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Inequality em981
oldbeginner   16
N 37 minutes ago by mihaig
Source: Own
Let $a, b, c>0, a+b+c=3$. Prove that
\[\sqrt{a+\frac{9}{b+2c}}+\sqrt{b+\frac{9}{c+2a}}+\sqrt{c+\frac{9}{a+2b}}+\frac{2(ab+bc+ca)}{9}\ge\frac{20}{3}\]
16 replies
oldbeginner
Sep 22, 2016
mihaig
37 minutes ago
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Nice FE over R+
doanquangdang   4
N 38 minutes ago by jasperE3
Source: collect
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ such that
\[x+f(yf(x)+1)=xf(x+y)+yf(yf(x))\]for all $x,y>0.$
4 replies
doanquangdang
Jul 19, 2022
jasperE3
38 minutes ago
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right triangle, midpoints, two circles, find angle
star-1ord   0
40 minutes ago
Source: Estonia Final Round 2025 8-3
In the right triangle $ABC$, $M$ is the midpoint of the hypotenuse $AB$. Point $D$ is chosen on the leg $BC$ so that the line segment $DM$ meets $(ACD)$ again at $K$ ($K\neq D$). Let $L$ be the reflection of $K$ in $M$. The circles $(ACD)$ and $(BCL)$ meet again at $N$ ($N\neq C$). Find the measure of $\angle KNL$.
0 replies
star-1ord
40 minutes ago
0 replies
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interesting functional equation
tabel   3
N 43 minutes ago by waterbottle432
Source: random romanian contest
Determine all functions \( f : (0, \infty) \to (0, \infty) \) that satisfy the functional equation:
\[ f(f(x)(1 + y)) = f(x) + f(xy), \quad \forall x, y > 0. \]
3 replies
tabel
2 hours ago
waterbottle432
43 minutes ago
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Serbian selection contest for the IMO 2025 - P6
OgnjenTesic   4
N an hour ago by GreenTea2593
Source: Serbian selection contest for the IMO 2025
For an $n \times n$ table filled with natural numbers, we say it is a divisor table if:
- the numbers in the $i$-th row are exactly all the divisors of some natural number $r_i$,
- the numbers in the $j$-th column are exactly all the divisors of some natural number $c_j$,
- $r_i \ne r_j$ for every $i \ne j$.

A prime number $p$ is given. Determine the smallest natural number $n$, divisible by $p$, such that there exists an $n \times n$ divisor table, or prove that such $n$ does not exist.

Proposed by Pavle Martinović
4 replies
OgnjenTesic
May 22, 2025
GreenTea2593
an hour ago
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pairs (m, n) such that a fractional expression is an integer
cielblue   2
N an hour ago by cielblue
Find all pairs $(m,\ n)$ of positive integers such that $\frac{m^3-mn+1}{m^2+mn+2}$ is an integer.
2 replies
cielblue
Yesterday at 8:38 PM
cielblue
an hour ago
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Sociable set of people
jgnr   23
N an hour ago by quantam13
Source: RMM 2012 day 1 problem 1
Given a finite number of boys and girls, a sociable set of boys is a set of boys such that every girl knows at least one boy in that set; and a sociable set of girls is a set of girls such that every boy knows at least one girl in that set. Prove that the number of sociable sets of boys and the number of sociable sets of girls have the same parity. (Acquaintance is assumed to be mutual.)

(Poland) Marek Cygan
23 replies
jgnr
Mar 3, 2012
quantam13
an hour ago
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diophantine equation
m4thbl3nd3r   0
an hour ago
Find all positive integers $n,k$ such that $$5^{2n+1}-5^n+1=k^2$$
0 replies
m4thbl3nd3r
an hour ago
0 replies
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Functional equation
shobber   19
N 3 hours ago by Unique_solver
Source: Canada 2002
Let $\mathbb N = \{0,1,2,\ldots\}$. Determine all functions $f: \mathbb N \to \mathbb N$ such that
\[ xf(y) + yf(x) = (x+y) f(x^2+y^2) \]
for all $x$ and $y$ in $\mathbb N$.
19 replies
shobber
Mar 5, 2006
Unique_solver
3 hours ago
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Prove the inequality
Butterfly   0
3 hours ago
Let $a,b,c$ be real numbers such that $a+b+c=3$. Prove $$a^3b+b^3c+c^3a\le \frac{9}{32}(63+5\sqrt{105}).$$
0 replies
Butterfly
3 hours ago
0 replies
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Functional equation
shactal   1
N 3 hours ago by ariopro1387
Let $f:\mathbb R\to \mathbb R$ a function satifying $$f(x+2xy) = f(x) + 2f(xy)$$for all $x,y\in \mathbb R$.
If $f(1991)=a$, then what is $f(1992)$, the answer is in terms of $a$.
1 reply
shactal
5 hours ago
ariopro1387
3 hours ago
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Minimum with natural numbers
giangtruong13   1
N Apr 10, 2025 by Ianis
Let $x,y,z,t$ be natural numbers such that: $x^2-y^2+t^2=21$ and $x^2+3y^2+4z^2=101$. Find the min: $$M=x^2+y^2+2z^2+t^2$$
1 reply
giangtruong13
Apr 10, 2025
Ianis
Apr 10, 2025
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Minimum with natural numbers
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giangtruong13
148 posts
giangtruong13
#1 h  Apr 10, 2025, 1:51 PM
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Let $x,y,z,t$ be natural numbers such that: $x^2-y^2+t^2=21$ and $x^2+3y^2+4z^2=101$. Find the min: $$M=x^2+y^2+2z^2+t^2$$
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Ianis
416 posts
Ianis
#2 h  Apr 10, 2025, 2:40 PM
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We have$$x^2-y^2\equiv x^2+3y^2+4z^2=101\equiv 1\pmod 4,$$so $x$ is odd and $y$ is even. Say $y=2b$, with $b\in \mathbb{N}$, then$$x^2+12b^2+4z^2=101,$$so $b=1$ or $b=2$.
If $b=1$ then$$x^2+4z^2=89,$$so $z\leq 4$, and only $z=4$ gives a solution in natural numbers. In this case we get $(x,y,z)=(5,2,4)$, which gives $t=0$, so this doesn't work.
Hence $b=2$, and so $y=4$. Then$$x^2+4z^2=53,$$so $z\leq 3$, and only $z=1$ gives a solution in natural numbers. In this case we get $(x,y,z)=(7,4,1)$, which gives $t^2=-12$, so this doesn't work.

Hence I assume you mean whole numbers rather than natural numbers.
In this case $b=1$ gives the solution $(x,y,z,t)=(5,2,4,0)$ and $M=61$.
We also have to check the case $b=0$. Here we have$$x^2+4z^2=101,$$so $z\leq 5$, and only $z=5$ gives a solution in whole numbers. In this case we get $(x,y,z)=(1,0,5)$, which gives $t^2=20$, so this doesn't work.

Hence there are no solutions in natural numbers and only one solution in whole numbers.
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