Inspired by 2012 Romania and 2021 BH

Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ ,
$f(x+yf(x))+y = xy + f(x+y).$
Proposed by Giannis Galamatis, Greece

31 replies

1 viewing

falantrng

Apr 27, 2025

Matematikus

a few seconds ago

P>2D

gwen01 3

N 35 minutes ago by AshAuktober

Source: Baltic Way 1992 #18

Show that in a non-obtuse triangle the perimeter of the triangle is always greater than two times the diameter of the circumcircle.

3 replies

gwen01

Feb 18, 2009

AshAuktober

35 minutes ago

inequality with interesting conditions

Cobedangiu 2

N an hour ago by musava_ribica

Let $x,y,z>0$ :

2 replies

Cobedangiu

an hour ago

musava_ribica

an hour ago

Hard geometry proof

radhoan_rikto- 1

N an hour ago by GreekIdiot

Source: BDMO 2025

Let ABC be an acute triangle and D the foot of the altitude from A onto BC. A semicircle with diameter BC intersects segments AB,AC and AD in the points F,E and X respectively.The circumcircles of the triangles DEX and DFX intersect BC in L and N respectively, other than D. Prove that BN=LC.

1 reply

radhoan_rikto-

Apr 25, 2025

GreekIdiot

an hour ago

Inspired by JK1603JK

sqing 0

an hour ago

Source: Own

Let $a,b,c$ be reals such that $abc\neq 0$ and $a+b+c=0.$ Prove that
$\left|\frac{a-b}{c}\right|+k\left|\frac{b-c}{a} \right|+k^2\left|\frac{c-a}{b} \right|\ge 3(k+1)$ Where $k>0.$
$\left|\frac{a-b}{c}\right|+2\left|\frac{b-c}{a} \right|+4\left|\frac{c-a}{b} \right|\ge 9$

0 replies

sqing

an hour ago

0 replies

problem interesting

Cobedangiu 9

N an hour ago by Cobedangiu

Let $a=3k^2+3k+1 (a,k \in N)$
$i)$ Prove that: $a^2$ is the sum of $3$ square numbers
$ii)$ Let $b \vdots a$ and $b$ is the sum of $3$ square numbers. Prove that: $b^n$ is the sum of $3$ square numbers

9 replies

Cobedangiu

Yesterday at 5:06 AM

Cobedangiu

an hour ago

4-var inequality

RainbowNeos 0

2 hours ago

Given $a,b,c,d>0$ , show that
$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.$

0 replies

RainbowNeos

2 hours ago

0 replies

Find all integer pairs (m,n) such that 2^n! + 1 | 2^m! + 19

Goblik 0

2 hours ago

Find all positive integer pairs $(m,n)$ such that $2^{n!} + 1 | 2^{m!} + 19$

0 replies

Goblik

2 hours ago

0 replies

Junior Balkan Mathematical Olympiad 2024- P3

Lukaluce 15

N 2 hours ago by MATHS_ENTUSIAST

Source: JBMO 2024

Find all triples of positive integers $(x, y, z)$ that satisfy the equation

$2020^x + 2^y = 2024^z.$
Proposed by Ognjen Tešić, Serbia

15 replies

Lukaluce

Jun 27, 2024

MATHS_ENTUSIAST

2 hours ago

AD is Euler line of triangle IKL

VicKmath7 16

N 2 hours ago by ErTeeEs06

Source: IGO 2021 Advanced P5

Given a triangle $ABC$ with incenter $I$ . The incircle of triangle $ABC$ is tangent to $BC$ at $D$ . Let $P$ and $Q$ be points on the side BC such that $\angle PAB = \angle BCA$ and $\angle QAC = \angle ABC$ , respectively. Let $K$ and $L$ be the incenter of triangles $ABP$ and $ACQ$ , respectively. Prove that $AD$ is the Euler line of triangle $IKL$ .

Proposed by Le Viet An, Vietnam

16 replies

+1 w

VicKmath7

Dec 30, 2021

ErTeeEs06

2 hours ago

Twin Prime Diophantine

awesomeming327. 22

N 2 hours ago by MATHS_ENTUSIAST

Source: CMO 2025

Determine all positive integers $a$ , $b$ , $c$ , $p$ , where $p$ and $p+2$ are odd primes and
$2^ap^b=(p+2)^c-1.$

22 replies

awesomeming327.

Mar 7, 2025

MATHS_ENTUSIAST

2 hours ago

Inspired by 2012 Romania and 2021 BH

sqing 0

Apr 6, 2025

Source: Own

Let $a, b, c, d\geq 0 , bc + d + a = 5, cd + a + b = 2$ and $da + b + c = 6.$ Prove that
$3\leq ab + c + d\leq 2\sqrt{13}-1$ $5\leq a+ b+ c +d \leq\frac{1}{2}(11+\sqrt{13})$ $\sqrt{13}+1 \leq a b +bc+ c d+d a \leq 6$