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Israel Number Theory
mathisreaI   63
N 43 minutes ago by Maximilian113
Source: IMO 2022 Problem 5
Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]
63 replies
mathisreaI
Jul 13, 2022
Maximilian113
43 minutes ago
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I need help for British maths olympiads
RCY   1
N an hour ago by Miquel-point
I’m a year ten student who’s going to take the bmo in one year.
However I have no experience in maths olympiads and the best results I have achieved so far was 25/60 in intermediate maths olympiads.
What shall I do?
I really need help!
1 reply
RCY
2 hours ago
Miquel-point
an hour ago
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Value of the sum
fermion13pi   0
2 hours ago
Source: Australia
Calculate the value of the sum

\sum_{k=1}^{9999999} \frac{1}{(k+1)^{3/2} + (k^2-1)^{1/3} + (k-1)^{2/3}}.
0 replies
fermion13pi
2 hours ago
0 replies
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NT Functional Equation
mkultra42   0
2 hours ago
Find all strictly increasing functions \(f: \mathbb{N} \to \mathbb{N}\) satsfying \(f(1)=1\) and:

\[ f(2n)f(2n+1)=9f(n)^2+3f(n)\]
0 replies
1 viewing
mkultra42
2 hours ago
0 replies
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Cyclic sum of 1/((3-c)(4-c))
v_Enhance   22
N 2 hours ago by Aiden-1089
Source: ELMO Shortlist 2013: Problem A6, by David Stoner
Let $a, b, c$ be positive reals such that $a+b+c=3$. Prove that \[18\sum_{\text{cyc}}\frac{1}{(3-c)(4-c)}+2(ab+bc+ca)\ge 15. \]Proposed by David Stoner
22 replies
v_Enhance
Jul 23, 2013
Aiden-1089
2 hours ago
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x^101=1 find 1/1+x+x^2+1/1+x^2+x^4+...+1/1+x^100+x^200
Mathmick51   6
N 2 hours ago by pi_quadrat_sechstel
Let $x^{101}=1$ such that $x\neq 1$. Find the value of $$\frac{1}{1+x+x^2}+\frac{1}{1+x^2+x^4}+\frac{1}{1+x^3+x^6}+\dots+\frac{1}{1+x^{100}+x^{200}}$$
6 replies
Mathmick51
Jun 22, 2021
pi_quadrat_sechstel
2 hours ago
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IMO Shortlist 2014 N5
hajimbrak   60
N 3 hours ago by sansgankrsngupta
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$.

Proposed by Belgium
60 replies
hajimbrak
Jul 11, 2015
sansgankrsngupta
3 hours ago
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n variables with n-gon sides
mihaig   0
3 hours ago
Source: Own
Let $n\geq3$ and let $a_1,a_2,\ldots, a_n\geq0$ be reals such that $\sum_{i=1}^{n}{\frac{1}{2a_i+n-2}}=1.$
Prove
$$\frac{24}{(n-1)(n-2)}\cdot\sum_{1\leq i<j<k\leq n}{a_ia_ja_k}\geq3\sum_{i=1}^{n}{a_i}+n.$$
0 replies
mihaig
3 hours ago
0 replies
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4 variables with quadrilateral sides
mihaig   3
N 3 hours ago by mihaig
Source: VL
Let $a,b,c,d\geq0$ satisfying
$$\frac1{a+1}+\frac1{b+1}+\frac1{c+1}+\frac1{d+1}=2.$$Prove
$$4\left(abc+abd+acd+bcd\right)\geq3\left(a+b+c+d\right)+4.$$
3 replies
mihaig
Today at 5:11 AM
mihaig
3 hours ago
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Calculate the distance of chess king!!
egxa   5
N 4 hours ago by Tesla12
Source: All Russian 2025 9.4
A chess king was placed on a square of an \(8 \times 8\) board and made $64$ moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called $\emph{pleasant}$ if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)
5 replies
egxa
Apr 18, 2025
Tesla12
4 hours ago
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Midpoint lies on Radical axis
Ankoganit   2
N Jul 1, 2016 by Ghost_rider
Source: My friend WizardMath
$C_1$ and $C_2$ are two disjoint circles. $AB$ is an external common tangent and $CD$ is an internal common tangent to them, such that $C$ is nearer to $AB$ than $D$. Suppose $P$ is the pole of $BC$ wrt $C_2$ and $M$ is the intersection point of the polars of $P$ wrt $C_1$ and $C_2$. Prove that the midpoint of $PM$ lies on the radical axis of $C_1$ and $C_2$.
2 replies
Ankoganit
May 6, 2016
Ghost_rider
Jul 1, 2016
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Midpoint lies on Radical axis
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Reveal topic tags
geometry
radical axis
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Source: My friend WizardMath
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Ankoganit
3070 posts
Ankoganit
#1 h  May 6, 2016, 3:09 PM • 3 Y
Y by sarthak7, Adventure10, Mango247
$C_1$ and $C_2$ are two disjoint circles. $AB$ is an external common tangent and $CD$ is an internal common tangent to them, such that $C$ is nearer to $AB$ than $D$. Suppose $P$ is the pole of $BC$ wrt $C_2$ and $M$ is the intersection point of the polars of $P$ wrt $C_1$ and $C_2$. Prove that the midpoint of $PM$ lies on the radical axis of $C_1$ and $C_2$.
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Ankoganit
3070 posts
Ankoganit
#2 h  May 25, 2016, 2:18 PM • 1 Y
Y by Adventure10
Any solutions? :roll:
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Ghost_rider
35 posts
Ghost_rider
#3 h  Jul 1, 2016, 2:48 AM • 2 Y
Y by Adventure10, Mango247
Let $DC$ $\cap$ $AB$ = $P$, $CB$ $\cap$ $AD$ = $M$ and midpoint of $PM$ = $R$. It´s easy to see that $P$ and $M$ are conjugate points respect to $C_1$ and $C_2$.Then the circle of diameter $MP$ is ortogonal to $C_1$ and $C_2$ and his radio is the tangent of $C_1$ and $C_2$ .Then $R$ lies on the radical axis of $C_1$ and $C_2$.
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