4 variables with quadrilateral sides 2

by mihaig, Apr 29, 2025, 8:47 PM

Let $a,b,c,d\geq0$ satisfying
$\frac1{a+1}+\frac1{b+1}+\frac1{c+1}+\frac1{d+1}=2.$ Prove
$\left(a+b+c+d-2\right)^2+8\geq3\left(abc+abd+acd+bcd\right).$

inequalities

geometry

D1025 : Can you do that?

by Dattier, Apr 29, 2025, 8:24 PM

Let $x_{n+1}=x_n^3$ and $x_0=3$ .

Can you calculate $\sum\limits_{i=1}^{2^{2025}} x_i \mod 10^{30}$ ?

Computer solutions

number theory

Number theory

by MuradSafarli, Apr 29, 2025, 7:39 PM

Prove that for any natural number $n$ :

$1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n + 1) \mid (4n + 3)(4n + 5) \cdot \ldots \cdot (8n + 3).$

This post has been edited 1 time. Last edited by MuradSafarli, 3 hours ago

number theory

Easy Combinatorics

by MuradSafarli, Apr 29, 2025, 6:40 PM

A student firstly wrote $x=3$ on the board. For each procces, the stutent deletes the number x and replaces it with either $(2x+4)$ or $(3x+8)$ or $(x^2+5x)$ . Is this possible to make the number $(20^{25}+2024)$ on the board?

combinatorics

My Unsolved Problem

by MinhDucDangCHL2000, Apr 29, 2025, 4:53 PM

Let triangle $ABC$ be inscribed in the circle $(O)$ . A line through point $O$ intersects $AC$ and $AB$ at points $E$ and $F$ , respectively. Let $P$ be the reflection of $E$ across the midpoint of $AC$ , and $Q$ be the reflection of $F$ across the midpoint of $AB$ . Prove that:
a) the reflection of the orthocenter $H$ of triangle $ABC$ across line $PQ$ lies on the circle $(O)$ .
b) the orthocenters of triangles $AEF$ and $HPQ$ coincide.

Im looking for a solution used complex bashing

geometry

geometry unsolved

Good divisors and special numbers.

by Nuran2010, Apr 29, 2025, 4:52 PM

$N$ is a positive integer. Call all positive divisors of $N$ which are different from $1$ and $N$ beautiful divisors.We call $N$ a special number when it has at least $2$ beautiful divisors and difference of any $2$ beautiful divisors divides $N$ as well. Find all special numbers.

number theory

Find points with sames integer distances as given

by nAalniaOMliO, Jul 17, 2024, 9:44 PM

Points $A_1, \ldots A_n$ with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points $B_1, \ldots ,B_n$ with integer coordinates such that $A_iA_j=B_iB_j$ for every pair $1 \leq i \leq j \leq n$
N. Sheshko, D. Zmiaikou

This post has been edited 1 time. Last edited by nAalniaOMliO, Oct 31, 2024, 10:12 AM

combinatorics

number theory

Geometry tangent circles

by Stefan4024, Apr 13, 2016, 11:18 AM

Two circles $\omega_1$ and $\omega_2$ , of equal radius intersect at different points $X_1$ and $X_2$ . Consider a circle $\omega$ externally tangent to $\omega_1$ at $T_1$ and internally tangent to $\omega_2$ at point $T_2$ . Prove that lines $X_1T_1$ and $X_2T_2$ intersect at a point lying on $\omega$ .

This post has been edited 2 times. Last edited by djmathman, Sep 12, 2020, 1:59 AM

The number of integers

by Fang-jh, Apr 4, 2009, 10:20 AM

Prove that for any odd prime number $p,$ the number of positive integer $n$ satisfying $p|n! + 1$ is less than or equal to $cp^\frac{2}{3}.$ where $c$ is a constant independent of $p.$

This post has been edited 2 times. Last edited by Fang-jh, Apr 4, 2009, 2:23 PM

modular arithmetic

inequalities

number theory proposed

number theory

Perpendicularity

by April, Dec 28, 2008, 4:09 AM

Point $D$ lies inside triangle $ABC$ such that $\angle DAC = \angle DCA = 30^{\circ}$ and $\angle DBA = 60^{\circ}$ . Point $E$ is the midpoint of segment $BC$ . Point $F$ lies on segment $AC$ with $AF = 2FC$ . Prove that $DE \perp EF$ .