nice geo

by Melid, Apr 23, 2025, 3:01 PM

Let ABCD be a cyclic quadrilateral, which is AB=7 and BC=6. Let E be a point on segment CD so that BE=9. Line BE and AD intersect at F. Suppose that A, D, and F lie in order. If AF=11 and DF:DE=7:6, find the length of segment CD.

geometry

Normal but good inequality

by giangtruong13, Mar 31, 2025, 4:04 PM

Let $a,b,c> 0$ satisfy that $a+b+c=3abc$ . Prove that: $\sum_{cyc} \frac{ab}{3c+ab+abc} \geq \frac{3}{5}$

inequalities

Function on positive integers with two inputs

by Assassino9931, Jan 27, 2025, 10:03 AM

The function $f: \mathbb{Z}_{>0} \times \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is such that $f(a,b) + f(b,c) = f(ac, b^2) + 1$ for any positive integers $a,b,c$ . Assume there exists a positive integer $n$ such that $f(n, m) \leq f(n, m + 1)$ for all positive integers $m$ . Determine all possible values of $f(2025, 2025)$ .

This post has been edited 1 time. Last edited by Assassino9931, Jan 27, 2025, 10:06 AM

function

number theory

functional equation

Inequalities

by idomybest, Oct 15, 2021, 9:16 PM

The problem is in the attachment below.

Attachments:

Inequality

inequalities

inequalities unsolved

algebra unsolved

algebra

product of all integers of form i^3+1 is a perfect square

by AlastorMoody, Apr 6, 2020, 12:09 PM

Determine all integers $1 \le m, 1 \le n \le 2009$ , for which
$\begin{align*} \prod_{i=1}^n \left( i^3 +1 \right) = m^2 \end{align*}$

number theory

Concurrency

by Dadgarnia, Mar 12, 2020, 10:54 AM

Let $ABC$ be an isosceles triangle ( $AB=AC$ ) with incenter $I$ . Circle $\omega$ passes through $C$ and $I$ and is tangent to $AI$ . $\omega$ intersects $AC$ and circumcircle of $ABC$ at $Q$ and $D$ , respectively. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $CQ$ . Prove that $AD$ , $MN$ and $BC$ are concurrent.

Proposed by Alireza Dadgarnia

Nice inequality

by sqing, Apr 24, 2019, 1:01 PM

Let $a_1,a_2,\cdots,a_n (n\ge 2)$ be real numbers . Prove that : There exist positive integer $k\in \{1,2,\cdots,n\}$ such that $\sum_{i=1}^{n}\{kx_i\}(1-\{kx_i\})<\frac{n-1}{6}.$ Where $\{x\}=x-\left \lfloor x \right \rfloor.$

inequalities

China

Tiling rectangle with smaller rectangles.

by MarkBcc168, Jul 10, 2018, 11:25 AM

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.

Proposed by Jeck Lim, Singapore

This post has been edited 2 times. Last edited by MarkBcc168, Jul 15, 2018, 12:57 PM

A magician has one hundred cards numbered 1 to 100

by Valentin Vornicu, Oct 24, 2005, 10:21 AM

A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician locates the box from which no card has been drawn.

How many ways are there to put the cards in the three boxes so that the trick works?

IMO ShortList 1998, combinatorics theory problem 1

by orl, Oct 22, 2004, 3:22 PM

A rectangular array of numbers is given. In each row and each column, the sum of all numbers is an integer. Prove that each nonintegral number $x$ in the array can be changed into either $\lceil x\rceil$ or $\lfloor x\rfloor$ so that the row-sums and column-sums remain unchanged. (Note that $\lceil x\rceil$ is the least integer greater than or equal to $x$ , while $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$ .)

Attachments: