Stay ahead of learning milestones! Enroll in a class over the summer!

G
Topic
First Poster
Last Poster
IMO 2009, Problem 5
orl   91
N an hour ago by maromex
Source: IMO 2009, Problem 5
Determine all functions $ f$ from the set of positive integers to the set of positive integers such that, for all positive integers $ a$ and $ b$, there exists a non-degenerate triangle with sides of lengths
\[ a, f(b) \text{ and } f(b + f(a) - 1).\]
(A triangle is non-degenerate if its vertices are not collinear.)

Proposed by Bruno Le Floch, France
91 replies
1 viewing
orl
Jul 16, 2009
maromex
an hour ago
What is thiss
EeEeRUT   4
N an hour ago by lksb
Source: Thailand MO 2025 P6
Find all function $f: \mathbb{R}^+ \rightarrow \mathbb{R}$,such that the inequality $$f(x) + f\left(\frac{y}{x}\right) \leqslant \frac{x^3}{y^2} + \frac{y}{x^3}$$holds for all positive reals $x,y$ and for every positive real $x$, there exist positive reals $y$, such that the equality holds.
4 replies
EeEeRUT
Wednesday at 6:45 AM
lksb
an hour ago
problem about equation
jred   2
N 2 hours ago by Truly_for_maths
Source: China south east mathematical Olympiad 2006 problem4
Given any positive integer $n$, let $a_n$ be the real root of equation $x^3+\dfrac{x}{n}=1$. Prove that
(1) $a_{n+1}>a_n$;
(2) $\sum_{i=1}^{n}\frac{1}{(i+1)^2a_i} <a_n$.
2 replies
jred
Jul 4, 2013
Truly_for_maths
2 hours ago
number theory and combinatoric sets of integers relations
trying_to_solve_br   40
N 2 hours ago by MathLuis
Source: IMO 2021 P6
Let $m\ge 2$ be an integer, $A$ a finite set of integers (not necessarily positive) and $B_1,B_2,...,B_m$ subsets of $A$. Suppose that, for every $k=1,2,...,m$, the sum of the elements of $B_k$ is $m^k$. Prove that $A$ contains at least $\dfrac{m}{2}$ elements.
40 replies
trying_to_solve_br
Jul 20, 2021
MathLuis
2 hours ago
Interesting inequalities
sqing   10
N 3 hours ago by ytChen
Source: Own
Let $ a,b,c\geq 0 , (a+k )(b+c)=k+1.$ Prove that
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq  \frac{2k-3+2\sqrt{k+1}}{3k-1}$$Where $ k\geq \frac{2}{3}.$
Let $ a,b,c\geq 0 , (a+1)(b+c)=2.$ Prove that
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq 2\sqrt{2}-1$$Let $ a,b,c\geq 0 , (a+3)(b+c)=4.$ Prove that
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq  \frac{7}{4}$$Let $ a,b,c\geq 0 , (3a+2)(b+c)= 5.$ Prove that
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq  \frac{2(2\sqrt{15}-5)}{3}$$
10 replies
1 viewing
sqing
May 10, 2025
ytChen
3 hours ago
GMO 2024 P1
Z4ADies   5
N 4 hours ago by awesomeming327.
Source: Geometry Mains Olympiad (GMO) 2024 P1
Let \( ABC \) be an acute triangle. Define \( I \) as its incenter. Let \( D \) and \( E \) be the incircle's tangent points to \( AC \) and \( AB \), respectively. Let \( M \) be the midpoint of \( BC \). Let \( G \) be the intersection point of a perpendicular line passing through \( M \) to \( DE \). Line \( AM \) intersects the circumcircle of \( \triangle ABC \) at \( H \). The circumcircle of \( \triangle AGH \) intersects line \( GM \) at \( J \). Prove that quadrilateral \( BGCJ \) is cyclic.

