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Inspired by lgx57
sqing   2
N a minute ago by sqing
Source: Own
Let $ a,b>0, a^4+ab+b^4=10 $. Prove that
$$ \sqrt{10}\leq a^2+ab+b^2 \leq 6$$$$ 2\leq a^2-ab+b^2 \leq \sqrt{10}$$$$ 4\sqrt{10}\leq 4a^2+ab+4b^2 \leq18$$$$ 12<4a^2-ab+4b^2 \leq14$$
2 replies
+2 w
sqing
33 minutes ago
sqing
a minute ago
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a^2-bc square implies 2a+b+c composite
v_Enhance   39
N 15 minutes ago by SimplisticFormulas
Source: ELMO 2009, Problem 1
Let $a,b,c$ be positive integers such that $a^2 - bc$ is a square. Prove that $2a + b + c$ is not prime.

Evan o'Dorney
39 replies
v_Enhance
Dec 31, 2012
SimplisticFormulas
15 minutes ago
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Vincent's Theorem
EthanWYX2009   0
15 minutes ago
Source: Vincent's Theorem
Let $p(x)$ be a real polynomial of degree $\deg(p)$ that has only simple roots. It is possible to determine a positive quantity $\delta$ so that for every pair of positive real numbers $a$, $b$ with ${\displaystyle |b-a|<\delta }$, every transformed polynomial of the form $${\displaystyle f(x)=(1+x)^{\deg(p)}p\left({\frac {a+bx}{1+x}}\right)}$$has exactly $0$ or $1$ sign variations.
0 replies
EthanWYX2009
15 minutes ago
0 replies
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JBMO Shortlist 2019 N5
Steve12345   11
N 20 minutes ago by MR.1
Find all positive integers $x, y, z$ such that $45^x-6^y=2019^z$

Proposed by Dorlir Ahmeti, Albania
11 replies
Steve12345
Sep 12, 2020
MR.1
20 minutes ago
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polonomials
Ducksohappi   1
N 26 minutes ago by top1vien
$P\in \mathbb{R}[x] $ with even-degree
Prove that there is a non-negative integer k such that
$Q_k(x)=P(x)+P(x+1)+...+P(x+k)$
has no real root
1 reply
Ducksohappi
Today at 8:36 AM
top1vien
26 minutes ago
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Inspired by Bet667
sqing   3
N an hour ago by sqing
Source: Own
Let $ a,b $ be a real numbers such that $a^3+kab+b^3\ge a^4+b^4.$Prove that
$$1-\sqrt{k+1} \leq a+b\leq 1+\sqrt{k+1} $$Where $ k\geq 0. $
3 replies
sqing
2 hours ago
sqing
an hour ago
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Geometry marathon
HoRI_DA_GRe8   846
N an hour ago by ItzsleepyXD
Ok so there's been no geo marathon here for more than 2 years,so lets start one,rules remain same.
1st problem.
Let $PQRS$ be a cyclic quadrilateral with $\angle PSR=90°$ and let $H$ and $K$ be the feet of altitudes from $Q$ to the lines $PR$ and $PS$,.Prove $HK$ bisects $QS$.
P.s._eeezy ,try without ss line.
846 replies
HoRI_DA_GRe8
Sep 5, 2021
ItzsleepyXD
an hour ago
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Find all functions $f$ is strictly increasing : \(\mathbb{R^+}\) \(\rightarrow\)
guramuta   0
an hour ago
Find all functions $f$ is strictly increasing : \(\mathbb{R^+}\) \(\rightarrow\) \(\mathbb{R^+}\) such that:
i) $f(2x)$ \(\geq\) $2f(x)$
ii) $f(f(x)f(y)+x) = f(xf(y)) + f(x) $
0 replies
guramuta
an hour ago
0 replies
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Partitioning coprime integers to arithmetic sequences
sevket12   3
N an hour ago by quacksaysduck
Source: 2025 Turkey EGMO TST P3
For a positive integer $n$, let $S_n$ be the set of positive integers that do not exceed $n$ and are coprime to $n$. Define $f(n)$ as the smallest positive integer that allows $S_n$ to be partitioned into $f(n)$ disjoint subsets, each forming an arithmetic progression.

