Drawing equilateral triangle

by xeroxia, May 11, 2025, 7:14 AM

Equilateral triangle $ABC$ is given. Let $M_a$ and $M_c$ be the midpoints of $BC$ and $AB$, respectively.
A point $D$ on segment $BM_c$ is given. Draw equilateral $\triangle DEF$ such that $E$ is on $BC$ and $F$ is on $AM_a$.
This post has been edited 1 time. Last edited by xeroxia, 35 minutes ago
geometry
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n-variable inequality

by bakkune, May 11, 2025, 6:48 AM

Prove that the following inequality holds for all positive integer $n$ and all real numbers $x_1, x_2, \dots, x_n\neq 0$:
$$ \sum_{1\leq i < j \leq n} \dfrac{x_ix_j}{x_i^2 + x_j^2} \ge -\dfrac{n}{4}. $$
This post has been edited 1 time. Last edited by bakkune, an hour ago
Inequality
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Divisibility NT

by reni_wee, May 11, 2025, 5:11 AM

Suppose that $a$ and $b$ are natural numbers such that
$$p = \frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}$$is a prime number. Find all possible values of $a$,$b$,$p$.
number theory
Divisibility
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Calculus

by youochange, May 10, 2025, 2:38 PM

Find the area enclosed by the curves $e^x,e^{-x},x^2+y^2=1$
calculus
geometry
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Arbitrary point on BC and its relation with orthocenter

by falantrng, Apr 27, 2025, 11:47 AM

In an acute-angled triangle \(ABC\), \(H\) be the orthocenter of it and \(D\) be any point on the side \(BC\). The points \(E, F\) are on the segments \(AB, AC\), respectively, such that the points \(A, B, D, F\) and \(A, C, D, E\) are cyclic. The segments \(BF\) and \(CE\) intersect at \(P.\) \(L\) is a point on \(HA\) such that \(LC\) is tangent to the circumcircle of triangle \(PBC\) at \(C.\) \(BH\) and \(CP\) intersect at \(X\). Prove that the points \(D, X, \) and \(L\) lie on the same line.

Proposed by Theoklitos Parayiou, Cyprus
This post has been edited 1 time. Last edited by falantrng, Apr 27, 2025, 4:38 PM
geometry
BMO
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Asymmetric FE

by sman96, Feb 8, 2025, 5:11 PM

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that$$f(xf(y)-y) + f(xy-x) + f(x+y) = 2xy$$for all $x, y \in \mathbb{R}$.
algebra
functional equation
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Kosovo MO 2021 Grade 12, Problem 2

by bsf714, Feb 27, 2021, 6:28 PM

Find all functions $f:\mathbb R\to\mathbb R$ so that the following relation holds for all $x, y\in\mathbb R$.

$$f(f(x)f(y)-1) = xy - 1$$
This post has been edited 1 time. Last edited by bsf714, Feb 27, 2021, 6:29 PM
function
algebra
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5-th powers is a no-go - JBMO Shortlist

by WakeUp, Oct 30, 2010, 7:30 PM

Prove that there are are no positive integers $x$ and $y$ such that $x^5+y^5+1=(x+2)^5+(y-3)^5$.

Note
modular arithmetic
number theory proposed
number theory
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IMO 2008, Question 1

by orl, Jul 16, 2008, 1:24 PM

Let $ H$ be the orthocenter of an acute-angled triangle $ ABC$. The circle $ \Gamma_{A}$ centered at the midpoint of $ BC$ and passing through $ H$ intersects the sideline $ BC$ at points $ A_{1}$ and $ A_{2}$. Similarly, define the points $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$.

Prove that the six points $ A_{1}$, $ A_{2}$, $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$ are concyclic.

Author: Andrey Gavrilyuk, Russia
This post has been edited 4 times. Last edited by orl, Jul 20, 2008, 8:14 AM
geometry
circumcircle
trigonometry
IMO 2008
radical axis
IMO
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a set of $9$ distinct integers

by N.T.TUAN, Mar 31, 2007, 5:13 AM

Let $S$ be a set of $9$ distinct integers all of whose prime factors are at most $3.$ Prove that $S$ contains $3$ distinct integers such that their product is a perfect cube.
geometry
analytic geometry
vector
pigeonhole principle
modular arithmetic
combinatorics unsolved
L

Notable algebra methods with proofs and examples

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fungarwai
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  • thanks for this

    by jasperE3, Dec 3, 2021, 10:01 PM

  • I am working as accountant and studying as ACCA student now.
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    by fungarwai, Aug 28, 2021, 4:54 AM

  • Awesome algebra blog :)

    by Euler1728, Mar 22, 2021, 5:37 AM

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    by Hamroldt, Jan 14, 2021, 10:55 AM

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    by freeman66, May 31, 2020, 5:40 AM

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  • Nice method :)

    by Feridimo, Jan 23, 2020, 5:05 PM

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    by mufree, May 26, 2019, 6:40 AM

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    by lkarhat, Dec 25, 2018, 11:53 PM

  • rip 0 shouts and 0 comments until now

    by harry1234, Nov 17, 2018, 8:56 PM

20 shouts
Tags
number theory
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