Author:Ismayil Ismayilzada (Azerbaijan)
5 replies
Z4ADies
Oct 20, 2024
awesomeming327.
4 hours ago
Power sequence
TheUltimate123   7
N 4 hours ago by MathLuis
Source: ELMO Shortlist 2023 N2
Determine the greatest positive integer \(n\) for which there exists a sequence of distinct positive integers \(s_1\), \(s_2\), \(\ldots\), \(s_n\) satisfying \[s_1^{s_2}=s_2^{s_3}=\cdots=s_{n-1}^{s_n}.\]
Proposed by Holden Mui
7 replies
TheUltimate123
Jun 29, 2023
MathLuis
4 hours ago
IMO Shortlist 2013, Combinatorics #4
lyukhson   21
N 6 hours ago by Ciobi_
Source: IMO Shortlist 2013, Combinatorics #4
Let $n$ be a positive integer, and let $A$ be a subset of $\{ 1,\cdots ,n\}$. An $A$-partition of $n$ into $k$ parts is a representation of n as a sum $n = a_1 + \cdots + a_k$, where the parts $a_1 , \cdots , a_k $ belong to $A$ and are not necessarily distinct. The number of different parts in such a partition is the number of (distinct) elements in the set $\{ a_1 , a_2 , \cdots , a_k \} $.
We say that an $A$-partition of $n$ into $k$ parts is optimal if there is no $A$-partition of $n$ into $r$ parts with $r<k$. Prove that any optimal $A$-partition of $n$ contains at most $\sqrt[3]{6n}$ different parts.
21 replies
lyukhson
Jul 9, 2014
Ciobi_
6 hours ago
Cycle in a graph with a minimal number of chords
GeorgeRP   4
N Yesterday at 8:29 PM by CBMaster
Source: Bulgaria IMO TST 2025 P3
In King Arthur's court every knight is friends with at least $d>2$ other knights where friendship is mutual. Prove that King Arthur can place some of his knights around a round table in such a way that every knight is friends with the $2$ people adjacent to him and between them there are at least $\frac{d^2}{10}$ friendships of knights that are not adjacent to each other.
4 replies
GeorgeRP
Wednesday at 7:51 AM
CBMaster
Yesterday at 8:29 PM
amazing balkan combi
egxa   8
N Yesterday at 7:55 PM by Gausikaci
Source: BMO 2025 P4
There are $n$ cities in a country, where $n \geq 100$ is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities $A$ and $B$ we define:

$(i)$ A $\emph{path}$ between $A$ and $B$ as a sequence of distinct cities $A = C_0, C_1, \dots, C_k, C_{k+1} = B$, $k \geq 0$, such that there are direct flights between $C_i$ and $C_{i+1}$ for every $0 \leq i \leq k$;
$(ii)$ A $\emph{long path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has more cities;
$(iii)$ A $\emph{short path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has fewer cities.
Assume that for any pair of cities $A$ and $B$ in the country, there exist a long path and a short path between them that have no cities in common (except $A$ and $B$). Let $F$ be the total number of pairs of cities in the country that are connected by direct flights. In terms of $n$, find all possible values $F$

Proposed by David-Andrei Anghel, Romania.
8 replies
egxa
Apr 27, 2025
Gausikaci
Yesterday at 7:55 PM
Prove that AX, BY, CZ are concurrent
phongzel34-vietnam   1
N Jul 1, 2022 by ancamagelqueme
Let $\Delta ABC$, incircle $(I)$ and $A$ - excircle $(I_a)$. $(I_a)$ touches $BC$ at $D$. $ID$ cuts $(I_a)$ at $X$. Define $Y, Z$ similarly. Prove that $AX, BY, CZ$ are concurrent
1 reply
phongzel34-vietnam
Jul 1, 2022
ancamagelqueme
Jul 1, 2022
Prove that AX, BY, CZ are concurrent
G H J
G H BBookmark kLocked kLocked NReply
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
phongzel34-vietnam
66 posts
#1 • 1 Y
Y by tiendung2006
Let $\Delta ABC$, incircle $(I)$ and $A$ - excircle $(I_a)$. $(I_a)$ touches $BC$ at $D$. $ID$ cuts $(I_a)$ at $X$. Define $Y, Z$ similarly. Prove that $AX, BY, CZ$ are concurrent
Attachments:
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
ancamagelqueme
104 posts
#2 • 1 Y
Y by Mango247
$X$ is the point in which the A-excircle is tangent to the circle OA that passes through vertices B and C, described in X(5423).
Z K Y
N Quick Reply
G
H
=
a