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functional equation
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Easy but unusual junior ineq
Maths_VC   1
N an hour ago by blug
Source: Serbia JBMO TST 2025, Problem 2
Real numbers $x, y$ $\ge$ $0$ satisfy $1$ $\le$ $x^2 + y^2$ $\le$ $5$. Determine the minimal and the maximal value of the expression $2x + y$
1 reply
Maths_VC
2 hours ago
blug
an hour ago
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Bosnia and Herzegovina JBMO TST 2009 Problem 1
gobathegreat   1
N an hour ago by FishkoBiH
Source: Bosnia and Herzegovina Junior Balkan Mathematical Olympiad TST 2009
Lengths of sides of triangle $ABC$ are positive integers, and smallest side is equal to $2$. Determine the area of triangle $P$ if $v_c = v_a + v_b$, where $v_a$, $v_b$ and $v_c$ are lengths of altitudes in triangle $ABC$ from vertices $A$, $B$ and $C$, respectively.
1 reply
gobathegreat
Sep 17, 2018
FishkoBiH
an hour ago
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USAMO 2001 Problem 2
MithsApprentice   53
N an hour ago by lksb
Let $ABC$ be a triangle and let $\omega$ be its incircle. Denote by $D_1$ and $E_1$ the points where $\omega$ is tangent to sides $BC$ and $AC$, respectively. Denote by $D_2$ and $E_2$ the points on sides $BC$ and $AC$, respectively, such that $CD_2=BD_1$ and $CE_2=AE_1$, and denote by $P$ the point of intersection of segments $AD_2$ and $BE_2$. Circle $\omega$ intersects segment $AD_2$ at two points, the closer of which to the vertex $A$ is denoted by $Q$. Prove that $AQ=D_2P$.
53 replies
MithsApprentice
Sep 30, 2005
lksb
an hour ago
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A=b
k2c901_1   89
N an hour ago by reni_wee
Source: Taiwan 1st TST 2006, 1st day, problem 3
Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$.

Proposed by Mohsen Jamali, Iran
89 replies
k2c901_1
Mar 29, 2006
reni_wee
an hour ago
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Strange angle condition and concyclic points
lminsl   129
N an hour ago by Aiden-1089
Source: IMO 2019 Problem 2
In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C=\angle BAC$. Similarly, let $Q_1$ be the point on line $QA_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle CQ_1Q=\angle CBA$.

Prove that points $P,Q,P_1$, and $Q_1$ are concyclic.

Proposed by Anton Trygub, Ukraine
129 replies
lminsl
Jul 16, 2019
Aiden-1089
an hour ago
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Simple inequality
sqing   12
N an hour ago by Rayvhs
Source: MEMO 2018 T1
Let $a,b$ and $c$ be positive real numbers satisfying $abc=1.$ Prove that$$\frac{a^2-b^2}{a+bc}+\frac{b^2-c^2}{b+ca}+\frac{c^2-a^2}{c+ab}\leq a+b+c-3.$$
12 replies
sqing
Sep 2, 2018
Rayvhs
an hour ago
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Random concyclicity in a square config
Maths_VC   2
N 2 hours ago by Maths_VC
Source: Serbia JBMO TST 2025, Problem 1
Let $M$ be a random point on the smaller arc $AB$ of the circumcircle of square $ABCD$, and let $N$ be the intersection point of segments $AC$ and $DM$. The feet of the tangents from point $D$ to the circumcircle of the triangle $OMN$ are $P$ and $Q$ , where $O$ is the center of the square. Prove that points $A$, $C$, $P$ and $Q$ lie on a single circle.
2 replies
Maths_VC
2 hours ago
Maths_VC
2 hours ago
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Serbian selection contest for the IMO 2025 - P3
OgnjenTesic   3
N 2 hours ago by atdaotlohbh
Source: Serbian selection contest for the IMO 2025
Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that:
- $f$ is strictly increasing,
- there exists $M \in \mathbb{N}$ such that $f(x+1) - f(x) < M$ for all $x \in \mathbb{N}$,
- for every $x \in \mathbb{Z}$, there exists $y \in \mathbb{Z}$ such that
\[ f(y) = \frac{f(x) + f(x + 2024)}{2}. \]Proposed by Pavle Martinović
3 replies
OgnjenTesic
May 22, 2025
atdaotlohbh
2 hours ago
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Easy P4 combi game with nt flavour
Maths_VC   0
2 hours ago
Source: Serbia JBMO TST 2025, Problem 4
Two players, Alice and Bob, play the following game, taking turns. In the beginning, the number $1$ is written on the board. A move consists of adding either $1$, $2$ or $3$ to the number written on the board, but only if the chosen number is coprime with the current number (for example, if the current number is $10$, then in a move a player can't choose the number $2$, but he can choose either $1$ or $3$). The player who first writes a perfect square on the board loses. Prove that one of the players has a winning strategy and determine who wins in the game.
0 replies
Maths_VC
2 hours ago
0 replies
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USAMO 2003 Problem 1
MithsApprentice   70
N 2 hours ago by endless_abyss
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.
70 replies
MithsApprentice
Sep 27, 2005
endless_abyss
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
11385 posts
jasperE3
#4 h  Apr 21, 2025, 10:44 PM
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