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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Apr 2, 2025
0 replies
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JBMO Shortlist 2023 A5
Orestis_Lignos   11
N a few seconds ago by sqing
Source: JBMO Shortlist 2023, A5
Let $a \geq b \geq 1 \geq c \geq 0$ be real numbers such that $a+b+c=3$. Show that

$$3 \left( \frac{a}{b}+\frac{b}{a} \right ) \geq 4c^2+\frac{a^2}{b}+\frac{b^2}{a}$$
11 replies
+1 w
Orestis_Lignos
Jun 28, 2024
sqing
a few seconds ago
J
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Classical factorial number theory
Orestis_Lignos   21
N 3 minutes ago by Pseudo_Matter
Source: JBMO 2023 Problem 1
Find all pairs $(a,b)$ of positive integers such that $a!+b$ and $b!+a$ are both powers of $5$.

Nikola Velov, North Macedonia
21 replies
1 viewing
Orestis_Lignos
Jun 26, 2023
Pseudo_Matter
3 minutes ago
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problem....
Cobedangiu   4
N 10 minutes ago by Jackson0423
$a,b,c>0$ and $a+b+c=1$
Prove that: $Q=ab+bc+ca-2abc\le\dfrac{7}{27}$
4 replies
Cobedangiu
5 hours ago
Jackson0423
10 minutes ago
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Math problem
Glist   2
N 11 minutes ago by Gggvds1
Given six distinct points on a plane, all pairwise distances between which are different. Prove that there exists a line segment connecting two of these points which is the longest side in one triangle formed by three of the points, and the shortest side in another triangle formed by three of the points.
2 replies
1 viewing
Glist
2 hours ago
Gggvds1
11 minutes ago
J
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A Loggy Problem from Pythagoras
Mathzeus1024   2
N an hour ago by wh0nix
Prove or disprove: $\exists x \in \mathbb{R}^{+}$ such that $\ln(x), \ln(2x), \ln(3x)$ are the lengths of a right triangle.
2 replies
1 viewing
Mathzeus1024
5 hours ago
wh0nix
an hour ago
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Polynomials
CuriousBabu   10
N an hour ago by wh0nix
\[ \frac{(x+y+z)^5 - x^5 - y^5 - z^5}{(x+y)(y+z)(z+x)} = 0 \]
Find the number of real solutions.
10 replies
CuriousBabu
Apr 14, 2025
wh0nix
an hour ago
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Combinatorics
AzSolver257   1
N an hour ago by SomeonecoolLovesMaths

1)Two players $A$ and $B$ play a series of of $2n$ games. Each game either results in a win or loss for $A$. Total number of ways in which $A$ can win the series is:
A) $ \frac{1}{2} ( 2^{2n} - \binom{2n}{n})$
B) $ \frac{1}{2} ( 2^{2n} - 2\cdot(\binom{2n}{n}))$
C) $ \frac{1}{2} ( 2^{n} - \binom{2n}{n})$
D) $ \frac{1}{2} ( 2^{n} - 2 \cdot \binom{2n}{n})$
2) A person predicts the outcome of 20 cricket matches hof his home team. Each match can either result in a win , loss or tie for the home team. The total number of ways in which he can make the predictions so that 10 predictions is correct is equal to:

A) $ \binom{20}{10} \times 2^{10} $
B) $ \binom{20}{10} \times 3^{10} $
C) $ \binom{20}{10} \times 3^{20} $
D) $ \binom{20}{10} \times 2^{20} $

Please mention the solutions properly.
1 reply
AzSolver257
2 hours ago
SomeonecoolLovesMaths
an hour ago
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Algebra Problems
ilikemath247365   9
N 2 hours ago by SomeonecoolLovesMaths
Find all real $(a, b)$ with $a + b = 1$ such that

$(a + \frac{1}{a})^{2} + (b + \frac{1}{b})^{2} = \frac{25}{2}$.
9 replies
ilikemath247365
Apr 14, 2025
SomeonecoolLovesMaths
2 hours ago
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How to prove one-one function
Vulch   7
N 3 hours ago by SomeonecoolLovesMaths
Hello everyone,
I am learning functional equations.
To prove the below problem one -one function,I have taken two non-negative real numbers $ (1,2)$ from the domain $\Bbb R_{*},$ and put those numbers into the given function f(x)=1/x.It gives us 1=1/2.But it's not true.So ,it can't be one-one function.But in the answer,it is one-one function.Would anyone enlighten me where is my fault? Thank you!
7 replies
Vulch
Apr 11, 2025
SomeonecoolLovesMaths
3 hours ago
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Inequalities
lgx57   5
N 5 hours ago by jjmmxx
Let $0 < a,b,c < 1$. Prove that

$$a(1-b)+b(1-c)+c(1-a)<1$$
5 replies
lgx57
Mar 19, 2025
jjmmxx
5 hours ago
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ez problem....
Cobedangiu   0
5 hours ago
Let $x,y \in Z$ and $xy \cancel \vdots7$
Find $n \in Z^+$.
$x^2+y^2+xy=7^n$
0 replies
Cobedangiu
5 hours ago
0 replies
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hard number theory
eric201291   2
N Today at 9:22 AM by eric201291
Prove:There are no integers x, y, that y^2+9998587980=x^3.
2 replies
eric201291
Apr 16, 2025
eric201291
Today at 9:22 AM
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Inequalities
sqing   9
N Today at 8:54 AM by sqing
Let $ a,b $ be reals such that $ a^2+ab+b^2 =3$ . Prove that
$$ \frac{4}{ 3}\geq \frac{1}{ a^2+5 }+ \frac{1}{ b^2+5 }+ab \geq -\frac{11}{4 }$$$$ \frac{13}{ 4}\geq \frac{1}{ a^2+5 }+ \frac{1}{ b^2+5 }+ab \geq -\frac{2}{3 }$$$$ \frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq -\frac{17}{6 }$$$$ \frac{19}{ 6}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2}$$Let $ a,b $ be reals such that $ a^2-ab+b^2 =1 $ . Prove that
$$ \frac{3}{ 2}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }+ab \geq \frac{4}{15 }$$$$ \frac{14}{ 15}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }-ab \geq -\frac{1}{2 }$$$$ \frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq \frac{13}{42 }$$$$ \frac{41}{ 42}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2 }$$
9 replies
sqing
Apr 16, 2025
sqing
Today at 8:54 AM
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Geometry
AlexCenteno2007   4
N Today at 8:40 AM by sunken rock
Let ABC be an isosceles triangle with AB = AC and M the midpoint of BC. Consider a point E outside the triangle such that BE = BM and CE perpendicular to AB. The point of intersection of the perpendicular bisector of segment EB with the circumcircle of triangle AMB, which is on the same side as A with respect to BE, is point F. Show that angle FME = 90°
4 replies
AlexCenteno2007
Yesterday at 3:49 AM
sunken rock
Today at 8:40 AM
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Sequence of functions
mathlover1231   0
Mar 30, 2025
Source: Own
Let f:N->N be a function such that f(1) = 1, f(n+1) = f(n) + 2^f(n) for every positive integer n. Prove that all numbers f(1), f(2), …, f(3^2023) give different remainders when divided by 3^2023
0 replies
mathlover1231
Mar 30, 2025
0 replies
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Sequence of functions
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mathlover1231
#1 h  Mar 30, 2025, 2:11 PM
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Let f:N->N be a function such that f(1) = 1, f(n+1) = f(n) + 2^f(n) for every positive integer n. Prove that all numbers f(1), f(2), …, f(3^2023) give different remainders when divided by 3^2023
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