynomial with complex coefficients such that there exist a polynomial $Q(x)$ with complex coefficients for which $P(x^2-2)=P(x)Q(x).$ Determine all complex numbers that could be the root of $P(x)$ .

Proposed by Valentio Iverson, Indonesia

This post has been edited 1 time. Last edited by GreenTea2593, 3 hours ago

polynomial

algebra

Inequality with three conditions

by oVlad, Apr 21, 2025, 1:48 PM

Let $a,b,c$ be non-negative real numbers such that $b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.$ Prove that $a^2+b^2+c^2\leqslant 2abc+1.$

algebra

inequalities

real+ FE

by pomodor_ap, Apr 21, 2025, 11:24 AM

Let $f : \mathbb{R}^+ \to \mathbb{R}^+$ be a function such that
$f(x)f(x^2 + y f(y)) = f(x)f(y^2) + x^3$ for all $x, y \in \mathbb{R}^+$ . Determine all such functions $f$ .

functional equation

algebra

Is this FE solvable?

by ItzsleepyXD, Apr 21, 2025, 3:02 AM

Let $c_1,c_2 \in \mathbb{R^+}$ . Find all $f : \mathbb{R^+} \rightarrow \mathbb{R^+}$ such that for all $x,y \in \mathbb{R^+}$ $f(x+c_1f(y))=f(x)+c_2f(y)$

function

standard Q FE

by jasperE3, Apr 20, 2025, 6:27 PM

Find all functions $f:\mathbb Q\to\mathbb Q$ such that for any $x,y\in\mathbb Q$ :
$f(xf(x)+f(x+2y))=f(x)^2+f(y)+y.$

functional equation

algebra

functional equation in Q

FE over Q

rational

function

Let $ a, b, c $ be positive real numbers satisfying \[ a^2 + c^2 = b(a + c). \

by Jackson0423, Apr 16, 2025, 2:58 PM

Let $a, b, c$ be positive real numbers satisfying
$a^2 + c^2 = b(a + c).$ Let
$m = \min \left( \frac{a^2 + ab + b^2}{ab + bc + ca} \right).$ Find the value of $2024m$ .

algebra

PQ bisects AC if <BCD=90^o, A, B,C,D concyclic

by parmenides51, Sep 7, 2022, 12:18 AM

Let $A$ , $B$ , $C$ and $D$ be points on the same circumference with $\angle BCD=90^\circ$ . Let $P$ and $Q$ be the projections of $A$ onto $BD$ and $CD$ , respectively. Prove that $PQ$ cuts the segment $AC$ into equal parts.

This post has been edited 1 time. Last edited by parmenides51, Sep 7, 2022, 12:40 AM

bisects segment

geometry

USEMO P6 (Idk what to say here)

by franzliszt, Oct 25, 2020, 11:03 PM

Prove that for every odd integer $n > 1$ , there exist integers $a, b > 0$ such that, if we let $Q(x) = (x + a)^ 2 + b$ , then the following conditions hold:
$\bullet$ we have $\gcd(a, n) = gcd(b, n) = 1$ ;
$\bullet$ the number $Q(0)$ is divisible by $n$ ; and
$\bullet$ the numbers $Q(1), Q(2), Q(3), \dots$ each have a prime factor not dividing $n$ .

This post has been edited 1 time. Last edited by v_Enhance, Oct 26, 2020, 1:40 AM
Reason: fix typo

number theory

USEMO

USEMO 2020