Number theory

by spiderman0, Mar 30, 2025, 4:51 PM

Find all n such that $3^n + 1$ is divisibly by $n^2$ .
I want a solution that uses order or a solution like “let p be the least prime divisor of n”

number theory

Order

Fixed point config on external similar isosceles triangles

by Assassino9931, Mar 30, 2025, 12:41 PM

Let $AB$ be an acute scalene triangle. A point $D$ varies on its side $BC$ . The points $P$ and $Q$ are the midpoints of the arcs $\widehat{AB}$ and $\widehat{AC}$ (not containing $D$ ) of the circumcircles of triangles $ABD$ and $ACD$ , respectively. Prove that the circumcircle of triangle $PQD$ passes through a fixed point, independent of the choice of $D$ on $BC$ .

This post has been edited 1 time. Last edited by Assassino9931, Today at 1:10 PM

Fixed point

geometry

nice problem

by hanzo.ei, Mar 29, 2025, 5:58 PM

Let triangle $ABC$ be inscribed in the circumcircle $(O)$ and circumscribed about the incircle $(I)$ , with $AB < AC$ . The incircle $(I)$ touches the sides $BC$ , $CA$ , and $AB$ at $D$ , $E$ , and $F$ , respectively. A line through $I$ , perpendicular to $AI$ , intersects $BC$ , $CA$ , and $AB$ at $X$ , $Y$ , and $Z$ , respectively. The line $AI$ meets $(O)$ at $M$ (distinct from $A$ ). The circumcircle of triangle $AYZ$ intersects $(O)$ at $N$ (distinct from $A$ ). Let $P$ be the midpoint of the arc $BAC$ of $(O)$ . The line $AI$ cuts segments $DF$ and $DE$ at $K$ and $L$ , respectively, and the tangents to the circle $(DKL)$ at $K$ and $L$ intersect at $T$ . Prove that $AT \perp BC$ .

geometry unsolved

geometry

Finding big a_i a_i+1

by nAalniaOMliO, Mar 28, 2025, 8:36 PM

Positive real numbers $a_1>a_2>\ldots>a_n$ with sum $s$ are such that the equation $nx^2-sx+1=0$ has a positive root $a_{n+1}$ smaller than $a_n$ .
Prove that there exists a positive integer $r \leq n$ such that the inequality $a_ra_{r+1} \geq \frac{1}{r}$ holds.

algebra

2025 Caucasus MO Seniors P7

by BR1F1SZ, Mar 26, 2025, 12:50 AM

From a point $O$ lying outside the circle $\omega$ , two tangents are drawn touching $\omega$ at points $M$ and $N$ . A point $K$ is chosen on the segment $MN$ . Let points $P$ and $Q$ be the midpoints of segments $KM$ and $OM$ respectively. The circumcircle of triangle $MPQ$ intersects $\omega$ again at point $L$ ( $L \neq M$ ). Prove that the line $LN$ passes through the centroid of triangle $KMO$ .

geometry

Escape from the room

by jannatiar, Mar 4, 2025, 6:51 AM

A person is locked in a room with a password-protected computer. If they enter the correct password, the door opens and they are freed. However, the password changes every time it is entered incorrectly. The person knows that the password is always a 10-digit number, and they also know that the password change follows a fixed pattern. This means that if the current password is $b$ and $a$ is entered, the new password is $c$ , which is determined by $b$ and $a$ (naturally, the person does not know $c$ or $b$ ). Prove that regardless of the characteristics of this computer, the prisoner can free themselves.

Proposed by Reza Tahernejad Karizi

This post has been edited 3 times. Last edited by jannatiar, Mar 10, 2025, 7:30 PM

combinatorics

Game Theory

AlborzMO

Special students

by BR1F1SZ, Dec 24, 2024, 5:47 PM

In a school with double schooling, in the morning the language teacher divided the students into $200$ groups for an activity. In the afternoon, the math teacher divided the same students into $300$ groups for another activity. A student is considered special if the group they belonged to in the afternoon is smaller than the group they belonged to in the morning. Find the minimum number of special students that can exist in the school.

Note: Each group has at least one student.

combinatorics

Midpoints of chords on a circle

by AwesomeToad, Sep 23, 2011, 1:53 AM

Let $C$ be a circle and $P$ a given point in the plane. Each line through $P$ which intersects $C$ determines a chord of $C$ . Show that the midpoints of these chords lie on a circle.

Canada Mathematical Olympiad

1991

Polish MO finals, problem 1

by michaj, Apr 10, 2008, 7:01 PM

In each cell of a matrix $n\times n$ a number from a set $\{1,2,\ldots,n^2\}$ is written --- in the first row numbers $1,2,\ldots,n$ , in the second $n+1,n+2,\ldots,2n$ and so on. Exactly $n$ of them have been chosen, no two from the same row or the same column. Let us denote by $a_i$ a number chosen from row number $i$ . Show that:

$\frac{1^2}{a_1}+\frac{2^2}{a_2}+\ldots +\frac{n^2}{a_n}\geq \frac{n+2}{2}-\frac{1}{n^2+1}$

linear algebra

matrix

inequalities proposed

inequalities

Poland

Question 2

by Valentin Vornicu, Jul 25, 2007, 7:34 AM

Consider five points $A$ , $B$ , $C$ , $D$ and $E$ such that $ABCD$ is a parallelogram and $BCED$ is a cyclic quadrilateral. Let $\ell$ be a line passing through $A$ . Suppose that $\ell$ intersects the interior of the segment $DC$ at $F$ and intersects line $BC$ at $G$ . Suppose also that $EF = EG = EC$ . Prove that $\ell$ is the bisector of angle $DAB$ .

Author: Charles Leytem, Luxembourg

Submit

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by spiderman0, Mar 30, 2025, 4:51 PM

by Assassino9931, Mar 30, 2025, 12:41 PM

by hanzo.ei, Mar 29, 2025, 5:58 PM

by nAalniaOMliO, Mar 28, 2025, 8:36 PM

by BR1F1SZ, Mar 26, 2025, 12:50 AM

by jannatiar, Mar 4, 2025, 6:51 AM

by BR1F1SZ, Dec 24, 2024, 5:47 PM

by AwesomeToad, Sep 23, 2011, 1:53 AM

by michaj, Apr 10, 2008, 7:01 PM

by Valentin Vornicu, Jul 25, 2007, 7:34 AM