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\textbf{A problem from LE ANH VINH book}
minhquannguyen   0
6 minutes ago
Source: LE ANH VINH, DINH HUONG BOI DUONG HOC SINH NANG KHIEU TOAN TAP 1 DAI SO
Let $n$ is a positive integer. Determine all functions $f:(1,+\infty)\to\mathbb{R}$ such that
\[f(x^{n+1}+y^{n+1})=x^nf(x)+y^nf(y),\forall x,y>1.\]
0 replies
minhquannguyen
6 minutes ago
0 replies
IMO ShortList 1999, algebra problem 1
orl   42
N an hour ago by ihategeo_1969
Source: IMO ShortList 1999, algebra problem 1
Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such the inequality

\[\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C
\left(\sum_{i}x_{i} \right)^4\]

holds for any $x_{1}, \ldots ,x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ summands). For this constant $C$, characterize the instances of equality.
42 replies
orl
Nov 13, 2004
ihategeo_1969
an hour ago
q(x) to be the product of all primes less than p(x)
orl   19
N an hour ago by ihategeo_1969
Source: IMO Shortlist 1995, S3
For an integer $x \geq 1$, let $p(x)$ be the least prime that does not divide $x$, and define $q(x)$ to be the product of all primes less than $p(x)$. In particular, $p(1) = 2.$ For $x$ having $p(x) = 2$, define $q(x) = 1$. Consider the sequence $x_0, x_1, x_2, \ldots$ defined by $x_0 = 1$ and \[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \] for $n \geq 0$. Find all $n$ such that $x_n = 1995$.
19 replies
orl
Aug 10, 2008
ihategeo_1969
an hour ago
Cyclic Quads and Parallel Lines
gracemoon124   16
N 3 hours ago by ohiorizzler1434
Source: 2015 British Mathematical Olympiad?
Let $ABCD$ be a cyclic quadrilateral. Let $F$ be the midpoint of the arc $AB$ of its circumcircle which does not contain $C$ or $D$. Let the lines $DF$ and $AC$ meet at $P$ and the lines $CF$ and $BD$ meet at $Q$. Prove that the lines $PQ$ and $AB$ are parallel.
16 replies
gracemoon124
Aug 16, 2023
ohiorizzler1434
3 hours ago
Radical Center on the Euler Line (USEMO 2020/3)
franzliszt   37
N 3 hours ago by Ilikeminecraft
Source: USEMO 2020/3
Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Let $\Gamma$ denote the circumcircle of triangle $ABC$, and $N$ the midpoint of $OH$. The tangents to $\Gamma$ at $B$ and $C$, and the line through $H$ perpendicular to line $AN$, determine a triangle whose circumcircle we denote by $\omega_A$. Define $\omega_B$ and $\omega_C$ similarly.
Prove that the common chords of $\omega_A$,$\omega_B$ and $\omega_C$ are concurrent on line $OH$.

Proposed by Anant Mudgal
37 replies
franzliszt
Oct 24, 2020
Ilikeminecraft
3 hours ago
Functional equation with powers
tapir1729   13
N 3 hours ago by ihategeo_1969
Source: TSTST 2024, problem 6
Determine whether there exists a function $f: \mathbb{Z}_{> 0} \rightarrow \mathbb{Z}_{> 0}$ such that for all positive integers $m$ and $n$,
\[f(m+nf(m))=f(n)^m+2024! \cdot m.\]Jaedon Whyte
13 replies
tapir1729
Jun 24, 2024
ihategeo_1969
3 hours ago
Powers of a Prime
numbertheorist17   34
N 3 hours ago by KevinYang2.71
Source: USA TSTST 2014, Problem 6
Suppose we have distinct positive integers $a, b, c, d$, and an odd prime $p$ not dividing any of them, and an integer $M$ such that if one considers the infinite sequence \begin{align*}
		ca &- db \\
		ca^2 &- db^2 \\
		ca^3 &- db^3 \\
		ca^4 &- db^4 \\
&\vdots
	\end{align*} and looks at the highest power of $p$ that divides each of them, these powers are not all zero, and are all at most $M$. Prove that there exists some $T$ (which may depend on $a,b,c,d,p,M$) such that whenever $p$ divides an element of this sequence, the maximum power of $p$ that divides that element is exactly $p^T$.
34 replies
numbertheorist17
Jul 16, 2014
KevinYang2.71
3 hours ago
IMO 2018 Problem 5
orthocentre   80
N 4 hours ago by OronSH
Source: IMO 2018
Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of positive integers. Suppose that there is an integer $N > 1$ such that, for each $n \geq N$, the number
$$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$$is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m+1}$ for all $m \geq M$.

