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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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Geometry
AlexCenteno2007   5
N a few seconds ago by AlexCenteno2007
Let DYC be the circumscribed circle of triangle DYC. Three tangent lines are drawn through D, Y and C, such that the lines passing through D and Y are perpendicular at A. Let B be the intersection of the tangent of Y and C. DX is drawn in such a way that AX = BY. Show that angle XDA = angle CDY.
5 replies
AlexCenteno2007
Wednesday at 5:26 PM
AlexCenteno2007
a few seconds ago
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Geometry
AlexCenteno2007   3
N 2 minutes ago by AlexCenteno2007
Let C be a point on a semicircle of diameter AB and let D be the mid-length of arc AC. Let E be the projection of point D on BC and F the intersection of AE with the semicircle. Prove that BF bisects segment DE.
3 replies
AlexCenteno2007
Yesterday at 3:54 AM
AlexCenteno2007
2 minutes ago
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Number of solutions
Ecrin_eren   4
N 41 minutes ago by Ecrin_eren
The given equation is:

x³ + 4y³ + 2y = (2024 + 2y)(xy + 1)

The question asks for the number of integer solutions.

4 replies
Ecrin_eren
Apr 3, 2025
Ecrin_eren
41 minutes ago
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Geometry
AlexCenteno2007   1
N 42 minutes ago by vanstraelen
Let A, B, C, and D be four distinct points on a straight line, in that order. The circles with diameters AC and BD intersect at X and Y. The straight line XY intersects BC at Z. Let P be a point on XY distinct from Z. The straight line CP intersects the circle with diameter AC at C and M, and the straight line BP intersects the circle with diameter BD at B and N. Show that AM, DN, and XY are aligned.
1 reply
AlexCenteno2007
Yesterday at 6:28 PM
vanstraelen
42 minutes ago
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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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Source of a combinatorics problem
isodynamicappolonius2903   0
Apr 5, 2025
Does anyone know exactly the source of this problem?? I just remember that it from a combinatorics book of Russia.
0 replies
isodynamicappolonius2903
Apr 5, 2025
0 replies
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Source of a combinatorics problem
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isodynamicappolonius2903
#1 h  Apr 5, 2025, 6:47 AM
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Does anyone know exactly the source of this problem?? I just remember that it from a combinatorics book of Russia.
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