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JBMO TST Bosnia and Herzegovina 2023 P1
FishkoBiH   1
N 8 minutes ago by clarkculus
Source: JBMO TST Bosnia and Herzegovina 2023 P1
Determine all real numbers $a, b, c, d$ for which

$ab+cd=6$
$ac+bd=3$
$ad+bc=2$
$a+b+c+d=6$
1 reply
FishkoBiH
an hour ago
clarkculus
8 minutes ago
JBMO TST Bosnia and Herzegovina 2024 P1
FishkoBiH   0
20 minutes ago
Source: JBMO TST Bosnia and Herzegovina 2024 P1
Let $a$,$b$,$c$ be real numbers different from 0 for which $ab$ + $bc$+ $ca$ = 0 holds
a) Prove that ($a$+$b$)($b$+$c$)($c$+$a$)≠ 0
b) Let $X$ = $a$ + $b$ + $c$ and $Y$ = $\frac{1}{a+b}$ + $\frac{1}{b+c}$ + $\frac{1}{c+a}$. Prove that numbers $X$ and $Y$ are both positive or both negative.
0 replies
FishkoBiH
20 minutes ago
0 replies
Inspired by 2025 Beijing
sqing   7
N 31 minutes ago by ytChen
Source: Own
Let $ a,b,c,d >0  $ and $ (a^2+b^2+c^2)(b^2+c^2+d^2)=36. $ Prove that
$$ab^2c^2d \leq 8$$$$a^2bcd^2 \leq 16$$$$ ab^3c^3d \leq \frac{2187}{128}$$$$ a^3bcd^3 \leq \frac{2187}{32}$$
7 replies
sqing
Yesterday at 4:56 PM
ytChen
31 minutes ago
Inequality em981
oldbeginner   18
N 36 minutes ago by xzlbq
Source: Own
Let $a, b, c>0, a+b+c=3$. Prove that
\[\sqrt{a+\frac{9}{b+2c}}+\sqrt{b+\frac{9}{c+2a}}+\sqrt{c+\frac{9}{a+2b}}+\frac{2(ab+bc+ca)}{9}\ge\frac{20}{3}\]
18 replies
oldbeginner
Sep 22, 2016
xzlbq
36 minutes ago
JBMO TST Bosnia and Herzegovina 2023 P2
FishkoBiH   1
N 36 minutes ago by clarkculus
Source: JBMO TST Bosnia and Herzegovina 2023 P2
Determine all non negative integers $x$ and $y$ such that $6^x$ + $2^y$ + 2 is a perfect square.
1 reply
FishkoBiH
an hour ago
clarkculus
36 minutes ago
Divisiblity...
TUAN2k8   1
N 39 minutes ago by Natrium
Source: Own
Let $m$ and $n$ be two positive integer numbers such that $m \le n$.Prove that $\binom{n}{m}$ divides $lcm(1,2,...,n)$
1 reply
TUAN2k8
Today at 6:13 AM
Natrium
39 minutes ago
JBMO TST Bosnia and Herzegovina 2023 P4
FishkoBiH   0
an hour ago
Source: JBMO TST Bosnia and Herzegovina 2023 P4
Let $n$ be a positive integer. A board with a format $n*n$ is divided in $n*n$ equal squares.Determine all integers $n$3 such that the board can be covered in $2*1$ (or $1*2$) pieces so that there is exactly one empty square in each row and each column.
0 replies
FishkoBiH
an hour ago
0 replies
JBMO TST Bosnia and Herzegovina 2023 P3
FishkoBiH   0
an hour ago
Source: JBMO TST Bosnia and Herzegovina 2023 P3
Let ABC be an acute triangle with an incenter $I$.The Incircle touches sides $AC$ and $AB$ in $E$ and $F$ ,respectively. Lines CI and EF intersect at $S$. The point $T$$I$ is on the line AI so that $EI$=$ET$.If $K$ is the foot of the altitude from $C$ in triangle $ABC$,prove that points $K$,$S$ and $T$ are colinear.
0 replies
FishkoBiH
an hour ago
0 replies
Locus of points P in triangle ABC
v_Enhance   25
N an hour ago by alexanderchew
Source: USA January TST for IMO 2016, Problem 3
Let $ABC$ be an acute scalene triangle and let $P$ be a point in its interior. Let $A_1$, $B_1$, $C_1$ be projections of $P$ onto triangle sides $BC$, $CA$, $AB$, respectively. Find the locus of points $P$ such that $AA_1$, $BB_1$, $CC_1$ are concurrent and $\angle PAB + \angle PBC + \angle PCA = 90^{\circ}$.
25 replies
v_Enhance
May 17, 2016
alexanderchew
an hour ago
number theory diophantic with factorials and primes
skellyrah   4
N 2 hours ago by skellyrah
Source: by me
find all triplets of non negative integers (a,b,p) where p is prime such that $$ a! + b! + 7ab = p^2 $$
4 replies
skellyrah
Feb 16, 2025
skellyrah
2 hours ago
primes,exponentials,factorials
skellyrah   6
N 2 hours ago by skellyrah
find all primes p,q such that $$ \frac{p^q+q^p-p-q}{p!-q!} $$is a prime number
6 replies
skellyrah
Apr 30, 2025
skellyrah
2 hours ago
Serbian selection contest for the IMO 2025 - P5
OgnjenTesic   2
N 2 hours ago by GreenTea2593
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
2 replies
OgnjenTesic
May 22, 2025
GreenTea2593
2 hours ago
Inequality with three conditions
oVlad   3
N Apr 22, 2025 by sqing
Source: Romania EGMO TST 2019 Day 1 P3
Let $a,b,c$ be non-negative real numbers such that \[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]Prove that $a^2+b^2+c^2\leqslant 2abc+1.$
3 replies
oVlad
Apr 21, 2025
sqing
Apr 22, 2025
Inequality with three conditions
G H J
G H BBookmark kLocked kLocked NReply
Source: Romania EGMO TST 2019 Day 1 P3
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oVlad
1746 posts
#1
Y by
Let $a,b,c$ be non-negative real numbers such that \[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]Prove that $a^2+b^2+c^2\leqslant 2abc+1.$
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Haris1
77 posts
#2
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Nice ineq,
$3(a+1)(b+1)(c+1)\geq (a+1)(b+1)(a+b)+(a+1)(a+c)(c+1)+(b+c)(b+1)(c+1)$
and using $(a+1)(b+1)(c+1)\geq (a+b)(b+c)(c+a)$ completes it.
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Quantum-Phantom
276 posts
#3
Y by
Because
\begin{align*}8abc+4-4\sum_{\rm cyc}a^2=&\left(1+\sum_{\rm cyc}a\right)\sum_{\rm cyc}(a+1-b-c)(b+1-c-a)\\&+\prod_{\rm cyc}(a+1-b-c)\ge0.\end{align*}
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sqing
42398 posts
#4
Y by
oVlad wrote:
Let $a,b,c$ be non-negative real numbers such that \[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]Prove that $a^2+b^2+c^2\leqslant 2abc+1.$
Indian 2007
https://artofproblemsolving.com/community/c6h1754566p11450296
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