$$ac=bd$$

by sqing, Mar 30, 2025, 2:25 PM

Let $a,b,c,d$ be reals such that $a^2+b^2=4,c^2+d^2=9$ and $abcd\ge 9.$ Prove that $ac=bd$ Let $a,b,c,d$ be reals such that $a^2+b^2=4,c^2+d^2=9$ and $ad+bc \ge 6.$ Prove that $ac=bd$ Let $a,b,c,d$ be reals such that $a^2+b^2=4,c^2+d^2=9$ and $ab+cd \geq \frac{13}{2}.$ Prove that $ac=bd$

algebra

possible triangle inequality

by sunshine_12, Mar 30, 2025, 2:12 PM

a, b, c are real numbers
|a| + |b| + |c| − |a + b| − |b + c| − |c + a| + |a + b + c| ≥ 0
hey everyone, so I came across this inequality, and I did make some progress:
Let (a+b), (b+c), (c+a) be three sums T1, T2 and T3. As there are 3 sums, but they can be of only 2 signs, by pigeon hole principle, atleast 2 of the 3 sums must be of the same sign.
But I'm getting stuck on the case work. Can anyone help?
Thnx a lot

This post has been edited 2 times. Last edited by sunshine_12, Yesterday at 2:41 PM

inequalities

Polynomials and their shift with all real roots and in common

by Assassino9931, Mar 30, 2025, 1:12 PM

We call two non-constant polynomials friendly if each of them has only real roots, and every root of one polynomial is also a root of the other. For two friendly polynomials $P(x), Q(x)$ and a constant $C \in \mathbb{R}, C \neq 0$ , it is given that $P(x) + C$ and $Q(x) + C$ are also friendly polynomials. Prove that $P(x) \equiv Q(x)$ .

This post has been edited 1 time. Last edited by Assassino9931, Yesterday at 1:13 PM

algebra

polynomial

Thanks u!

by Ruji2018252, Mar 30, 2025, 11:07 AM

Let $x,y,z,t\in\mathbb{R}$ and $\begin{cases}x^2+y^2=4\\z^2+t^2=9\\xt+yz\geqslant 6\end{cases}$ .
$1,$ Prove $xz=yt$
$2,$ Find maximum $P=x+z$

inequalities

VERY HARD MATH PROBLEM!

by slimshadyyy.3.60, Mar 29, 2025, 10:49 PM

Let a ≥b ≥c ≥0 be real numbers such that a^2 +b^2 +c^2 +abc = 4. Prove that
a+b+c+(√a−√c)^2 ≥3.

Inequality

inequalities

Harmonic Series and Infinite Sequences

by steven_zhang123, Mar 29, 2025, 12:41 AM

Let $\left \{ x_n \right \} _{n\ge 1}$ and $\left \{ y_n \right \} _{n\ge 1}$ be two infinite sequences of integers. Prove that there exists an infinite sequence of integers $\left \{ z_n \right \} _{n\ge 1}$ such that for any positive integer $n$ , the following holds:

$\sum_{k|n} k \cdot z_k^{\frac{n}{k}} = \left( \sum_{k|n} k \cdot x_k^{\frac{n}{k}} \right) \cdot \left( \sum_{k|n} k \cdot y_k^{\frac{n}{k}} \right).$

number theory

Inspired by old results

by sqing, Mar 27, 2025, 12:35 PM

Let $a,b,c > 0$ and $a+b+c +abc =4.$ Prove that
$a^2 + b^2 + c^2 + 3 \geq 2( ab+bc + ca )$ Let $a,b,c > 0$ and $ab+bc+ca+abc=4.$ Prove that
$a^2 + b^2 + c^2 + 2abc \geq 5$

This post has been edited 1 time. Last edited by sqing, Mar 27, 2025, 12:44 PM

inequalities proposed

algebra

inequalities

A cute FE

by Aritra12, Mar 23, 2021, 3:17 PM

Hope so not prediscovered

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all reals $x,y,$
$f(f(x+f(y)))(f(x)+y)=xf(x)+yf(y)+2f(xy)$ Proposed by Aritra12, India

Click to reveal hidden text
Its simple but yet cute acc to me

This post has been edited 6 times. Last edited by Aritra12, Jun 13, 2021, 12:47 PM

algebra

x is rational implies y is rational

by pohoatza, Jun 28, 2007, 7:24 PM

For $x \in (0, 1)$ let $y \in (0, 1)$ be the number whose $n$ -th digit after the decimal point is the $2^{n}$ -th digit after the decimal point of $x$ . Show that if $x$ is rational then so is $y$ .

Proposed by J.P. Grossman, Canada

number theory

rational

decimal representation

IMO Shortlist

USAMO 1995

by paul_mathematics, Dec 31, 2004, 1:01 PM

Given a nonisosceles, nonright triangle ABC, let O denote the center of its circumscribed circle, and let $A_1$ , $B_1$ , and $C_1$ be the midpoints of sides BC, CA, and AB, respectively. Point $A_2$ is located on the ray $OA_1$ so that $OAA_1$ is similar to $OA_2A$ . Points $B_2$ and $C_2$ on rays $OB_1$ and $OC_1$ , respectively, are defined similarly. Prove that lines $AA_2$ , $BB_2$ , and $CC_2$ are concurrent, i.e. these three lines intersect at a point.

perpendicular bisector

Submit

done!
$~~~~$
by pupitrethebean, Mar 19, 2025, 2:05 PM
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686 shouts

pupitre

by sqing, Mar 30, 2025, 2:25 PM

by sunshine_12, Mar 30, 2025, 2:12 PM

by Assassino9931, Mar 30, 2025, 1:12 PM

by Ruji2018252, Mar 30, 2025, 11:07 AM

by slimshadyyy.3.60, Mar 29, 2025, 10:49 PM

by steven_zhang123, Mar 29, 2025, 12:41 AM

by sqing, Mar 27, 2025, 12:35 PM

by Aritra12, Mar 23, 2021, 3:17 PM

by pohoatza, Jun 28, 2007, 7:24 PM

by paul_mathematics, Dec 31, 2004, 1:01 PM