BDF tangent to EF

For each integer $C>1$ decide whether there exist pairwise distinct positive integers $a_1,a_2,a_3,...$ such that for every $k\ge 1$ , $a_{k+1}^k$ divides $C^ka_1a_2...a_k$ .

Proposed by Daniel Liu

17 replies

tastymath75025

Jul 3, 2017

quantam13

32 minutes ago

IMO 2011 Problem 1

Amir Hossein 102

N 32 minutes ago by Jupiterballs

Given any set $A = \{a_1, a_2, a_3, a_4\}$ of four distinct positive integers, we denote the sum $a_1 +a_2 +a_3 +a_4$ by $s_A$ . Let $n_A$ denote the number of pairs $(i, j)$ with $1 \leq i < j \leq 4$ for which $a_i +a_j$ divides $s_A$ . Find all sets $A$ of four distinct positive integers which achieve the largest possible value of $n_A$ .

Proposed by Fernando Campos, Mexico

102 replies

Amir Hossein

Jul 18, 2011

Jupiterballs

32 minutes ago

harmonic quadrilateral

Lukariman 2

N 33 minutes ago by Lukariman

Given quadrilateral ABCD inscribed in a circle with center O. CA:CB= DA:DB are satisfied. M is any point and d is a line parallel to MC. Radial projection M transforms A,B,D onto line d into A',B',D'. Prove that B' is the midpoint of A'D'.

2 replies

Lukariman

Today at 6:36 AM

Lukariman

33 minutes ago

Find the angle alpha [Iran Second Round 1994]

Amir Hossein 4

N 35 minutes ago by Mysteriouxxx

In the following diagram, $O$ is the center of the circle. If three angles $\alpha, \beta$ and $\gamma$ be equal, find $\alpha.$
IMAGE

4 replies

Amir Hossein

Nov 26, 2010

Mysteriouxxx

35 minutes ago

Inequality

lgx57 1

N 41 minutes ago by DAVROS

Source: Own

$a,b>0$ , $a^4+a^2b^2+b^4=k$ .Find the min of $4a^2-ab+4b^2$ .

$a,b>0$ , $a^4-a^2b^2+b^4=k$ .Find the min of $4a^2-ab+4b^2$ .

1 reply

lgx57

5 hours ago

DAVROS

41 minutes ago

Calculating sum of the numbers

Sadigly 2

N an hour ago by Gggvds1

Source: Azerbaijan Junior MO 2025 P4

A $3\times3$ square is filled with numbers $1;2;3...;9$ .The numbers inside four $2\times2$ squares is summed,and arranged in an increasing order. Is it possible to obtain the following sequences as a result of this operation?

$\text{a)}$ $24,24,25,25$

$\text{b)}$ $20,23,26,29$

2 replies

Sadigly

Today at 7:56 AM

Gggvds1

an hour ago

Help me identify what should i focus in alcumus for contest's each problem

Hope_and_fight 0

an hour ago

So this file i have attached is a sample test from my upcoming regional schools' math contest. This is just a test sample, i was looking for problems as similar as to these so i have more material to practice with. Unfortunately i don't have much time to read the whole books the contest is already soon. I want to test myself as much as possible with problems, also Alcumus shows from what section of the book it is. so i can kinda cram the pages

TLDR: I WOULD BE REALLY GRATEFUL IF YOU COULD POINT ON WHAT TO FOCUS ON ALCUMUS FOR EACH OF THESE CONTEST PROBLEMS. AND SORRY FOR MY ENGLISH. I DON'T KNOW MUCH OF MATH IN THERE:)

0 replies

Hope_and_fight

an hour ago

0 replies

Divisibility..

Sadigly 1

N an hour ago by Mathzeus1024

Source: Azerbaijan Junior MO 2025 P2

Find all $4$ consecutive even numbers, such that the square of their product is divisible by the sum of their squares.

1 reply

Sadigly

Today at 7:37 AM

Mathzeus1024

an hour ago

official solution of IGO

ABCD1728 7

N 2 hours ago by ABCD1728

Source: IGO official website

Where can I get the official solution of IGO for 2023 and 2024, there are some inhttps://imogeometry.blogspot.com/p/iranian-geometry-olympiad.html, but where can I find them on the official website, thanks :)

7 replies

ABCD1728

May 4, 2025

ABCD1728

2 hours ago

Combo geo with circles

a_507_bc 10

N 2 hours ago by EthanWYX2009

Source: 239 MO 2024 S8

There are $2n$ points on the plane. No three of them lie on the same straight line and no four lie on the same circle. Prove that it is possible to split these points into $n$ pairs and cover each pair of points with a circle containing no other points.

10 replies

a_507_bc

May 22, 2024

EthanWYX2009

2 hours ago

BDF tangent to EF

NicoN9 0

Apr 10, 2025

Source: Japan Junior MO Preliminary 2022 P4

Let $ABC$ be a triangle with $AB=5$ , $BC=7$ , $CA=6$ . Let $D, E$ , and $F$ be points lying on sides $BC, CA, AB$ , respectively. Given that $A, B, D, E$ , and $B, C, E, F$ are cyclic respectively, and the circumcircle of $BDF$ are tangent to line $EF$ , find the length of segment $AE$ .