Basic, Part B

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

CarSa

#1 h Apr 24, 2025, 8:45 PM • 1 Y

Y by PikaPika999

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

Math-lover1

#2 h Apr 26, 2025, 10:41 PM • 2 Y

Y by PikaPika999, CarSa

My solution

First, we observe that $a, b, c, d$ must be either powers of $2$ or $0$ .

Notice that the cases $(2, 2, 2, 0), (2, 2, 0, 0), (2, 0, 0, 0), (4, 4, 4, 0).... (8, 0, 0, 0)$ work, where each element is either a fixed $a = 2^m$ or $0$ . By counting the cases and keeping in mind that $a \neq 0$ , we get $(3+3+1)\cdot3 = \boxed{21}$

Hopefully I'm not too late

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