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Serbian selection contest for the IMO 2025 - P5
OgnjenTesic   3
N 35 minutes ago by atdaotlohbh
Source: Serbian selection contest for the IMO 2025
Determine the smallest positive real number $\alpha$ such that there exists a sequence of positive real numbers $(a_n)$, $n \in \mathbb{N}$, with the property that for every $n \in \mathbb{N}$ it holds that:
\[ a_1 + \cdots + a_{n+1} < \alpha \cdot a_n. \]Proposed by Pavle Martinović
3 replies
OgnjenTesic
May 22, 2025
atdaotlohbh
35 minutes ago
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Equation in integers with gcd and lcm
skellyrah   1
N 40 minutes ago by frost23
Find all integers \( x \) and \( y \) such that
\[ \frac{1}{\gcd(x, y)} + \frac{3}{xy} + \frac{y}{\operatorname{lcm}(x, y)} = y, \]where \( \gcd(x, y) \) denotes the greatest common divisor of \( x \) and \( y \), and \( \operatorname{lcm}(x, y) \) denotes their least common multiple.
1 reply
skellyrah
an hour ago
frost23
40 minutes ago
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set construction nt
top1vien   1
N an hour ago by alexheinis
Is there a set of 2025 positive integers $S$ that satisfies: for all different $a,b,c,d\in S$, we have $\gcd(ab+1000,cd+1000)=1$?
1 reply
top1vien
Today at 10:04 AM
alexheinis
an hour ago
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A,P,Q lies on the Radical Axis
MrCriminal   3
N an hour ago by Blackbeam999
Source: Power Of a Point -Yufei Zhao #P8
Hint Needed
Let \(ABC\) be a triangle and let \(D\) and \(E\) be points on the sides \(AB\) and \(AC\), respectively , such that \(DE\) is parallel to \(BC\). Let \(P\) be any point interior to triangle \(ADE\) , and let \(F\) and \(G\) be the intersections of \(DE\) with the lines \(BP\) and \(CP\), respectively. Let \(Q\) be the second intersection points of the circumcircles of triangles \(PDG\) and \(PFE\) . Prove that the points \(A, P, \text{and } Q\) are collinear .
3 replies
MrCriminal
May 15, 2021
Blackbeam999
an hour ago
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Hellopoo
Bet667   8
N an hour ago by Aiden-1089
In how many ways can you tile a $3 \cdot n$ rectangle with $2 \cdot 1$ dominoes?
8 replies
Bet667
Yesterday at 11:22 AM
Aiden-1089
an hour ago
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Clifford's chain of circles, concurrent Simson lines
kosmonauten3114   0
an hour ago
Source: My own, but most likely already known
Let $C_1$, $C_2$, $C_3$, $C_4$ be circles having a common point $P$. Denote by $P_{ij}$ the intersection point, other than $P$, of $C_i$ and $C_j$ ($\{i,j\} \in \{1,2,3,4\},i<j$). Let $Q$ be the common point of the 4 circles $\odot(P_{23}P_{24}P_{34})$, $\odot(P_{13}P_{14}P_{34})$, $\odot(P_{12}P_{14}P_{24})$, $\odot(P_{12}P_{13}P_{23})$.
Let $\ell_1$ be the Simson line of $P$ with respect to $\triangle{P_{12}P_{13}P_{14}}$. Define $\ell_2$, $\ell_3$, $\ell_4$ cyclically.
Let $\ell_1'$ be the Simson line of $Q$ with respect to $\triangle{P_{23}P_{24}P_{34}}$. Define $\ell_2'$, $\ell_3'$, $\ell_4'$ cyclically.
Prove that if $\ell_1$, $\ell_2$, $\ell_3$, $\ell_4$ are concurrent, then, $\ell_1'$, $\ell_2'$, $\ell_3'$, $\ell_4'$ are also concurrent.
0 replies
kosmonauten3114
an hour ago
0 replies
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Inequality
srnjbr   4
N an hour ago by sqing
For real numbers a, b, c and d that a+d=b+c prove the following:
(a-b)(c-d)+(a-c)(b-d)+(d-a)(b-c)>=0
4 replies
srnjbr
Oct 30, 2024
sqing
an hour ago
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Solution of an interesting inequality
imnotgoodatmathsorry   1
N an hour ago by imnotgoodatmathsorry
Source: @Alphabetamath on Facebook
$\text{The problem:}$
1 reply
imnotgoodatmathsorry
an hour ago
imnotgoodatmathsorry
an hour ago
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Similarity of two triangles!
ariopro1387   1
N an hour ago by Mahdi_Mashayekhi
Source: Iran Team selection test 2025 - P5
In a scalene triangle $ABC$, $D$ is the point of tangency of the incircle with the side $BC$. Points $T_B$ and $T_C$ are the intersections of the angle bisectors of $\angle ABC$ and $\angle ACB$ with the circumcircle of $ABC$, respectively. Let $X_B$ be the antipodal point of $A$ in the circumcircle of $ACD$, and let $X_C$ be the antipodal point of $A$ in the circumcircle of $ABD$.
Prove that triangles $B T_C X_C$ and $C T_B X_B$ are similar.
1 reply
ariopro1387
Today at 10:34 AM
Mahdi_Mashayekhi
an hour ago
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trigonometric inequality
MATH1945   7
N 2 hours ago by sqing
Source: ?
In triangle $ABC$, prove that $$sin^2(A)+sin^2(B)+sin^2(C) \leq \frac{9}{4}$$
7 replies
MATH1945
May 26, 2016
sqing
2 hours ago
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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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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
23515 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
578 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
11384 posts
jasperE3
#4 h  Apr 21, 2025, 10:44 PM
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