Proposed by Bayarmagnai Gombodorj, Mongolia
80 replies
orthocentre
Jul 10, 2018
OronSH
4 hours ago
Line passes through fixed point, as point varies
Jalil_Huseynov   60
N 5 hours ago by Rayvhs
Source: APMO 2022 P2
Let $ABC$ be a right triangle with $\angle B=90^{\circ}$. Point $D$ lies on the line $CB$ such that $B$ is between $D$ and $C$. Let $E$ be the midpoint of $AD$ and let $F$ be the seconf intersection point of the circumcircle of $\triangle ACD$ and the circumcircle of $\triangle BDE$. Prove that as $D$ varies, the line $EF$ passes through a fixed point.
60 replies
Jalil_Huseynov
May 17, 2022
Rayvhs
5 hours ago
Tangent to two circles
Mamadi   2
N 5 hours ago by A22-
Source: Own
Two circles \( w_1 \) and \( w_2 \) intersect each other at \( M \) and \( N \). The common tangent to two circles nearer to \( M \) touch \( w_1 \) and \( w_2 \) at \( A \) and \( B \) respectively. Let \( C \) and \( D \) be the reflection of \( A \) and \( B \) respectively with respect to \( M \). The circumcircle of the triangle \( DCM \) intersect circles \( w_1 \) and \( w_2 \) respectively at points \( E \) and \( F \) (both distinct from \( M \)). Show that the line \( EF \) is the second tangent to \( w_1 \) and \( w_2 \).
2 replies
Mamadi
May 2, 2025
A22-
5 hours ago
Deduction card battle
anantmudgal09   55
N 6 hours ago by deduck
Source: INMO 2021 Problem 4
A Magician and a Detective play a game. The Magician lays down cards numbered from $1$ to $52$ face-down on a table. On each move, the Detective can point to two cards and inquire if the numbers on them are consecutive. The Magician replies truthfully. After a finite number of moves, the Detective points to two cards. She wins if the numbers on these two cards are consecutive, and loses otherwise.

Prove that the Detective can guarantee a win if and only if she is allowed to ask at least $50$ questions.

