3D geometry theorem

by KAME06, Apr 21, 2025, 10:18 PM

Let $M$ a point in the space and $G$ the centroid of a tetrahedron $ABCD$ . Prove that:
$\frac{1}{4}(AB^2+AC^2+AD^2+BC^2+BD^2+CD^2)+4MG^2=MA^2+MB^2+MC^2+MD^2$

geometry

3D geometry

domino question

by kjhgyuio, Apr 21, 2025, 10:02 PM

........

Attachments:

domino

algebra

demonic monic polynomial problem

by iStud, Apr 21, 2025, 9:51 PM

(a) Let $P(x)$ be a monic polynomial so that there exists another real coefficients $Q(x)$ that satisfy
$P(x^2-2)=P(x)Q(x)$ Determine all complex roots that are possible from $P(x)$
(b) For arbitrary polynomial $P(x)$ that satisfies (a), determine whether $P(x)$ should have real coefficients or not.

algebra

polynomial

monic

fun set problem

by iStud, Apr 21, 2025, 9:47 PM

Given a set $S$ with exactly 9 elements that is subset of $\{1,2,\dots,72\}$ . Prove that there exist two subsets $A$ and $B$ that satisfy the following:
- $A$ and $B$ are non-empty subsets from $S$ ,
- the sum of all elements in each of $A$ and $B$ are equal, and
- $A\cap B$ is an empty subset.

Sets

combinatorics

Funny easy transcendental geo

by qwerty123456asdfgzxcvb, Apr 21, 2025, 7:23 PM

Let $\mathcal{S}$ be a logarithmic spiral centered at the origin (ie curve satisfying for any point $X$ on it, line $OX$ makes a fixed angle with the tangent to $\mathcal{S}$ at $X$ ). Let $\mathcal{H}$ be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.

Prove that for a point $P$ on the spiral, the polar of $P$ wrt. $\mathcal{H}$ is tangent to the spiral.

This post has been edited 3 times. Last edited by qwerty123456asdfgzxcvb, 4 hours ago

An easy FE

by oVlad, Apr 21, 2025, 1:36 PM

Determine all functions $f:\mathbb R\to\mathbb R$ such that $f(xy-1)+f(x)f(y)=2xy-1,$ for any real numbers $x{}$ and $y{}.$

algebra

function

functional equation

Interesting F.E

by Jackson0423, Apr 18, 2025, 4:12 PM

Show that there does not exist a function
$f : \mathbb{R}^+ \to \mathbb{R}$ satisfying the condition that for all $x, y \in \mathbb{R}^+$ ,
$f(x + y^2) \geq f(x) + y.$

~Korea 2017 P7

This post has been edited 3 times. Last edited by Jackson0423, Today at 3:23 PM
Reason: Sorry guys..

algebra

functional equation

two tangent circles

by KPBY0507, May 8, 2021, 1:19 PM

The incenter and $A$ -excenter of $\triangle{ABC}$ is $I$ and $O$ . The foot from $A,I$ to $BC$ is $D$ and $E$ . The intersection of $AD$ and $EO$ is $X$ . The circumcenter of $\triangle{BXC}$ is $P$ .
Show that the circumcircle of $\triangle{BPC}$ is tangent to the $A$ -excircle if $X$ is on the incircle of $\triangle{ABC}$ .

GCD Functional Equation

by pinetree1, Jun 25, 2019, 5:36 PM

Let $f: \mathbb Z\to \{1, 2, \dots, 10^{100}\}$ be a function satisfying
$\gcd(f(x), f(y)) = \gcd(f(x), x-y)$ for all integers $x$ and $y$ . Show that there exist positive integers $m$ and $n$ such that $f(x) = \gcd(m+x, n)$ for all integers $x$ .

Ankan Bhattacharya

number theory

greatest common divisor

algebra

functional equation

tstst 2019

p^3 divides (a + b)^p - a^p - b^p

by 62861, Feb 23, 2017, 5:14 PM

Prove that there are infinitely many triples $(a, b, p)$ of positive integers with $p$ prime, $a < p$ , and $b < p$ , such that $(a + b)^p - a^p - b^p$ is a multiple of $p^3$ .

Noam Elkies

This post has been edited 1 time. Last edited by 62861, May 18, 2018, 11:58 PM

number theory

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thx bcp123, I guess its because I only have 10 posts, not much to brag on about
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why no one is writing here?
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by bcp123, Aug 15, 2014, 10:14 PM
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by liberator, Aug 14, 2014, 2:56 PM
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by bcp123, Aug 14, 2014, 2:39 PM