Inspired by qrxz17

by sqing, May 29, 2025, 8:35 AM

Let $a,b,c$ be reals such that $(a^2+b^2)^2 + (b^2+c^2)^2 +(c^2+a^2)^2 = 28$ and $(a^2+b^2+c^2)^2 =16.$ Find the value of $a^2(a^2-1) + b^2(b^2-1)+c^2(c^2-1).$

algebra

One old Long List No.1

by prof., May 29, 2025, 7:57 AM

A circle is inscribed in a rhombus. In each corner of the rhombus, a circle is inscribed such that it touches two sides of the rhombus and inscribed circle. These corner circles have radii $r_1$ and $r_2$ and the radius of the inscribed circle is $r$ . If $r_1$ and $r_2$ are natural numbers and $r=r_1\cdot r_2$ , find the value of $r_1,r_2$ and $r$ .

4 var inequality

by SunnyEvan, May 29, 2025, 7:07 AM

Let $x,y,z,t \in R^+ ,$ such that : $(x+y+z+t)^2 = x+y+z+t + (x+z)(y+t)$ and $x \geq y \geq z \geq t .$
Try to prove or disprove : $\frac{2 \sqrt{x+y+z+t +(x+t)(y+z)}}{x^2+y^2+z^2+t^2 +3xz+3yt+xt+yz} \geq \frac{11(x+z)(z+t)-(x+y+z+t)}{x+y+z+t +(x+z)(y+t)}$

inequalities

Inspired by a9opsow_

by sqing, May 29, 2025, 1:49 AM

Let $a,b > 0 .$ Prove that
$\frac{(ka^2 - kab-b)^2 + (kb^2 - kab-a)^2 + (ab-ka-kb )^2}{ (ka+b)^2 + (kb+a)^2+(a - b)^2 }\geq \frac {1}{(k+1)^2}$ Where $k\geq 0.37088 .$
$\frac{(a^2 - ab-b)^2 + (b^2 - ab-a)^2 + ( ab-a-b)^2}{a^2 +b^2+(a - b)^2 } \geq 1$ $\frac{(2a^2 - 2ab-b)^2 + (2b^2 - 2ab-a)^2 + (ab-2a-2b )^2}{ (2a+b)^2 + (2b+a)^2+(a - b)^2 }\geq \frac 19$

This post has been edited 2 times. Last edited by sqing, Today at 2:28 AM

inequalities proposed

algebra

inequalities

Inspired by Adhyayan Jana

by sqing, May 28, 2025, 2:38 AM

Let $a,b,c,d>0,a^2 + d^2-ad = (b + c)^2$ aand $a^2 +c^2 = b^2 + d^2.$ Prove that $\frac{ab+cd}{ad+bc} \geq \frac{ 4}{5}$ Let $a,b,c,d>0,a^2 + d^2-ad = b^2 + c^2 + bc$ aand $a^2 +c^2 = b^2 + d^2.$ Prove that $\frac{ab+cd}{ad+bc} \geq \frac{\sqrt 3}{2}$ Let $a,b,c,d>0,a^2 + d^2 - ad = b^2 + c^2 + bc$ aand $a^2 + b^2 = c^2 + d^2.$ Prove that $\frac{ab+cd}{ad+bc} =\frac{\sqrt 3}{2}$

This post has been edited 4 times. Last edited by sqing, Yesterday at 2:59 AM

algebra

inequalities proposed

inequalities

Cup of Combinatorics

by M11100111001Y1R, May 27, 2025, 7:24 AM

There are $n$ cups labeled $1, 2, \dots, n$ , where the $i$ -th cup has capacity $i$ liters. In total, there are $n$ liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.

$a)$ Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most $\frac{4n}{3}$ steps.

$b)$ Prove that in at most $\frac{5n}{3}$ steps, one can go from any configuration with integer water amounts to any other configuration with the same property.

This post has been edited 1 time. Last edited by M11100111001Y1R, May 27, 2025, 7:26 AM

combinatorics

an equation from the a contest

by alpha31415, May 21, 2025, 11:23 AM

Find all (complex) roots of the equation:
(z^2-z)(1-z+z^2)^2=-1/7

This post has been edited 1 time. Last edited by alpha31415, May 21, 2025, 11:24 AM
Reason: correct the problem

algebra

Shortest number theory you might've seen in your life

by AlperenINAN, May 11, 2025, 7:51 PM

Let $p$ and $q$ be prime numbers. Prove that if $pq(p+1)(q+1)+1$ is a perfect square, then $pq + 1$ is also a perfect square.

This post has been edited 3 times. Last edited by AlperenINAN, May 12, 2025, 10:09 AM
Reason: Typo

number theory

TST

Equation with primes

by oVlad, Jan 8, 2024, 6:50 PM

Let $p,q$ and $s{}$ be prime numbers such that $2^sq =p^y-1$ where $y > 1.$ Find all possible values of $p.$

number theory

prime numbers

IMO ShortList 1999, geometry problem 2

by orl, Nov 13, 2004, 11:47 PM

A circle is called a separator for a set of five points in a plane if it passes through three of these points, it contains a fourth point inside and the fifth point is outside the circle. Prove that every set of five points such that no three are collinear and no four are concyclic has exactly four separators.

Attachments:

This post has been edited 1 time. Last edited by orl, Nov 14, 2004, 10:19 PM

geometry

point set

circles

combinatorial geometry

IMO Shortlist

Submit

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math_exploration

by sqing, May 29, 2025, 8:35 AM

by prof., May 29, 2025, 7:57 AM

by SunnyEvan, May 29, 2025, 7:07 AM

by sqing, May 29, 2025, 1:49 AM

by sqing, May 28, 2025, 2:38 AM

by M11100111001Y1R, May 27, 2025, 7:24 AM

by alpha31415, May 21, 2025, 11:23 AM

by AlperenINAN, May 11, 2025, 7:51 PM

by oVlad, Jan 8, 2024, 6:50 PM

by orl, Nov 13, 2004, 11:47 PM