Quadric function

by soryn, Apr 18, 2025, 2:47 AM

If f(x)=ax^2+bx+c, a,b,c integers, |a|>=3, and M îs the set of integers x for which f(x) is a prime number and f has exactly one integer solution,prove that M has at most three elements.

This post has been edited 1 time. Last edited by soryn, an hour ago

function

number theory

2 var inquality

by sqing, Apr 18, 2025, 2:39 AM

Let $a,b\geq 0$ and $a+ b+ab=3 .$ Prove that
$(a^2+b^2+a+b-4)(2 - ab)\ge\sqrt 7(a-1)(b-1)(a-b)$ $3 (a^2+b^2+a+b-4)(5 - 2ab)\ge 20(a-1)(b-1)(a-b)$ $6(a^2+b^2-2)(5 - 2ab)\ge 35(a-1)(b-1)(a-b)$ $2(a^2+b^2-2)(3 - ab)\ge 7(a-1)(b-1)(a-b)$

inequalities proposed

inequalities

2 var inquality

by sqing, Apr 18, 2025, 1:50 AM

Let $a,b\geq 0$ and $a+ b+2ab=4 .$ Prove that
$3(a+b-2)(2 - ab) \ge (a-1)(b-1)(a-b)$ $9 (a+b-2)(3 - 2ab) \ge 2\sqrt 5(a-1)(b-1)(a-b)$ $9(a+b-2)(6 - 5ab) \ge2\sqrt {14} (a-1)(b-1)(a-b)$

inequalities proposed

inequalities

Another factorisation problem

by kjhgyuio, Apr 17, 2025, 11:35 PM

........

Attachments:

Function equation

by luci1337, Apr 17, 2025, 3:01 PM

find all function $f:R \rightarrow R$ such that:
$2f(x)f(x+y)-f(x^2)=\frac{x}{2}(f(2x)+f(f(y)))$ with all $x,y$ is real number

functional equation

algebra

Inspired by JK1603JK

by sqing, Apr 17, 2025, 2:11 PM

Let $a,b,c\geq 0$ and $ab+bc+ca=3.$ Prove that
$(a+b+c-3)(12-5abc)\ge 2(a-b)(b-c)(c-a)$ $6(a+b+c-3)(5-2abc)\ge 5(a-b)(b-c)(c-a)$ $2(a+b+c-3)(9-5abc)\ge 3(a-b)(b-c)(c-a)$ $3(a+b+c-3)(14-5abc)\ge 7(a-b)(b-c)(c-a)$

This post has been edited 2 times. Last edited by sqing, Yesterday at 2:56 PM

inequalities proposed

algebra

inequalities

The old one is gone.

by EeEeRUT, Apr 16, 2025, 1:37 AM

An infinite increasing sequence $a_1 < a_2 < a_3 < \cdots$ of positive integers is called central if for every positive integer $n$ , the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$ .

Show that there exists an infinite sequence $b_1, b_2, b_3, \dots$ of positive integers such that for every central sequence $a_1, a_2, a_3, \dots,$ there are infinitely many positive integers $n$ with $a_n = b_n$ .

This post has been edited 2 times. Last edited by EeEeRUT, Apr 16, 2025, 1:39 AM

algebra

EGMO 2025

Inequalities

by hn111009, Apr 13, 2025, 1:22 PM

Let $a,b,c>0;r,s\in\mathbb{R}$ satisfied $a+b+c=1.$ Find minimum and maximum of $P=a^rb^s+b^rc^s+c^ra^s.$

inequalities

Integer-Valued FE comes again

by lminsl, Jul 16, 2019, 12:08 PM

Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$ , $f(2a)+2f(b)=f(f(a+b)).$ Proposed by Liam Baker, South Africa

This post has been edited 2 times. Last edited by djmathman, Jul 18, 2019, 4:41 AM

(204)
(+)

