Maximizing the Area

by steven_zhang123, Mar 29, 2025, 12:45 AM

Given a circle $\omega$ and two points $A$ and $B$ outside $\omega$ , a quadrilateral $PQRS$ is defined as "good" if $P, Q, R, S$ are four distinct points on $\omega$ in order, and lines $PQ$ and $RS$ intersect at $A$ and lines $PS$ and $QR$ intersect at $B$ .

For a quadrilateral $T$ , let $S_T$ denote its area. If there exists a good quadrilateral, prove that there exists good quadrilateral $T$ such that for any good quadrilateral $T_1 (T_1 \neq T)$ , $S_{T_1} < S_T$ .

This post has been edited 3 times. Last edited by steven_zhang123, 2 hours ago

geometry

Modular Matching Pairs

by steven_zhang123, Mar 29, 2025, 12:42 AM

Let $n$ be an odd integer, $m = \frac{n+1}{2}$ . Consider $2m$ integers $a_1, a_2, \ldots, a_m, b_1, b_2, \ldots, b_m$ such that for any $1 \leq i < j \leq m$ , $a_i \not\equiv a_j \pmod{n}$ and $b_i \not\equiv b_j \pmod{n}$ . Prove that the number of $k \in \{0, 1, \ldots, n-1\}$ for which satisfy $a_i + b_j \equiv k \pmod{n}$ for some $i \neq j$ , $i, j \in \left \{ 1,2,\cdots,m \right \}$ is greater than $n - \sqrt{n} - \frac{1}{2}$ .

This post has been edited 1 time. Last edited by steven_zhang123, 2 hours ago

number theory

Harmonic Series and Infinite Sequences

by steven_zhang123, Mar 29, 2025, 12:41 AM

Let $\left \{ x_n \right \} _{n\ge 1}$ and $\left \{ y_n \right \} _{n\ge 1}$ be two infinite sequences of integers. Prove that there exists an infinite sequence of integers $\left \{ z_n \right \} _{n\ge 1}$ such that for any positive integer $n$ , the following holds:

$\sum_{k|n} k \cdot z_k^{\frac{n}{k}} = \left( \sum_{k|n} k \cdot x_k^{\frac{n}{k}} \right) \cdot \left( \sum_{k|n} k \cdot y_k^{\frac{n}{k}} \right).$

number theory

An almost identity polynomial

by nAalniaOMliO, Mar 28, 2025, 8:28 PM

Let $n$ be a positive integer and $P(x)$ be a polynomial with integer coefficients such that $P(1)=1,P(2)=2,\ldots,P(n)=n$ .
Prove that $P(0)$ is divisible $2 \cdot 3 \cdot \ldots \cdot n$ .

algebra

polynomial

A lot of numbers and statements

by nAalniaOMliO, Mar 28, 2025, 8:20 PM

101 numbers are written in a circle. Near the first number the statement "This number is bigger than the next one" is written, near the second "This number is bigger that the next two" and etc, near the 100th "This number is bigger than the next 100 numbers".
What is the maximum possible amount of the statements that can be true?

combinatorics

number theory

by MuradSafarli, Mar 28, 2025, 8:03 PM

Find all prime numbers $p$ and $q$ such that $2q$ divides $\phi(p+q)$ and $2p$ divides $\phi(p+q)$ .

number theory

Infinite integer sequence problem

by mathlover1231, Mar 28, 2025, 6:04 PM

Let a_1, a_2, … be an infinite sequence of pairwise distinct positive integers and c be a real number such that 0 < c < 3/2. Prove that there exist infinitely many positive integers k such that lcm(a_k, a_{k+1}) > ck.

This post has been edited 1 time. Last edited by mathlover1231, Yesterday at 6:12 PM

f(x+f(y))=f(x+y)+y

by John_Mgr, Mar 28, 2025, 5:14 PM

Determine with proof all functions $f: \mathbb{R} \to \mathbb{R}$ such that for all real $x,y:$
$f(x+f(y))=f(x+y)+y$

Function equations

algebra

IMO 2018 Problem 6

by m.candales, Jul 10, 2018, 11:21 AM

A convex quadrilateral $ABCD$ satisfies $AB\cdot CD = BC\cdot DA$ . Point $X$ lies inside $ABCD$ so that $\angle{XAB} = \angle{XCD}\quad\,\,\text{and}\quad\,\,\angle{XBC} = \angle{XDA}.$ Prove that $\angle{BXA} + \angle{DXC} = 180^\circ$ .

Proposed by Tomasz Ciesla, Poland

This post has been edited 3 times. Last edited by djmathman, Jun 16, 2020, 4:03 AM

Simple inequality

by sqing, Jul 24, 2017, 3:30 AM

In $\triangle ABC$ with length-side $a,b,c$ , prove that $\frac{a}{b+c-a}+\frac{b}{c+a-b}+\frac{c}{a+b-c}\ge\frac{b+c-a}{a}+\frac{c+a-b}{b}+\frac{a+b-c}{c}\ge3 .$