Locus of a point on the side of a square

by EmersonSoriano, Apr 2, 2025, 9:58 PM

Let $ABCD$ be a fixed square and $K$ a variable point on segment $AD$ . The square $KLMN$ is constructed such that $B$ is on segment $LM$ and $C$ is on segment $MN$ . Let $T$ be the intersection point of lines $LA$ and $ND$ . Find the locus of $T$ as $K$ varies along segment $AD$ .

Locus

geometry

Chess queens on a cylindrical board

by EmersonSoriano, Apr 2, 2025, 9:56 PM

Let $n$ be a positive integer. In an $n \times n$ board, two opposite sides have been joined, forming a cylinder. Determine whether it is possible to place $n$ queens on the board such that no two threaten each other when:

$a)\:$ $n=14$ .

$b)\:$ $n=15$ .

GCD of x^2-y, y^2-z and z^2-x

by EmersonSoriano, Apr 2, 2025, 9:38 PM

Find all positive integers $d$ that can be written in the form
$d = \gcd(|x^2 - y| , |y^2 - z| , |z^2 - x|),$ where $x, y, z$ are pairwise coprime positive integers such that $x^2 \neq y$ , $y^2 \neq z$ , and $z^2 \neq x$ .

This post has been edited 2 times. Last edited by EmersonSoriano, 23 minutes ago
Reason: change subject

GCD

number theory

kind of well known?

by dotscom26, Apr 1, 2025, 4:11 AM

Let $y_1, y_2, ..., y_{2025}$ be real numbers satisfying
$y_1^2 + y_2^2 + \cdots + y_{2025}^2 = 1.$
Find the maximum value of
$|y_1 - y_2| + |y_2 - y_3| + \cdots + |y_{2025} - y_1|.$

I have seen many problems with the same structure, Id really appreciate if someone could explain which approach is suitable here

This post has been edited 1 time. Last edited by dotscom26, Yesterday at 4:20 AM

algebra

algebra-geometry

inequalities hard

by Cobedangiu, Mar 31, 2025, 11:45 AM

problem

Attachments:

This post has been edited 1 time. Last edited by Cobedangiu, Mar 31, 2025, 2:50 PM

inequalities

Geo Final but hard to solve with Conics...

by Seungjun_Lee, Jan 18, 2025, 7:13 AM

Let $\omega$ be the circumcircle of triangle $ABC$ with center $O$ , and the $A$ inmixtilinear circle is tangent to $AB, AC, \omega$ at $D,E,T$ respectively. $P$ is the intersection of $TO$ and $DE$ and $X$ is the intersection of $AP$ and $\omega$ . Prove that the isogonal conjugate of $P$ lies on the line passing through the midpoint of $BC$ and $X$ .

This post has been edited 1 time. Last edited by Seungjun_Lee, Jan 18, 2025, 12:44 PM

geometry

Polynomial

by EtacticToe, Dec 14, 2024, 6:43 PM

Let $f(x)$ be a monic polynomial with integer coefficient. And suppose there exist 4 distinct integer $a,b,c,d$ such that $f(a)=…=f(d)=5$ .

Find all $k$ such that $f(k)=8$

This post has been edited 1 time. Last edited by EtacticToe, Dec 14, 2024, 6:44 PM
Reason: The row

algebra

polynomial

calculate the perimeter of triangle MNP

by PennyLane_31, Oct 16, 2024, 8:26 PM

Let $ABCD$ be a convex quadrilateral, and $M$ , $N$ , and $P$ be the midpoints of diagonals $AC$ and $BD$ , and side $AD$ , respectively. Also, suppose that $\angle{ABC} + \angle{DCB} = 90$ and that $AB = 6$ , $CD = 8$ . Calculate the perimeter of triangle $MNP$ .

geometry

perimeter

egmo 2018 p4

by microsoft_office_word, Apr 12, 2018, 11:02 AM

A domino is a $1 \times 2$ or $2 \times 1$ tile.
Let $n \ge 3$ be an integer. Dominoes are placed on an $n \times n$ board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. The value of a row or column is the number of dominoes that cover at least one cell of this row or column. The configuration is called balanced if there exists some $k \ge 1$ such that each row and each column has a value of $k$ . Prove that a balanced configuration exists for every $n \ge 3$ , and find the minimum number of dominoes needed in such a configuration.

This post has been edited 3 times. Last edited by microsoft_office_word, Feb 18, 2020, 9:47 PM

Convex and concave functions in Real numbers -- Basic 1

by adityaguharoy, Mar 1, 2018, 1:44 PM

Convex functions
Let $f : \mathbb{R} \to \mathbb{R}$ be a function, and let $a,b$ be two real numbers with $a<b$ . Then we say that $f$ is a convex function on the interval $[a,b]$ if and only if the following is true :
Given any $t \in [0,1]$ , and , any $x_1 , x_2 \in [a,b]$ then,
$\boxed{f(tx_1 + (1-t)x_2) \le t \cdot f(x_1) + (1-t) \cdot f(x_2)}$ And we say that $f$ is strictly convex on $[a,b]$ if the above inequality is strict whenever $x_1 \ne x_2$ and $t \in (0,1)$ .

Concave functions
Let $f : \mathbb{R} \to \mathbb{R}$ be a function, and let $a,b$ be two real numbers with $a<b$ . Then we say that $f$ is a concave function on the interval $[a,b]$ if and only if the following is true :
Given any $t \in [0,1]$ , and , any $x_1 , x_2 \in [a,b]$ then,
$\boxed{f(tx_1 + (1-t)x_2) \ge t \cdot f(x_1) + (1-t) \cdot f(x_2)}$ And we say that $f$ is strictly concave on $[a,b]$ if the above inequality is strict whenever $x_1 \ne x_2$ and $t \in (0,1)$ .

Quick exercises

Let us celebrate

This post has been edited 2 times. Last edited by adityaguharoy, Mar 4, 2018, 7:25 AM

2015 solutions for quotient function!

by raxu, Jun 26, 2015, 1:45 AM

Let $\varphi(n)$ denote the number of positive integers less than $n$ that are relatively prime to $n$ . Prove that there exists a positive integer $m$ for which the equation $\varphi(n)=m$ has at least $2015$ solutions in $n$ .

Proposed by Iurie Boreico

This post has been edited 2 times. Last edited by v_Enhance, Aug 23, 2016, 12:47 AM

number theory

relatively prime

function

Analytic Number Theory

Submit

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An Introduction to the Theory of Mathematics

by EmersonSoriano, Apr 2, 2025, 9:58 PM

by EmersonSoriano, Apr 2, 2025, 9:56 PM

by EmersonSoriano, Apr 2, 2025, 9:38 PM

by dotscom26, Apr 1, 2025, 4:11 AM

by Cobedangiu, Mar 31, 2025, 11:45 AM

by Seungjun_Lee, Jan 18, 2025, 7:13 AM

by EtacticToe, Dec 14, 2024, 6:43 PM

by PennyLane_31, Oct 16, 2024, 8:26 PM

by microsoft_office_word, Apr 12, 2018, 11:02 AM

by adityaguharoy, Mar 1, 2018, 1:44 PM

by raxu, Jun 26, 2015, 1:45 AM