Unexpecredly Quick-Solve Inequality

by Primeniyazidayi, May 28, 2025, 5:18 AM

If $a, b, c>0$ , prove that $\frac{a^5}{b^2}+\frac{b}{c}+\frac{c^3}{a^2}>2a$

inequalities

Inspired by Adhyayan Jana

by sqing, May 28, 2025, 4:35 AM

Let $a,b,c,d>0,a^2 + d^2+ad = b^2 + c^2$ aand $a^2 + b^2 = c^2 + d^2+cd$ Prove that $\frac{ab+cd}{ad+bc} =1$

algebra

Inspired by Adhyayan Jana

by sqing, May 28, 2025, 2:38 AM

Let $a,b,c,d>0,a^2 + d^2-ad = (b + c)^2$ aand $a^2 +c^2 = b^2 + d^2.$ Prove that $\frac{ab+cd}{ad+bc} \geq \frac{ 4}{5}$ Let $a,b,c,d>0,a^2 + d^2-ad = b^2 + c^2 + bc$ aand $a^2 +c^2 = b^2 + d^2.$ Prove that $\frac{ab+cd}{ad+bc} \geq \frac{\sqrt 3}{2}$ Let $a,b,c,d>0,a^2 + d^2 - ad = b^2 + c^2 + bc$ aand $a^2 + b^2 = c^2 + d^2.$ Prove that $\frac{ab+cd}{ad+bc} =\frac{\sqrt 3}{2}$

This post has been edited 4 times. Last edited by sqing, 4 hours ago

algebra

inequalities proposed

inequalities

Strange circles in an orthocenter config

by VideoCake, May 26, 2025, 5:10 PM

Let $\overline{AD}$ and $\overline{BE}$ be altitudes in an acute triangle $ABC$ which meet at $H$ . Suppose that $DE$ meets the circumcircle of $ABC$ at $P$ and $Q$ such that $P$ lies on the shorter arc of $BC$ and $Q$ lies on the shorter arc of $CA$ . Let $AQ$ and $BE$ meet at $S$ . Show that the circumcircles of $BPE$ and $QHS$ and the line $PH$ concur.

geometry

Problem 7

by SlovEcience, May 14, 2025, 11:03 AM

Consider the sequence $(u_n)$ defined by $u_0 = 5$ and
$u_{n+1} = \frac{1}{2}u_n^2 - 4 \quad \text{for all } n \in \mathbb{N}.$ a) Prove that there exist infinitely many positive integers $n$ such that $u_n > 2020n$ .

b) Compute
$\lim_{n \to \infty} \frac{2u_{n+1}}{u_0u_1\cdots u_n}.$

arithmetic sequence

algebra

Impossible Infinite Sequence

by Rijul saini, May 31, 2024, 4:16 AM

Let $P(x) \in \mathbb{Q}[x]$ be a polynomial with rational coefficients and degree $d\ge 2$ . Prove there is no infinite sequence $a_0, a_1, \ldots$ of rational numbers such that $P(a_i)=a_{i-1}+i$ for all $i\ge 1$ .

Proposed by Pranjal Srivastava and Rohan Goyal

This post has been edited 1 time. Last edited by Rijul saini, May 31, 2024, 6:52 AM

number theory

Easy Taiwanese Geometry

by USJL, Jan 31, 2024, 6:27 AM

Suppose $O$ is the circumcenter of $\Delta ABC$ , and $E, F$ are points on segments $CA$ and $AB$ respectively with $E, F \neq A$ . Let $P$ be a point such that $PB = PF$ and $PC = PE$ .
Let $OP$ intersect $CA$ and $AB$ at points $Q$ and $R$ respectively. Let the line passing through $P$ and perpendicular to $EF$ intersect $CA$ and $AB$ at points $S$ and $T$ respectively. Prove that points $Q, R, S$ , and $T$ are concyclic.

Proposed by Li4 and usjl

This post has been edited 1 time. Last edited by USJL, Jan 31, 2024, 6:28 AM

Taiwan

geometry

Functional xf(x+f(y))=(y-x)f(f(x)) for all reals x,y

by cretanman, May 10, 2023, 3:50 PM

Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ ,
$xf(x+f(y))=(y-x)f(f(x)).$
Proposed by Nikola Velov, Macedonia

This post has been edited 4 times. Last edited by Amir Hossein, May 13, 2023, 1:00 AM
Reason: Fixed source

Concurrent lines

by syk0526, May 17, 2014, 9:41 AM

The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $BC, CA, AB$ at $A_1 , B_1 , C_1$ respectively. The line $AI$ meets the circumcircle of $ABC$ at $A_2$ . The line $B_1 C_1$ meets the line $BC$ at $A_3$ and the line $A_2 A_3$ meets the circumcircle of $ABC$ at $A_4 (\ne A_2 )$ . Define $B_4 , C_4$ similarly. Prove that the lines $AA_4 , BB_4 , CC_4$ are concurrent.

Equal angles (a very old problem)

by April, Jul 13, 2008, 1:45 AM

The diagonals of a trapezoid $ABCD$ intersect at point $P$ . Point $Q$ lies between the parallel lines $BC$ and $AD$ such that $\angle AQD = \angle CQB$ , and line $CD$ separates points $P$ and $Q$ . Prove that $\angle BQP = \angle DAQ$ .

Author: Vyacheslav Yasinskiy, Ukraine