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Problem 2
delegat   147
N an hour ago by math-olympiad-clown
Source: 0
Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that
\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]

Proposed by Angelo Di Pasquale, Australia
147 replies
delegat
Jul 10, 2012
math-olympiad-clown
an hour ago
Coloring points of a square, finding a monochromatic hexagon
goodar2006   6
N 2 hours ago by quantam13
Source: Iran 3rd round 2012-Special Lesson exam-Part 2-P1
Prove that for each coloring of the points inside or on the boundary of a square with $1391$ colors, there exists a monochromatic regular hexagon.
6 replies
goodar2006
Sep 15, 2012
quantam13
2 hours ago
Van der Warden Theorem!
goodar2006   7
N 2 hours ago by quantam13
Source: Iran 3rd round 2012-Special Lesson exam-Part 2-P2
Suppose $W(k,2)$ is the smallest number such that if $n\ge W(k,2)$, for each coloring of the set $\{1,2,...,n\}$ with two colors there exists a monochromatic arithmetic progression of length $k$. Prove that


$W(k,2)=\Omega (2^{\frac{k}{2}})$.
7 replies
goodar2006
Sep 15, 2012
quantam13
2 hours ago
Maxi-inequality
giangtruong13   0
2 hours ago
Let $a,b,c >0$ and $a+b+c=2abc$. Find max: $$P= \sum_{cyc} \frac{a+2}{\sqrt{6(a^2+2)}}$$
0 replies
giangtruong13
2 hours ago
0 replies
Isosceles triangles among a group of points
goodar2006   2
N 2 hours ago by quantam13
Source: Iran 3rd round 2012-Special Lesson exam-Part1-P2
Consider a set of $n$ points in plane. Prove that the number of isosceles triangles having their vertices among these $n$ points is $\mathcal O (n^{\frac{7}{3}})$. Find a configuration of $n$ points in plane such that the number of equilateral triangles with vertices among these $n$ points is $\Omega (n^2)$.
2 replies
1 viewing
goodar2006
Jul 27, 2012
quantam13
2 hours ago
APMO Number Theory
somebodyyouusedtoknow   12
N 2 hours ago by math-olympiad-clown
Source: APMO 2023 Problem 2
Find all integers $n$ satisfying $n \geq 2$ and $\dfrac{\sigma(n)}{p(n)-1} = n$, in which $\sigma(n)$ denotes the sum of all positive divisors of $n$, and $p(n)$ denotes the largest prime divisor of $n$.
12 replies
somebodyyouusedtoknow
Jul 5, 2023
math-olympiad-clown
2 hours ago
My Unsolved Problem
ZeltaQN2008   0
2 hours ago
Source: IDK
Let \( P(x) = x^{2024} + a_{2023}x^{2023} + \cdots + a_1x + a_0 \) be a polynomial with real coefficients.

(a) Suppose that \( 2023a_{2023}^2 - 4048a_{2022} < 0 \). Prove that the polynomial \( P(x) \) cannot have 2024 real roots.

(b) Suppose that \( a_0 = 1 \) and \( 2023(a_1^2 + a_2^2 + \cdots + a_{2023}^2) \leq 4 \). Prove that \( P(x) \geq 0 \) for all real numbers \( x \).
0 replies
ZeltaQN2008
2 hours ago
0 replies
Points of a grid
goodar2006   2
N 2 hours ago by quantam13
Source: Iran 3rd round 2012-Special Lesson exam-Part1-P4
Prove that from an $n\times n$ grid, one can find $\Omega (n^{\frac{5}{3}})$ points such that no four of them are vertices of a square with sides parallel to lines of the grid. Imagine yourself as Erdos (!) and guess what is the best exponent instead of $\frac{5}{3}$!
2 replies
goodar2006
Jul 27, 2012
quantam13
2 hours ago
Classical NT FE
Kimchiks926   6
N 3 hours ago by math-olympiad-clown
Source: Baltic Way 2022, Problem 16
Let $\mathbb{Z^+}$ denote the set of positive integers. Find all functions $f:\mathbb{Z^+} \to \mathbb{Z^+}$ satisfying the condition
$$ f(a) + f(b) \mid (a + b)^2$$for all $a,b \in \mathbb{Z^+}$
6 replies
Kimchiks926
Nov 12, 2022
math-olympiad-clown
3 hours ago
Hagge circle, Thomson cubic, coaxal
kosmonauten3114   0
3 hours ago
Source: My own (maybe well-known)
Let $\triangle{ABC}$ be a scalene triangle, $\triangle{M_AM_BM_C}$ its medial triangle, and $P$ a point on the Thomson cubic (= $\text{K002}$) of $\triangle{ABC}$. (Suppose that $P \notin \odot(ABC)$ ).
Let $\triangle{A'B'C'}$ be the circumcevian triangle of $P$ wrt $\triangle{ABC}$.
Let $\triangle{P_AP_BP_C}$ be the pedal triangle of $P$ wrt $\triangle{ABC}$.
Let $A_1$ be the reflection in $BC$ of $A'$. Define $B_1$, $C_1$ cyclically.
Let $A_2$ be the reflection in $M_A$ of $A'$. Define $B_2$, $C_2$ cyclically.
Let $A_3$ be the reflection in $P_A$ of $A'$. Define $B_3$, $C_3$ cyclically.

Prove that $\odot(A_1B_1C_1)$, $\odot(A_2B_2C_2)$, $\odot(A_3B_3C_3)$ and the orthocentroidal circle of $\triangle{ABC}$ are coaxal.
0 replies
kosmonauten3114
3 hours ago
0 replies
Basic, Part B
CarSa   1
N Apr 26, 2025 by Math-lover1
For each four--digit number $\overline {abcd}$, that is, with $a$ nonzero, let $P(\overline {abcd})$ be the product $(a+b)(a+c)(a+d)(b+c)(b+d)(c+d)$.
For example, $P(2022) = (2+0)(2+2)(2+2)(0+2)(0+2)(2+2) = 512$ and $P(1234) = (1+2)(1+3)(1+4)(2+3)(2+4)(3+4)$.
How many numbers $\overline {abcd}$ with at least one $0$ amoung their digits satisfy that $P(\overline {abcd})$ is a power of 2?
1 reply
CarSa
Apr 24, 2025
Math-lover1
Apr 26, 2025
Basic, Part B
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CarSa
3 posts
#1 • 1 Y
Y by PikaPika999
For each four--digit number $\overline {abcd}$, that is, with $a$ nonzero, let $P(\overline {abcd})$ be the product $(a+b)(a+c)(a+d)(b+c)(b+d)(c+d)$.
For example, $P(2022) = (2+0)(2+2)(2+2)(0+2)(0+2)(2+2) = 512$ and $P(1234) = (1+2)(1+3)(1+4)(2+3)(2+4)(3+4)$.
How many numbers $\overline {abcd}$ with at least one $0$ amoung their digits satisfy that $P(\overline {abcd})$ is a power of 2?
This post has been edited 2 times. Last edited by CarSa, Apr 24, 2025, 8:48 PM
Reason: yes
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Math-lover1
355 posts
#2 • 2 Y
Y by PikaPika999, CarSa
My solution

Hopefully I'm not too late :)
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