Sequence

by Titibuuu, May 10, 2025, 2:22 AM

Let $a_1 = a$ , and for all $n \geq 1$ , define the sequence $\{a_n\}$ by the recurrence
$a_{n+1} = a_n^2 + 1$ Prove that there is no natural number $n$ such that
$\prod_{k=1}^{n} \left( a_k^2 + a_k + 1 \right)$ is a perfect square.

This post has been edited 1 time. Last edited by Titibuuu, 16 minutes ago

algebra

Coolabra

by Titibuuu, May 10, 2025, 2:21 AM

Let $a, b, c$ be distinct real numbers such that
$a + b + c + \frac{1}{abc} = \frac{19}{2}$ Find the maximum possible value of $a$ .

This post has been edited 2 times. Last edited by Titibuuu, 16 minutes ago

algebra

A tangent problem

by hn111009, May 10, 2025, 2:09 AM

Let quadrilateral $ABCD$ with $P$ be the intersection of $AC$ and $BD.$ Let $\odot(APD)$ meet again $\odot(BPC)$ at $Q.$ Called $M$ be the midpoint of $BD.$ Assume that $\angle{DPQ}=\angle{CPM}.$ Prove that $AB$ is the tangent of $\odot(APD)$ and $BC$ is the tangent of $\odot(AQB).$

geometry

Inspired by Kosovo 2010

by sqing, May 9, 2025, 3:56 AM

Let $a,b>0 , a+b\leq k$ . Prove that
$\left(1+\frac{1}{a(b+1)}\right)\left(1+\frac{1}{b(a+1)}\right)\geq\left(1+\frac{4}{k(k+2)}\right)^2$ $\left(1+\frac {a}{b(a+1)}\right)\left(1+\frac {b}{a(b+1)}\right) \geq\left(1+\frac{2}{k+2}\right)^2$ Let $a,b>0 , a+b\leq 2$ . Prove that
$\left(1+\frac{1}{a(b+1)}\right)\left(1+\frac{1}{b(a+1)}\right)\geq \frac{9}{4}$ $\left(1+\frac {a}{b(a+1)}\right)\left(1+\frac {b}{a(b+1)}\right) \geq \frac{9}{4}$

This post has been edited 1 time. Last edited by sqing, Yesterday at 4:04 AM

inequalities proposed

algebra

inequalities

Inspired by Bet667

by sqing, May 8, 2025, 1:03 PM

Let $a,b$ be a real numbers such that $a^3+kab+b^3\ge a^4+b^4.$ Prove that
$1-\sqrt{k+1} \leq a+b\leq 1+\sqrt{k+1}$ Where $k\geq 0.$

inequalities proposed

algebra

inequalities

3-var inequality

by sqing, May 7, 2025, 12:43 PM

Let $a,b>0$ and $\frac{1}{a^2+3}+ \frac{1}{b^2+ 3} \leq \frac{1}{2} .$ Prove that
$a^2+ab+b^2\geq 3$ $a^2-ab+b^2 \geq 1$ Let $a,b>0$ and $\frac{1}{a^3+3}+ \frac{1}{b^3+ 3}\leq \frac{1}{2} .$ Prove that
$a^3+ab+b^3 \geq 3$ $a^3-ab+b^3\geq 1$

This post has been edited 1 time. Last edited by sqing, May 7, 2025, 12:43 PM

inequalities proposed

inequalities

Find all real numbers

by sqing, Aug 11, 2021, 1:57 PM

Find all real numbers x that satisfies $\sqrt{\sqrt{x-\frac{1}{x}}+\sqrt{1-\frac{1}{x}}-\frac{1}{\sqrt{x-\frac{1}{x}}+\sqrt{1-\frac{1}{x}}}}+\sqrt{1-\frac{1}{\sqrt{x-\frac{1}{x}}+\sqrt{1-\frac{1}{x}}}}=x.$ 2021 IMOC Problems

algebra

equation

I accidentally drew a 200-gon and ran out of time

by justin1228, Oct 25, 2020, 10:59 PM

The sides of a convex $200$ -gon $A_1 A_2 \dots A_{200}$ are colored red and blue in an alternating fashion.
Suppose the extensions of the red sides determine a regular $100$ -gon, as do the extensions of the blue sides.

Prove that the $50$ diagonals $\overline{A_1A_{101}},\ \overline{A_3A_{103}},\ \dots, \ \overline{A_{99}A_{199}}$ are concurrent.

Proposed by: Ankan Bhattacharya

This post has been edited 2 times. Last edited by justin1228, Oct 25, 2020, 11:10 PM
Reason: ankan

geometry

USEMO

USEMO 2020

Number theory sequences : sums divisible by n

by Muradjl, Jul 10, 2018, 11:15 AM

Determine all integers $n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$ , there exists an index $1 \leq i \leq n$ such that none of the numbers $a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$ is divisible by $n$ . Here, we let $a_i=a_{i-n}$ when $i >n$ .

Proposed by Warut Suksompong, Thailand

This post has been edited 6 times. Last edited by Muradjl, Jan 26, 2020, 6:23 PM

IMO Shortlist

number theory

IMO 2016 Problem 4

by termas, Jul 12, 2016, 5:50 AM

A set of positive integers is called fragrant if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n)=n^2+n+1$ . What is the least possible positive integer value of $b$ such that there exists a non-negative integer $a$ for which the set $\{P(a+1),P(a+2),\ldots,P(a+b)\}$ is fragrant?