4-vars inequality

by xytunghoanh, May 15, 2025, 2:10 PM

For $a,b,c,d \ge 0$ and $a\ge c$ , $b \ge d$ . Prove that
$a+b+c+d+ac+bd+8 \ge 2(\sqrt{ab}+\sqrt{bc}+\sqrt{cd}+\sqrt{da}+\sqrt{ac}+\sqrt{bd})$ .

This post has been edited 1 time. Last edited by xytunghoanh, 4 hours ago
Reason: add condition

inequalities

Find all p(x) such that p(p) is a power of 2

by truongphatt2668, May 15, 2025, 1:05 PM

Find all polynomial $P(x) \in \mathbb{R}[x]$ such that:
$P(p_i) = 2^{a_i}$ with $p_i$ is an $i$ th prime and $a_i$ is an arbitrary positive integer.

algebra

polynomial

Angle Relationships in Triangles

by steven_zhang123, May 14, 2025, 11:09 PM

In $\triangle ABC$ , $AB > AC$ . The internal angle bisector of $\angle BAC$ and the external angle bisector of $\angle BAC$ intersect the ray $BC$ at points $D$ and $E$ , respectively. Given that $CE - CD = 2AC$ , prove that $\angle ACB = 2\angle ABC$ .

angle

geometry

angle bisector

functional equation

by uaua, Jan 3, 2023, 2:52 AM

f:R--R
f(f(x)+xy) = xf(x) + f(x)

functional equation

algebra

Iranian TST 2019, first exam, day1, problem 2

by Hamper.r, Apr 7, 2019, 4:12 AM

$a, a_1,a_2,\dots ,a_n$ are natural numbers. We know that for any natural number $k$ which $ak+1$ is square, at least one of $a_1k+1,\dots ,a_n k+1$ is also square.
Prove $a$ is one of $a_1,\dots ,a_n$
Proposed by Mohsen Jamali

This post has been edited 6 times. Last edited by Hamper.r, May 12, 2020, 9:54 PM
Reason: :-|

Iranian TST

TST

Iran

Easy functional equation

by fattypiggy123, Jul 5, 2014, 8:41 AM

Find all functions from the reals to the reals satisfying
$f(xf(y) + x) = xy + f(x)$

function

algebra proposed

algebra

incircle and excircles

by micliva, May 16, 2014, 2:47 PM

The incircle of triangle $ABC$ has centre $I$ and touches the sides $BC$ , $CA$ , $AB$ at points $A_1$ , $B_1$ , $C_1$ , respectively. Let $I_a$ , $I_b$ , $I_c$ be excentres of triangle $ABC$ , touching the sides $BC$ , $CA$ , $AB$ respectively. The segments $I_aB_1$ and $I_bA_1$ intersect at $C_2$ . Similarly, segments $I_bC_1$ and $I_cB_1$ intersect at $A_2$ , and the segments $I_cA_1$ and $I_aC_1$ at $B_2$ . Prove that $I$ is the center of the circumcircle of the triangle $A_2B_2C_2$ .

L. Emelyanov, A. Polyansky

geometry

circumcircle

geometric transformation

Prove angles are equal

by BigSams, May 13, 2011, 2:20 AM

Let $ABC$ be an acute triangle. Let $AD$ be the altitude on $BC$ , and let $H$ be any interior point on $AD$ . Lines $BH,CH$ , when extended, intersect $AC,AB$ at $E,F$ respectively. Prove that $\angle EDH=\angle FDH$ .

Symmetric squares wrt centre of 4x4 square add to 17

by Goutham, Dec 6, 2010, 8:00 AM

Numbers $1, 2,\cdots, 16$ are written in a $4\times 4$ square matrix so that the sum of the numbers in every row, every column, and every diagonal is the same and furthermore that the numbers $1$ and $16$ lie in opposite corners. Prove that the sum of any two numbers symmetric with respect to the center of the square equals $17$ .

linear algebra

matrix

number theory unsolved

number theory

Two circles, a tangent line and a parallel

by Valentin Vornicu, Oct 24, 2005, 10:15 AM

Two circles $G_1$ and $G_2$ intersect at two points $M$ and $N$ . Let $AB$ be the line tangent to these circles at $A$ and $B$ , respectively, so that $M$ lies closer to $AB$ than $N$ . Let $CD$ be the line parallel to $AB$ and passing through the point $M$ , with $C$ on $G_1$ and $D$ on $G_2$ . Lines $AC$ and $BD$ meet at $E$ ; lines $AN$ and $CD$ meet at $P$ ; lines $BN$ and $CD$ meet at $Q$ . Show that $EP = EQ$ .