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i love mordell
MR.1   1
N an hour ago by MR.1
Source: own
find all pairs of $(m,n)$ such that $n^2-79=m^3$
1 reply
MR.1
an hour ago
MR.1
an hour ago
J
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MM 2201 (Symmetric Inequality with Weird Sharp Case)
kgator   1
N an hour ago by CHESSR1DER
Source: Mathematics Magazine Volume 97 (2024), Issue 4: https://doi.org/10.1080/0025570X.2024.2393998
2201. Proposed by Leonard Giugiuc, Drobeta-Turnu Severin, Romania. Find all real numbers $K$ such that
$$a^2 + b^2 + c^2 - 3 \geq K(a + b + c - 3)$$for all nonnegative real numbers $a$, $b$, and $c$ with $abc \leq 1$.
1 reply
kgator
2 hours ago
CHESSR1DER
an hour ago
J
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Interesting Inequality Problem
Omerking   0
an hour ago
Let $a,b,c$ be three non-negative real numbers satisfying $a+b+c+abc=4.$
Prove that
$$\frac{a}{a^{2}+1}+\frac{b}{b^{2}+1}+\frac{c}{c^{2}+1} \leq\frac{6}{13-3ab-3bc-3ca}$$
0 replies
Omerking
an hour ago
0 replies
J
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[SEIF Q1] FE on x^3+xy...( ͡° ͜ʖ ͡°)
EmilXM   18
N an hour ago by jasperE3
Source: SEIF 2022
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that any real numbers $x$ and $y$ satisfy
$$x^3+f(x)f(y)=f(f(x^3)+f(xy)).$$Proposed by EmilXM
18 replies
EmilXM
Mar 12, 2022
jasperE3
an hour ago
J
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RGB chessboard
BR1F1SZ   1
N an hour ago by alfonsoramires
Source: 2025 Argentina TST P3
A $100 \times 100$ board has some of its cells coloured red, blue, or green. Each cell is coloured with at most one colour, and some cells may remain uncoloured. Additionally, there is at least one cell of each colour. Two coloured cells are said to be friends if they have different colours and lie in the same row or in the same column. The following conditions are satisfied:
[list=i]
[*]Each coloured cell has exactly three friends.
[*]All three friends of any given coloured cell lie in the same row or in the same column.
[/list]
Determine the maximum number of cells that can be coloured on the board.
1 reply
BR1F1SZ
Tuesday at 11:15 PM
alfonsoramires
an hour ago
J
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JBMO 2013 Problem 2
Igor   43
N 2 hours ago by EVS383
Source: Proposed by Macedonia
Let $ABC$ be an acute-angled triangle with $AB<AC$ and let $O$ be the centre of its circumcircle $\omega$. Let $D$ be a point on the line segment $BC$ such that $\angle BAD = \angle CAO$. Let $E$ be the second point of intersection of $\omega$ and the line $AD$. If $M$, $N$ and $P$ are the midpoints of the line segments $BE$, $OD$ and $AC$, respectively, show that the points $M$, $N$ and $P$ are collinear.
43 replies
Igor
Jun 23, 2013
EVS383
2 hours ago
J
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Prove \frac{x-y}{x^6-y^6}\leq \frac{4}{3}(x+y) if x^4-y^4=x-y
andria   11
N 2 hours ago by iliya8788
Source: Iran National Olympiad 2017, Second Round, Problem 4
Let $x,y$ be two positive real numbers such that $x^4-y^4=x-y$. Prove that
$$\frac{x-y}{x^6-y^6}\leq \frac{4}{3}(x+y).$$
11 replies
andria
Apr 21, 2017
iliya8788
2 hours ago
J
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[SEIF A3] Very elegant alg FE
gghx   6
N 2 hours ago by jasperE3
Source: SEIF 2022
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for any $x,y\in\mathbb{R}$,
\[f(xf(y)) + xf(x - y) = x^2 + f(2x).\]Proposed by hyay

6 replies
gghx
Mar 12, 2022
jasperE3
2 hours ago
J
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Sequence of numbers in form of a^2+b^2
TheOverlord   13
N 2 hours ago by bin_sherlo
Source: Iran TST 2015, exam 1, day 1 problem 3
Let $ b_1<b_2<b_3<\dots $ be the sequence of all natural numbers which are sum of squares of two natural numbers.
Prove that there exists infinite natural numbers like $m$ which $b_{m+1}-b_m=2015$ .
13 replies
TheOverlord
May 11, 2015
bin_sherlo
2 hours ago
J
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Nice problem
Tiks   24
N 2 hours ago by Nari_Tom
Source: IMO Shortlist 2000, G6
Let $ ABCD$ be a convex quadrilateral. The perpendicular bisectors of its sides $ AB$ and $ CD$ meet at $ Y$. Denote by $ X$ a point inside the quadrilateral $ ABCD$ such that $ \measuredangle ADX = \measuredangle BCX < 90^{\circ}$ and $ \measuredangle DAX = \measuredangle CBX < 90^{\circ}$. Show that $ \measuredangle AYB = 2\cdot\measuredangle ADX$.
24 replies
Tiks
Nov 2, 2005
Nari_Tom
2 hours ago
J
a