Polynomial

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies

jlacosta

May 1, 2025

0 replies

Hard Functional Equation in the Complex Numbers

yaybanana 5

N 7 minutes ago by aaravdodhia

Source: Own

Find all functions $f:\mathbb {C}\rightarrow \mathbb {C}$ , s.t :

$f(xf(y)) + f(x^2+y) = f(x+y)x + f(f(y))$

for all $x,y \in \mathbb{C}$

5 replies

yaybanana

Apr 9, 2025

aaravdodhia

7 minutes ago

Power tower sum

Rijul saini 9

N 22 minutes ago by ihategeo_1969

Source: India IMOTC 2024 Day 3 Problem 2

Let $a$ and $n$ be positive integers such that:
1. $a^{2^n}-a$ is divisible by $n$ ,
2. $\sum\limits_{k=1}^{n} k^{2024}a^{2^k}$ is not divisible by $n$ .

Prove that $n$ has a prime factor smaller than $2024$ .

Proposed by Shantanu Nene

9 replies

Rijul saini

May 31, 2024

ihategeo_1969

22 minutes ago

Nice "if and only if" function problem

ICE_CNME_4 7

N 25 minutes ago by ICE_CNME_4

Let $f : [0, \infty) \to [0, \infty)$ , $f(x) = \dfrac{ax + b}{cx + d}$ , with $a, d \in (0, \infty)$ , $b, c \in [0, \infty)$ . Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
$f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.$ (For $n \in \mathbb{N}^*$ and $x \geq 0$ , the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$ . )

Please do it at 9th grade level. Thank you!

7 replies

ICE_CNME_4

Yesterday at 7:23 PM

ICE_CNME_4

25 minutes ago

Cauchy-Schwarz 2

prtoi 8

N 25 minutes ago by mrtheory

Source: Handout by Samin Riasat

if $a^2+b^2+c^2+d^2=4$ , prove that:
$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{d}+\frac{d^2}{a}\ge4$

8 replies

prtoi

Mar 26, 2025

mrtheory

25 minutes ago

No more topics!

Polynomial

EtacticToe 3

N Apr 2, 2025 by EmersonSoriano

Source: Own

Let $f(x)$ be a monic polynomial with integer coefficient. And suppose there exist 4 distinct integer $a,b,c,d$ such that $f(a)=…=f(d)=5$ .

Find all $k$ such that $f(k)=8$

3 replies

EtacticToe

Dec 14, 2024

EmersonSoriano

Apr 2, 2025

Polynomial

G H J

Reveal topic tags

algebra

polynomial

G H BBookmark kLocked kLocked NReply

Source: Own

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EtacticToe

82 posts

EtacticToe

#1 h Dec 14, 2024, 6:43 PM

Y by

Let $f(x)$ be a monic polynomial with integer coefficient. And suppose there exist 4 distinct integer $a,b,c,d$ such that $f(a)=…=f(d)=5$ .

Find all $k$ such that $f(k)=8$

This post has been edited 1 time. Last edited by EtacticToe, Dec 14, 2024, 6:44 PM
Reason: The row

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RagvaloD

4918 posts

RagvaloD

#2 h Dec 15, 2024, 2:32 AM

Y by

$f(x)=(x-a)(x-b)(x-c)(x-d)g(x)+5$
$f(k)=(k-a)(k-b)(k-c)(k-d)g(k)+5=8 \to (k-a)(k-b)(k-c)(k-d) | 3$
But as $k-a,k-b,k-c,k-d$ should be different integers, there is no such $k$

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yuribogomolov

1 post

yuribogomolov

#3 h Apr 2, 2025, 3:00 PM

Y by

Quote:

RagvaloD wrote:

there is no such k

3 has four integer divisors (-3, -1, 1, 3).

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EmersonSoriano

47 posts

EmersonSoriano

#4 h Apr 2, 2025, 8:46 PM

Y by

yuribogomolov wrote:

Quote:

RagvaloD wrote:

there is no such k

3 has four integer divisors (-3, -1, 1, 3).

But it would have to be fulfilled that the product of those four integers is a divisor of 3, which cannot happen, since the product of those four numbers would be 9.

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