annoying algebra with sequence :/

by tabel, Jun 3, 2025, 4:55 PM

Let $a_1 = 1$ and $a_{n+1} = 1 + \frac{n}{a_n}$ for $n \geq 1$ . Prove that the sequence $(a_n)_{n \geq 1}$ is increasing.

Sequence

algebra

monotony

diophantine with factorials and exponents

by skellyrah, May 30, 2025, 7:56 PM

find all positive integers $a,b,c$ such that $a! + 5^b = c^3$

Sums of n mod k

by EthanWYX2009, May 26, 2025, 2:48 PM

Given $0<\varepsilon <1.$ Show that there exists a constant $c>0,$ such that for all positive integer $n,$
$\sum_{k\le n^{\varepsilon}}(n\text{ mod } k)>cn^{2\varepsilon}.$ Proposed by Cheng Jiang

number theory

The Return of Triangle Geometry

by peace09, Jul 17, 2024, 12:00 PM

Let $N$ be a positive integer. Prove that there exist three permutations $a_1,\dots,a_N$ , $b_1,\dots,b_N$ , and $c_1,\dots,c_N$ of $1,\dots,N$ such that $\left|\sqrt{a_k}+\sqrt{b_k}+\sqrt{c_k}-2\sqrt{N}\right|<2023$ for every $k=1,2,\dots,N$ .

This post has been edited 1 time. Last edited by peace09, Jul 17, 2024, 12:14 PM

P(z) and P(z)-1 have roots of magnitude 1

by anser, Jan 25, 2021, 5:00 PM

Find all nonconstant polynomials $P(z)$ with complex coefficients for which all complex roots of the polynomials $P(z)$ and $P(z) - 1$ have absolute value 1.

Ankan Bhattacharya

This post has been edited 3 times. Last edited by anser, Jan 25, 2021, 5:42 PM

f(1)f(2)...f(n) has at most n prime factors

by MarkBcc168, Jul 15, 2020, 2:22 AM

Let $f(x) = 3x^2 + 1$ . Prove that for any given positive integer $n$ , the product
$f(1)\cdot f(2)\cdot\dots\cdot f(n)$ has at most $n$ distinct prime divisors.

Proposed by Géza Kós

This post has been edited 1 time. Last edited by MarkBcc168, Jul 16, 2020, 1:53 PM

number theory

cyberspace

smallest a so that S(n)-S(n+a) = 2018, where S(n)=sum of digits

by parmenides51, Sep 13, 2018, 8:32 AM

For each positive integer $n$ , let $S(n)$ be the sum of the digits of $n$ . Determines the smallest positive integer $a$ such that there are infinite positive integers $n$ for which you have $S (n) -S (n + a) = 2018$ .

number theory

sum of digits

minimum

ABC is similar to XYZ

by Amir Hossein, May 20, 2011, 12:44 PM

Let $AA',BB',CC'$ be three diameters of the circumcircle of an acute triangle $ABC$ . Let $P$ be an arbitrary point in the interior of $\triangle ABC$ , and let $D,E,F$ be the orthogonal projection of $P$ on $BC,CA,AB$ , respectively. Let $X$ be the point such that $D$ is the midpoint of $A'X$ , let $Y$ be the point such that $E$ is the midpoint of $B'Y$ , and similarly let $Z$ be the point such that $F$ is the midpoint of $C'Z$ . Prove that triangle $XYZ$ is similar to triangle $ABC$ .

geometry

circumcircle

geometric transformation

reflection

perpendicular bisector

Russia 2001

by sisioyus, Aug 18, 2007, 10:08 AM

Find all odd positive integers $n > 1$ such that if $a$ and $b$ are relatively prime divisors of $n$ , then $a+b-1$ divides $n$ .

conditional sequence

by MithsApprentice, Oct 23, 2005, 12:25 AM

Suppose $\, q_{0}, \, q_{1}, \, q_{2}, \ldots \; \,$ is an infinite sequence of integers satisfying the following two conditions:

(i) $\, m-n \,$ divides $\, q_{m}-q_{n}\,$ for $\, m > n \geq 0,$
(ii) there is a polynomial $\, P \,$ such that $\, |q_{n}| < P(n) \,$ for all $\, n$

Prove that there is a polynomial $\, Q \,$ such that $\, q_{n}= Q(n) \,$ for all $\, n$ .

polynomial

modular arithmetic

number theory

least common multiple

greatest common divisor

induction