Proposed by Anant Mudgal
55 replies
anantmudgal09
Mar 7, 2021
deduck
6 hours ago
Geometry
Lukariman   7
N Yesterday at 8:13 PM by vanstraelen
Given circle (O) and point P outside (O). From P draw tangents PA and PB to (O) with contact points A, B. On the opposite ray of ray BP, take point M. The circle circumscribing triangle APM intersects (O) at the second point D. Let H be the projection of B on AM. Prove that $\angle HDM$ = 2∠AMP.
7 replies
Lukariman
Tuesday at 12:43 PM
vanstraelen
Yesterday at 8:13 PM
perpendicularity involving ex and incenter
Erken   20
N Yesterday at 7:48 PM by Baimukh
Source: Kazakhstan NO 2008 problem 2
Suppose that $ B_1$ is the midpoint of the arc $ AC$, containing $ B$, in the circumcircle of $ \triangle ABC$, and let $ I_b$ be the $ B$-excircle's center. Assume that the external angle bisector of $ \angle ABC$ intersects $ AC$ at $ B_2$. Prove that $ B_2I$ is perpendicular to $ B_1I_B$, where $ I$ is the incenter of $ \triangle ABC$.
20 replies
Erken
Dec 24, 2008
Baimukh
Yesterday at 7:48 PM
problem interesting
Cobedangiu   9
N May 1, 2025 by Cobedangiu
Let $a=3k^2+3k+1 (a,k \in N)$
$ii)$ Let $b \vdots a$ and $b$ is the sum of $3$ square numbers. Prove that: $b^n$ is the sum of $3$ square numbers
9 replies
Cobedangiu
Apr 30, 2025
Cobedangiu
May 1, 2025
problem interesting
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Cobedangiu
70 posts
#1
Y by
Let $a=3k^2+3k+1 (a,k \in N)$
$ii)$ Let $b \vdots a$ and $b$ is the sum of $3$ square numbers. Prove that: $b^n$ is the sum of $3$ square numbers
This post has been edited 2 times. Last edited by Cobedangiu, May 5, 2025, 4:02 PM
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Cobedangiu
70 posts
#2
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Cobedangiu wrote:
Let $a=3k^2+3k+1 (a,k \in N)$
$ii)$ Let $b \vdots a$ and $b$ is the sum of $3$ square numbers. Prove that: $b^n$ is the sum of $3$ square numbers

.....
This post has been edited 1 time. Last edited by Cobedangiu, Apr 30, 2025, 5:04 PM
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Cobedangiu
70 posts
#3
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Cobedangiu wrote:
Let $a=3k^2+3k+1 (a,k \in N)$
$ii)$ Let $b \vdots a$ and $b$ is the sum of $3$ square numbers. Prove that: $b^n$ is the sum of $3$ square numbers

no one?
This post has been edited 1 time. Last edited by Cobedangiu, Apr 30, 2025, 5:04 PM
Reason: .
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tom-nowy
124 posts
#4 • 1 Y
Y by Cobedangiu
$ i)$ here
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Cobedangiu
70 posts
#5
Y by
anyone have solution for next part?
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compoly2010
4 posts
#6
Y by
What do the three dots between b and a symbolise?
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Cobedangiu
70 posts
#7
Y by
compoly2010 wrote:
What do the three dots between b and a symbolise?

divisible
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vincentwant
1381 posts
#8
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https://en.wikipedia.org/wiki/Legendre%27s_three-square_theorem

Let $f(x)$ be equal to $x$ divided by the largest power of $4$ that divides $x$. Legendre's three-square theorem says that $x$ is expressible as the sum of three squares iff $f(x)$ is not $7$ mod $8$.

Observe that $f(x^2)$ is never $7$ mod $8$, so even $n$ always works. If $n$ is odd, if $f(b^n)$ is $7$ mod $8$, then $\nu_2(b^n)$ is even and thus $\nu_2(b)$ is even. Thus $f(b)$ is odd. Observe that if $f(b)$ is odd then $f(b^n)=f(b)^n$. Thus if $f(b^n)$ is $7$ mod $8$ then $f(b)^n$ is $7$ mod $8$. However, any odd number to an odd power is congruent to itself mod $8$, so this means $f(b)\equiv7\pmod8$, contradiction.
This post has been edited 1 time. Last edited by vincentwant, May 1, 2025, 1:45 AM
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tom-nowy
124 posts
#9
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Perhaps "three square numbers" means three positive squares. If zero were allowed, Question 1 would just be $a^2 = (3k^2 + 3k + 1)^2 + 0^2 + 0^2$, which would make it trivial.
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Cobedangiu
70 posts
#10
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tom-nowy wrote:
Perhaps "three square numbers" means three positive squares. If zero were allowed, Question 1 would just be $a^2 = (3k^2 + 3k + 1)^2 + 0^2 + 0^2$, which would make it trivial.

it is not allowed
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