Prove that there exist infinitely many pairs $(a, b)$ satisfying $a, b > 2025$, $a \mid b$, and $f(a) \nmid f(b)$.
3 replies
sevket12
Feb 8, 2025
quacksaysduck
an hour ago
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Inspired by Bet667
sqing   3
N an hour ago by sqing
Source: Own
Let $ a,b $ be a real numbers such that $a^2+kab+b^2\ge a^3+b^3.$Prove that$$a+b\leq k+2$$Where $ k\geq 0. $
3 replies
sqing
May 6, 2025
sqing
an hour ago
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F has at least n distinct values
nataliaonline75   0
2 hours ago

Let $n$ be natural number and $S$ be the set of $n$ distinct natural numbers. Define function $f: S \times S \rightarrow N$ with $f(x,y)=\frac{xy}{(gcd(x,y))^2}$. Prove that $f$ have at least $n$ distinct values.
0 replies
nataliaonline75
2 hours ago
0 replies
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An easy FE
oVlad   3
N Apr 21, 2025 by jasperE3
Source: Romania EGMO TST 2017 Day 1 P3
Determine all functions $f:\mathbb R\to\mathbb R$ such that \[f(xy-1)+f(x)f(y)=2xy-1,\]for any real numbers $x{}$ and $y{}.$
3 replies
oVlad
Apr 21, 2025
jasperE3
Apr 21, 2025
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An easy FE
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Reveal topic tags
algebra
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functional equation
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Source: Romania EGMO TST 2017 Day 1 P3
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oVlad
1746 posts
oVlad
#1 h  Apr 21, 2025, 1:36 PM
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Determine all functions $f:\mathbb R\to\mathbb R$ such that \[f(xy-1)+f(x)f(y)=2xy-1,\]for any real numbers $x{}$ and $y{}.$
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pco
23511 posts
pco
#2 h  Apr 21, 2025, 3:54 PM • 1 Y
Y by ATM_
oVlad wrote:
Determine all functions $f:\mathbb R\to\mathbb R$ such that \[f(xy-1)+f(x)f(y)=2xy-1,\]for any real numbers $x{}$ and $y{}.$
Let $P(x,y)$ be the assertion $f(xy-1)+f(x)f(y)=2xy-1$
Let $c=f(1)$

If $f(0)\ne 0$, $P(x,0)$ $\implies$ $f(x)$ constant, which is never a solution. So $f(0)=0$

$P(0,0)$ $\implies$ $f(-1)=-1$
$P(1,1)$ $\implies$ $c=\pm 1$
Subtracting $P(x,1)$ from $P(-x,-1)$, we get $f(-x)=-cf(x)$

Subtracting $P(x,y)$ from $P(xy,1)$, we get new assertion $Q(x,y)$ : $f(x)f(y)=cf(xy)$
If $f(u)=0$ for some $u\ne 0$, $Q(x,u)$ implies $f(ux)=0$ $\forall x$ and so $f\equiv 0$, which is not a solution.
So $f(x)=0$ $\iff$ $x=0$

$Q(x,x)$ implies $\frac{f(x)}c$ is multiplicative and positive $\forall x>0$ and so $g(x)=\ln \frac{f(e^x)}c$ is additive

If $g(x)$ is not linear, its graph is dense in $\mathbb R^2$ and so graph of $f(x)$ is :
Either dense in $\mathbb R_{>0}\times \mathbb R_{>0}$ if $c=1$
Either dense in $\mathbb R_{>0}\times \mathbb R_{<0}$ if $c=-1$

But $P(x,x)$ $\implies$ $f(x^2-1)\le 2x^2-1$ and so contradiction with both cases
So $g(x)$ is linear and $f(x)=cx^a$ $\forall x>0$ for some real $a$
Then $P(2,1)$ implies $c+2^a=3$ and so :

If $c=1$ : $a=1$ and $f(x)=x$ $\forall x\ge 0$ and $f(-x)=-cf(x)=-f(x)$ imply $\boxed{\text{S1 : }f(x)=x\quad\forall x}$, which indeed fits

If $c=-1$ : $a=2$ and $f(x)=-x^2$ $\forall x\ge 0$ and $f(-x)=-cf(x)=f(x)$ imply $\boxed{\text{S2 : }f(x)=-x^2\quad\forall x}$, which indeed fits
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BR1F1SZ
570 posts
BR1F1SZ
#3 h  Apr 21, 2025, 6:20 PM
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It is also 2015 Argentina TST P3
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jasperE3
11305 posts
jasperE3
#4 h  Apr 21, 2025, 10:44 PM
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