Quadratic system

by juckter, Jun 22, 2014, 4:27 PM

Let $n$ be a positive integer. Find all real solutions $(a_1, a_2, \dots, a_n)$ to the system:

$a_1^2 + a_1 - 1 = a_2$ $a_2^2 + a_2 - 1 = a_3$ $\hspace*{3.3em} \vdots$ $a_{n}^2 + a_n - 1 = a_1$

This post has been edited 1 time. Last edited by juckter, Dec 5, 2016, 2:01 AM

(31)
(+)

Submit

thank you both wahoo
by Helena_Liang, Feb 16, 2025, 3:45 AM
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by Moonlight11, Feb 16, 2025, 3:21 AM
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by contactbibliophile, Feb 16, 2025, 2:28 AM
you are peak
by Helena_Liang, Feb 15, 2025, 4:50 AM
peak css hehe
by mathical8, Feb 10, 2025, 8:27 AM
thank you sir but you should admit orz
by Helena_Liang, Jan 26, 2025, 2:22 AM
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by alice_inmathland, Jan 22, 2025, 7:49 PM
pls pls be active again :eyes:
by Hestu_the_Bestu, Nov 16, 2024, 12:45 AM
I have been inactive on aops
perhaps I'll be active again
by Helena_Liang, Oct 9, 2024, 3:10 AM
sirr when are u posting next lol
by mathical8, Sep 29, 2024, 8:12 PM
Do you have any tips for making AIME?
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by Hestu_the_Bestu, Sep 9, 2024, 9:15 PM
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also u should post something
by alice_inmathland, Sep 7, 2024, 11:36 PM
contrib

1434 orz xooks
by ChromeRaptor777, Aug 30, 2024, 4:04 AM
@2 below ay yes sir I shall
@below indeed
by Helena_Liang, Aug 22, 2024, 1:34 AM
first as well $\text{[math]}$
by madeleinelee, Nov 21, 2023, 8:26 PM
sapphire's css skills are pro
by Helena_Liang, Nov 20, 2023, 5:14 PM
Unfortunately, first
by awesomeming327., Nov 20, 2023, 4:59 AM
Lol Helena and Emma I think your blogs are essentially the same as mine ha cant top that
by SapphireDolphin9, Nov 20, 2023, 4:28 AM
first $\text{[math]}$
by awesomeming327., Nov 20, 2023, 3:54 AM
this blog would not exist without helena
by flec, Nov 20, 2023, 3:35 AM
I agree with that statement
this blog would not be complete without flec
by Helena_Liang, Nov 20, 2023, 3:34 AM
i had very valuable edits
by flec, Nov 20, 2023, 1:38 AM
lol emma I think your blog css is essentially the same as ours (with edits)
by Helena_Liang, Nov 19, 2023, 11:46 PM
fifth!! also very nice css and blog idea :D $~$
by Beast_Academic_Emma, Nov 19, 2023, 10:28 PM
fourth!!
by mathical8, Nov 19, 2023, 6:49 PM
third ! $\text{[math]}$
by purplepenguin2, Nov 19, 2023, 1:41 AM
second : $\text{[math]}$ D
by Helena_Liang, Nov 18, 2023, 6:26 AM
obviously i take the first shout
by flec, Nov 18, 2023, 6:07 AM

70 shouts

two up, one down

by soryn, Apr 18, 2025, 2:47 AM

by sqing, Apr 18, 2025, 2:39 AM

by sqing, Apr 18, 2025, 1:50 AM

by kjhgyuio, Apr 17, 2025, 11:35 PM

by luci1337, Apr 17, 2025, 3:01 PM

by sqing, Apr 17, 2025, 2:11 PM

by EeEeRUT, Apr 16, 2025, 1:37 AM

by hn111009, Apr 13, 2025, 1:22 PM

by lminsl, Jul 16, 2019, 12:08 PM

by juckter, Jun 22, 2014, 4:27 PM